Title: | Parametric Statistical Modelling and Inference for the 'spatstat' Family |
---|---|
Description: | Functionality for parametric statistical modelling and inference for spatial data, mainly spatial point patterns, in the 'spatstat' family of packages. (Excludes analysis of spatial data on a linear network, which is covered by the separate package 'spatstat.linnet'.) Supports parametric modelling, formal statistical inference, and model validation. Parametric models include Poisson point processes, Cox point processes, Neyman-Scott cluster processes, Gibbs point processes and determinantal point processes. Models can be fitted to data using maximum likelihood, maximum pseudolikelihood, maximum composite likelihood and the method of minimum contrast. Fitted models can be simulated and predicted. Formal inference includes hypothesis tests (quadrat counting tests, Cressie-Read tests, Clark-Evans test, Berman test, Diggle-Cressie-Loosmore-Ford test, scan test, studentised permutation test, segregation test, ANOVA tests of fitted models, adjusted composite likelihood ratio test, envelope tests, Dao-Genton test, balanced independent two-stage test), confidence intervals for parameters, and prediction intervals for point counts. Model validation techniques include leverage, influence, partial residuals, added variable plots, diagnostic plots, pseudoscore residual plots, model compensators and Q-Q plots. |
Authors: | Adrian Baddeley [aut, cre, cph]
|
Maintainer: | Adrian Baddeley <[email protected]> |
License: | GPL (>= 2) |
Version: | 3.3-1.002 |
Built: | 2024-09-01 06:18:03 UTC |
Source: | https://github.com/spatstat/spatstat.model |
The spatstat.model package belongs to the spatstat family of packages. It contains the core functionality for parametric statistical modelling of spatial data.
spatstat is a family of R packages for the statistical analysis of spatial data. Its main focus is the analysis of spatial patterns of points in two-dimensional space.
The original spatstat package has now been split into several sub-packages.
This sub-package spatstat.model contains all the main user-level functions that perform parametric statistical modelling of spatial data.
(The main exception is that functions for linear networks are in the separate sub-package spatstat.linnet.)
The orginal spatstat package grew to be very large. It has now been divided into several sub-packages:
spatstat.utils containing basic utilities
spatstat.sparse containing linear algebra utilities
spatstat.data containing datasets
spatstat.univar containing functions for estimating probability distributions of random variables
spatstat.geom containing geometrical objects and geometrical operations
spatstat.explore containing the functionality for exploratory analysis and nonparametric modelling of spatial data
spatstat.model containing the main functionality for parametric modelling, analysis and inference for spatial data
spatstat.linnet containing functions for spatial data on a linear network
spatstat, which simply loads the other sub-packages listed above, and provides documentation.
When you install spatstat, these sub-packages are also
installed. Then if you load the spatstat package by typing
library(spatstat)
, the other sub-packages listed above will
automatically be loaded or imported.
For an overview of all the functions available in
the sub-packages of spatstat,
see the help file for "spatstat-package"
in the spatstat package.
Additionally there are several extension packages:
spatstat.gui for interactive graphics
spatstat.local for local likelihood (including geographically weighted regression)
spatstat.Knet for additional, computationally efficient code for linear networks
spatstat.sphere (under development) for spatial data on a sphere, including spatial data on the earth's surface
The extension packages must be installed separately and loaded explicitly if needed. They also have separate documentation.
The spatstat family of packages is designed to support a complete statistical analysis of spatial data. It supports
creation, manipulation and plotting of point patterns;
exploratory data analysis;
spatial random sampling;
simulation of point process models;
parametric model-fitting;
non-parametric smoothing and regression;
formal inference (hypothesis tests, confidence intervals);
model diagnostics.
For an overview, see the help file for "spatstat-package"
in the spatstat package.
Following is a list of the functionality provided in the spatstat.model package only.
To simulate a random point pattern:
Functions for generating random point patterns are now contained in the spatstat.random package.
Exploratory analysis
Exploratory graphics, smoothing, and exploratory analysis of spatial data are now provided in the spatstat.explore package.
Model fitting (Cox and cluster models)
Cluster process models (with homogeneous or inhomogeneous intensity)
and Cox processes can be fitted by the function kppm
.
Its result is an object of class "kppm"
.
The fitted model can be printed, plotted, predicted, simulated
and updated.
kppm |
Fit model |
plot.kppm |
Plot the fitted model |
summary.kppm |
Summarise the fitted model |
fitted.kppm |
Compute fitted intensity |
predict.kppm |
Compute fitted intensity |
update.kppm |
Update the model |
improve.kppm |
Refine the estimate of trend |
simulate.kppm |
Generate simulated realisations |
vcov.kppm |
Variance-covariance matrix of coefficients |
coef.kppm
|
Extract trend coefficients |
formula.kppm
|
Extract trend formula |
parameters |
Extract all model parameters |
clusterfield.kppm |
Compute offspring density |
clusterradius.kppm |
Radius of support of offspring density |
Kmodel.kppm |
function of fitted model |
pcfmodel.kppm |
Pair correlation of fitted model |
For model selection, you can also use
the generic functions step
, drop1
and AIC
on fitted point process models.
For variable selection, see sdr
.
The theoretical models can also be simulated,
for any choice of parameter values,
using rThomas
, rMatClust
,
rCauchy
, rVarGamma
,
and rLGCP
.
Lower-level fitting functions include:
lgcp.estK |
fit a log-Gaussian Cox process model |
lgcp.estpcf |
fit a log-Gaussian Cox process model |
thomas.estK |
fit the Thomas process model |
thomas.estpcf |
fit the Thomas process model |
matclust.estK |
fit the Matern Cluster process model |
matclust.estpcf |
fit the Matern Cluster process model |
cauchy.estK |
fit a Neyman-Scott Cauchy cluster process |
cauchy.estpcf |
fit a Neyman-Scott Cauchy cluster process |
vargamma.estK |
fit a Neyman-Scott Variance Gamma process |
vargamma.estpcf |
fit a Neyman-Scott Variance Gamma process |
mincontrast |
low-level algorithm for fitting models |
by the method of minimum contrast |
Model fitting (Poisson and Gibbs models)
Poisson point processes are the simplest models for point patterns. A Poisson model assumes that the points are stochastically independent. It may allow the points to have a non-uniform spatial density. The special case of a Poisson process with a uniform spatial density is often called Complete Spatial Randomness.
Poisson point processes are included in the more general class of Gibbs point process models. In a Gibbs model, there is interaction or dependence between points. Many different types of interaction can be specified.
For a detailed explanation of how to fit Poisson or Gibbs point process models to point pattern data using spatstat, see Baddeley and Turner (2005b) or Baddeley (2008).
To fit a Poisson or Gibbs point process model:
Model fitting in spatstat is performed mainly by the function
ppm
. Its result is an object of class "ppm"
.
Here are some examples, where X
is a point pattern (class
"ppp"
):
command | model |
ppm(X) |
Complete Spatial Randomness |
ppm(X ~ 1) |
Complete Spatial Randomness |
ppm(X ~ x) |
Poisson process with |
intensity loglinear in coordinate |
|
ppm(X ~ 1, Strauss(0.1)) |
Stationary Strauss process |
ppm(X ~ x, Strauss(0.1)) |
Strauss process with |
conditional intensity loglinear in
|
It is also possible to fit models that depend on other covariates.
Manipulating the fitted model:
plot.ppm |
Plot the fitted model |
predict.ppm
|
Compute the spatial trend and conditional intensity |
of the fitted point process model | |
coef.ppm |
Extract the fitted model coefficients |
parameters |
Extract all model parameters |
formula.ppm |
Extract the trend formula |
intensity.ppm |
Compute fitted intensity |
Kmodel.ppm |
function of fitted model |
pcfmodel.ppm |
pair correlation of fitted model |
fitted.ppm |
Compute fitted conditional intensity at quadrature points |
residuals.ppm |
Compute point process residuals at quadrature points |
update.ppm |
Update the fit |
vcov.ppm |
Variance-covariance matrix of estimates |
rmh.ppm |
Simulate from fitted model |
simulate.ppm |
Simulate from fitted model |
print.ppm |
Print basic information about a fitted model |
summary.ppm |
Summarise a fitted model |
effectfun |
Compute the fitted effect of one covariate |
logLik.ppm |
log-likelihood or log-pseudolikelihood |
anova.ppm |
Analysis of deviance |
model.frame.ppm |
Extract data frame used to fit model |
model.images |
Extract spatial data used to fit model |
model.depends |
Identify variables in the model |
as.interact |
Interpoint interaction component of model |
fitin |
Extract fitted interpoint interaction |
is.hybrid |
Determine whether the model is a hybrid |
valid.ppm |
Check the model is a valid point process |
project.ppm |
Ensure the model is a valid point process |
For model selection, you can also use
the generic functions step
, drop1
and AIC
on fitted point process models.
For variable selection, see sdr
.
See spatstat.options
to control plotting of fitted model.
To specify a point process model:
The first order “trend” of the model is determined by an R language formula. The formula specifies the form of the logarithm of the trend.
X ~ 1 |
No trend (stationary) |
X ~ x |
Loglinear trend
|
where are Cartesian coordinates |
|
X ~ polynom(x,y,3) |
Log-cubic polynomial trend |
X ~ harmonic(x,y,2) |
Log-harmonic polynomial trend |
X ~ Z |
Loglinear function of covariate Z |
|
The higher order (“interaction”) components are described by
an object of class "interact"
. Such objects are created by:
Poisson() |
the Poisson point process |
AreaInter() |
Area-interaction process |
BadGey() |
multiscale Geyer process |
Concom() |
connected component interaction |
DiggleGratton() |
Diggle-Gratton potential |
DiggleGatesStibbard() |
Diggle-Gates-Stibbard potential |
Fiksel() |
Fiksel pairwise interaction process |
Geyer() |
Geyer's saturation process |
Hardcore() |
Hard core process |
HierHard() |
Hierarchical multiype hard core process |
HierStrauss() |
Hierarchical multiype Strauss process |
HierStraussHard() |
Hierarchical multiype Strauss-hard core process |
Hybrid() |
Hybrid of several interactions |
LennardJones() |
Lennard-Jones potential |
MultiHard() |
multitype hard core process |
MultiStrauss() |
multitype Strauss process |
MultiStraussHard() |
multitype Strauss/hard core process |
OrdThresh() |
Ord process, threshold potential |
Ord() |
Ord model, user-supplied potential |
PairPiece() |
pairwise interaction, piecewise constant |
Pairwise() |
pairwise interaction, user-supplied potential |
Penttinen() |
Penttinen pairwise interaction |
SatPiece() |
Saturated pair model, piecewise constant potential |
Saturated() |
Saturated pair model, user-supplied potential |
Softcore() |
pairwise interaction, soft core potential |
Strauss() |
Strauss process |
StraussHard() |
Strauss/hard core point process |
Triplets() |
Geyer triplets process |
Note that it is also possible to combine several such interactions
using Hybrid
.
Simulation and goodness-of-fit for fitted models:
rmh.ppm |
simulate realisations of a fitted model |
simulate.ppm |
simulate realisations of a fitted model |
envelope |
compute simulation envelopes for a fitted model |
Model fitting (determinantal point process models)
Code for fitting determinantal point process models has recently been added to spatstat.
For information, see the help file for dppm
.
Model fitting (spatial logistic regression)
Pixel-based spatial logistic regression is an alternative technique for analysing spatial point patterns that is widely used in Geographical Information Systems. It is approximately equivalent to fitting a Poisson point process model.
In pixel-based logistic regression, the spatial domain is divided into small pixels, the presence or absence of a data point in each pixel is recorded, and logistic regression is used to model the presence/absence indicators as a function of any covariates.
Facilities for performing spatial logistic regression are provided in spatstat for comparison purposes.
Fitting a spatial logistic regression
Spatial logistic regression is performed by the function
slrm
. Its result is an object of class "slrm"
.
There are many methods for this class, including methods for
print
, fitted
, predict
, simulate
,
anova
, coef
, logLik
, terms
,
update
, formula
and vcov
.
For example, if X
is a point pattern (class
"ppp"
):
command | model |
slrm(X ~ 1) |
Complete Spatial Randomness |
slrm(X ~ x) |
Poisson process with |
intensity loglinear in coordinate |
|
slrm(X ~ Z) |
Poisson process with |
intensity loglinear in covariate Z
|
Manipulating a fitted spatial logistic regression
anova.slrm |
Analysis of deviance |
coef.slrm |
Extract fitted coefficients |
vcov.slrm |
Variance-covariance matrix of fitted coefficients |
fitted.slrm |
Compute fitted probabilities or intensity |
logLik.slrm |
Evaluate loglikelihood of fitted model |
plot.slrm |
Plot fitted probabilities or intensity |
predict.slrm |
Compute predicted probabilities or intensity with new data |
simulate.slrm |
Simulate model |
There are many other undocumented methods for this class,
including methods for print
, update
, formula
and terms
. Stepwise model selection is
possible using step
or stepAIC
.
For variable selection, see sdr
.
Simulation
There are many ways to generate a random point pattern, line segment pattern, pixel image or tessellation in spatstat.
Random point patterns: Functions for random generation are now contained in the spatstat.random package.
See also varblock
for estimating the variance
of a summary statistic by block resampling, and
lohboot
for another bootstrap technique.
Fitted point process models:
If you have fitted a point process model to a point pattern dataset, the fitted model can be simulated.
Cluster process models
are fitted by the function kppm
yielding an
object of class "kppm"
. To generate one or more simulated
realisations of this fitted model, use
simulate.kppm
.
Gibbs point process models
are fitted by the function ppm
yielding an
object of class "ppm"
. To generate a simulated
realisation of this fitted model, use rmh.ppm
.
To generate one or more simulated realisations of the fitted model,
use simulate.ppm
.
Other random patterns: Functions for random generation are now contained in the spatstat.random package.
Simulation-based inference
Simulation-based inference including simulation envelopes and hypothesis tests is now supported by the package spatstat.explore.
Sensitivity diagnostics:
Classical measures of model sensitivity such as leverage and influence have been adapted to point process models.
leverage.ppm |
Leverage for point process model |
influence.ppm |
Influence for point process model |
dfbetas.ppm |
Parameter influence |
dffit.ppm |
Effect change diagnostic |
Diagnostics for covariate effect:
Classical diagnostics for covariate effects have been adapted to point process models.
parres |
Partial residual plot |
addvar |
Added variable plot |
rhohat.ppm |
Kernel estimate of covariate effect |
rho2hat |
Kernel estimate of covariate effect (bivariate) |
Residual diagnostics:
Residuals for a fitted point process model, and diagnostic plots based on the residuals, were introduced in Baddeley et al (2005) and Baddeley, Rubak and Moller (2011).
Type demo(diagnose)
for a demonstration of the diagnostics features.
diagnose.ppm |
diagnostic plots for spatial trend |
qqplot.ppm |
diagnostic Q-Q plot for interpoint interaction |
residualspaper |
examples from Baddeley et al (2005) |
Kcom |
model compensator of function |
Gcom |
model compensator of function |
Kres |
score residual of function |
Gres |
score residual of function |
psst |
pseudoscore residual of summary function |
psstA |
pseudoscore residual of empty space function |
psstG |
pseudoscore residual of function |
compareFit |
compare compensators of several fitted models |
Resampling and randomisation procedures
You can build your own tests based on randomisation and resampling using the following capabilities:
quadratresample |
block resampling |
rshift |
random shifting of (subsets of) points |
rthin |
random thinning |
This library and its documentation are usable under the terms of the "GNU General Public License", a copy of which is distributed with the package.
Kasper Klitgaard Berthelsen, Ottmar Cronie, Tilman Davies, Julian Gilbey, Yongtao Guan, Ute Hahn, Kassel Hingee, Abdollah Jalilian, Marie-Colette van Lieshout, Greg McSwiggan, Tuomas Rajala, Suman Rakshit, Dominic Schuhmacher, Rasmus Waagepetersen and Hangsheng Wang made substantial contributions of code.
For comments, corrections, bug alerts and suggestions, we thank Monsuru Adepeju, Corey Anderson, Ang Qi Wei, Ryan Arellano, Jens Astrom, Robert Aue, Marcel Austenfeld, Sandro Azaele, Malissa Baddeley, Guy Bayegnak, Colin Beale, Melanie Bell, Thomas Bendtsen, Ricardo Bernhardt, Andrew Bevan, Brad Biggerstaff, Anders Bilgrau, Leanne Bischof, Christophe Biscio, Roger Bivand, Jose M. Blanco Moreno, Florent Bonneu, Jordan Brown, Ian Buller, Julian Burgos, Simon Byers, Ya-Mei Chang, Jianbao Chen, Igor Chernayavsky, Y.C. Chin, Bjarke Christensen, Lucia Cobo Sanchez, Jean-Francois Coeurjolly, Kim Colyvas, Hadrien Commenges, Rochelle Constantine, Robin Corria Ainslie, Richard Cotton, Marcelino de la Cruz, Peter Dalgaard, Mario D'Antuono, Sourav Das, Peter Diggle, Patrick Donnelly, Ian Dryden, Stephen Eglen, Ahmed El-Gabbas, Belarmain Fandohan, Olivier Flores, David Ford, Peter Forbes, Shane Frank, Janet Franklin, Funwi-Gabga Neba, Oscar Garcia, Agnes Gault, Jonas Geldmann, Marc Genton, Shaaban Ghalandarayeshi, Jason Goldstick, Pavel Grabarnik, C. Graf, Ute Hahn, Andrew Hardegen, Martin Bogsted Hansen, Martin Hazelton, Juha Heikkinen, Mandy Hering, Markus Herrmann, Maximilian Hesselbarth, Paul Hewson, Hamidreza Heydarian, Kurt Hornik, Philipp Hunziker, Jack Hywood, Ross Ihaka, Cenk Icos, Aruna Jammalamadaka, Robert John-Chandran, Devin Johnson, Mahdieh Khanmohammadi, Bob Klaver, Lily Kozmian-Ledward, Peter Kovesi, Mike Kuhn, Jeff Laake, Robert Lamb, Frederic Lavancier, Tom Lawrence, Tomas Lazauskas, Jonathan Lee, George Leser, Angela Li, Li Haitao, George Limitsios, Andrew Lister, Nestor Luambua, Ben Madin, Martin Maechler, Kiran Marchikanti, Jeff Marcus, Robert Mark, Peter McCullagh, Monia Mahling, Jorge Mateu Mahiques, Ulf Mehlig, Frederico Mestre, Sebastian Wastl Meyer, Mi Xiangcheng, Lore De Middeleer, Robin Milne, Enrique Miranda, Jesper Moller, Annie Mollie, Ines Moncada, Mehdi Moradi, Virginia Morera Pujol, Erika Mudrak, Gopalan Nair, Nader Najari, Nicoletta Nava, Linda Stougaard Nielsen, Felipe Nunes, Jens Randel Nyengaard, Jens Oehlschlaegel, Thierry Onkelinx, Sean O'Riordan, Evgeni Parilov, Jeff Picka, Nicolas Picard, Tim Pollington, Mike Porter, Sergiy Protsiv, Adrian Raftery, Ben Ramage, Pablo Ramon, Xavier Raynaud, Nicholas Read, Matt Reiter, Ian Renner, Tom Richardson, Brian Ripley, Ted Rosenbaum, Barry Rowlingson, Jason Rudokas, Tyler Rudolph, John Rudge, Christopher Ryan, Farzaneh Safavimanesh, Aila Sarkka, Cody Schank, Katja Schladitz, Sebastian Schutte, Bryan Scott, Olivia Semboli, Francois Semecurbe, Vadim Shcherbakov, Shen Guochun, Shi Peijian, Harold-Jeffrey Ship, Tammy L Silva, Ida-Maria Sintorn, Yong Song, Malte Spiess, Mark Stevenson, Kaspar Stucki, Jan Sulavik, Michael Sumner, P. Surovy, Ben Taylor, Thordis Linda Thorarinsdottir, Leigh Torres, Berwin Turlach, Torben Tvedebrink, Kevin Ummer, Medha Uppala, Andrew van Burgel, Tobias Verbeke, Mikko Vihtakari, Alexendre Villers, Fabrice Vinatier, Maximilian Vogtland, Sasha Voss, Sven Wagner, Hao Wang, H. Wendrock, Jan Wild, Carl G. Witthoft, Selene Wong, Maxime Woringer, Luke Yates, Mike Zamboni and Achim Zeileis.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Computes the coordinates for an Added Variable Plot for a fitted point process model.
addvar(model, covariate, ..., subregion=NULL, bw="nrd0", adjust=1, from=NULL, to=NULL, n=512, bw.input = c("points", "quad"), bw.restrict = FALSE, covname, crosscheck=FALSE)
addvar(model, covariate, ..., subregion=NULL, bw="nrd0", adjust=1, from=NULL, to=NULL, n=512, bw.input = c("points", "quad"), bw.restrict = FALSE, covname, crosscheck=FALSE)
model |
Fitted point process model (object of class |
covariate |
The covariate to be added to the model. Either a
pixel image, a |
subregion |
Optional. A window (object of class |
bw |
Smoothing bandwidth or bandwidth rule
(passed to |
adjust |
Smoothing bandwidth adjustment factor
(passed to |
n , from , to
|
Arguments passed to |
... |
Additional arguments passed to |
bw.input |
Character string specifying the input data used for automatic bandwidth selection. |
bw.restrict |
Logical value, specifying whether bandwidth selection is performed using
data from the entire spatial domain or from the |
covname |
Optional. Character string to use as the name of the covariate. |
crosscheck |
For developers only. Logical value indicating whether to perform cross-checks on the validity of the calculation. |
This command generates the plot coordinates for an Added Variable Plot for a spatial point process model.
Added Variable Plots (Cox, 1958, sec 4.5; Wang, 1985)
are commonly used in linear models and generalized linear
models, to decide whether a model with response and predictors
would be improved by including another predictor
.
In a (generalised) linear model
with response and predictors
,
the Added Variable Plot for a new covariate
is a plot of the smoothed Pearson residuals from the original model
against the scaled residuals from a weighted linear
regression of
on
.
If this plot has nonzero slope, then the new covariate
is
needed. For general advice see Cook and Weisberg(1999); Harrell (2001).
Essentially the same technique can be used for a spatial point process model (Baddeley et al, 2012).
The argument model
should be a fitted spatial point process
model (object of class "ppm"
).
The argument covariate
identifies the covariate that is to be considered for addition to
the model. It should be either a pixel image (object of class
"im"
) or a function(x,y)
giving the values of the
covariate at any spatial location. Alternatively covariate
may be a character string, giving the name of a covariate that was
supplied (in the covariates
argument to ppm
)
when the model was fitted, but was not used in the model.
The result of addvar(model, covariate)
is an object belonging
to the classes "addvar"
and "fv"
. Plot this object to
generate the added variable plot.
Note that the plot method shows the pointwise significance bands for a test of the null model, i.e. the null hypothesis that the new covariate has no effect.
The smoothing bandwidth is controlled by the arguments
bw
, adjust
, bw.input
and bw.restrict
.
If bw
is a numeric value, then
the bandwidth is taken to be adjust * bw
.
If bw
is a string representing a bandwidth selection rule
(recognised by density.default
)
then the bandwidth is selected by this rule.
The data used for automatic bandwidth selection are
specified by bw.input
and bw.restrict
.
If bw.input="points"
(the default) then bandwidth selection is
based on the covariate values at the points of the original point
pattern dataset to which the model was fitted.
If bw.input="quad"
then bandwidth selection is
based on the covariate values at every quadrature point used to
fit the model.
If bw.restrict=TRUE
then the bandwidth selection is performed
using only data from inside the subregion
.
An object of class "addvar"
containing the coordinates
for the added variable plot. There is a plot
method.
In a large dataset, computation can be very slow if the default
settings are used, because the smoothing bandwidth is selected
automatically. To avoid this, specify a numerical value
for the bandwidth bw
. One strategy is to use a coarser
subset of the data to select bw
automatically.
The selected bandwidth can be read off the print output for
addvar
.
The return value has an attribute "spatial"
which contains
the internal data: the computed values of the residuals,
and of all relevant covariates,
at each quadrature point of the model. It is an object of class
"ppp"
with a data frame of marks.
Adrian Baddeley [email protected], Rolf Turner [email protected], Ya-Mei Chang and Yong Song.
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2013) Residual diagnostics for covariate effects in spatial point process models. Journal of Computational and Graphical Statistics, 22, 886–905.
Cook, R.D. and Weisberg, S. (1999) Applied regression, including computing and graphics. New York: Wiley.
Cox, D.R. (1958) Planning of Experiments. New York: Wiley.
Harrell, F. (2001) Regression Modeling Strategies. New York: Springer.
Wang, P. (1985) Adding a variable in generalized linear models. Technometrics 27, 273–276.
X <- rpoispp(function(x,y){exp(3+3*x)}) model <- ppm(X, ~y) adv <- addvar(model, "x") plot(adv) adv <- addvar(model, "x", subregion=square(0.5))
X <- rpoispp(function(x,y){exp(3+3*x)}) model <- ppm(X, ~y) adv <- addvar(model, "x") plot(adv) adv <- addvar(model, "x", subregion=square(0.5))
Performs analysis of deviance for one or more point process models fitted to replicated point pattern data.
## S3 method for class 'mppm' anova(object, ..., test=NULL, adjust=TRUE, fine=FALSE, warn=TRUE)
## S3 method for class 'mppm' anova(object, ..., test=NULL, adjust=TRUE, fine=FALSE, warn=TRUE)
object |
Object of class |
... |
Optional. Additional objects of class |
test |
Type of hypothesis test to perform.
A character string, partially matching one of
|
adjust |
Logical value indicating whether to correct the pseudolikelihood ratio when some of the models are not Poisson processes. |
fine |
Logical value passed to |
warn |
Logical value indicating whether to issue warnings if problems arise. |
This is a method for anova
for comparing several
fitted point process models of class "mppm"
,
usually generated by the model-fitting function mppm
).
If the fitted models are all Poisson point processes,
then this function performs an Analysis of Deviance of
the fitted models. The output shows the deviance differences
(i.e. 2 times log likelihood ratio),
the difference in degrees of freedom, and (if test="Chi"
)
the two-sided p-values for the chi-squared tests. Their interpretation
is very similar to that in anova.glm
.
If some of the fitted models are not Poisson point processes,
the ‘deviance’ differences in this table are
'pseudo-deviances' equal to 2 times the differences
in the maximised values of the log pseudolikelihood (see
ppm
). It is not valid to compare these
values to the chi-squared distribution. In this case,
if adjust=TRUE
(the default), the
pseudo-deviances will be adjusted using the method of Pace et al
(2011) and Baddeley, Turner and Rubak (2015)
so that the chi-squared test is valid.
It is strongly advisable to perform this adjustment.
The argument test
determines which hypothesis test, if any, will
be performed to compare the models. The argument test
should be a character string, partially matching one of
"Chisq"
, "F"
or "Cp"
,
or NULL
. The first option "Chisq"
gives
the likelihood ratio test based on the asymptotic chi-squared
distribution of the deviance difference.
The meaning of the other options is explained in
anova.glm
.
An object of class "anova"
, or NULL
.
For models with random effects
(i.e. where the call to mppm
included the argument random
),
analysis of deviance is currently not supported,
due to changes in the nlme package.
We will try to find a solution.
An error message that reports
system is computationally singular indicates that the
determinant of the Fisher information matrix of one of the models
was either too large or too small for reliable numerical calculation.
See vcov.ppm
for suggestions on how to handle this.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Baddeley, A., Turner, R. and Rubak, E. (2015) Adjusted composite likelihood ratio test for Gibbs point processes. Journal of Statistical Computation and Simulation 86 (5) 922–941. DOI: 10.1080/00949655.2015.1044530.
Pace, L., Salvan, A. and Sartori, N. (2011) Adjusting composite likelihood ratio statistics. Statistica Sinica 21, 129–148.
H <- hyperframe(X=waterstriders) #' test for loglinear trend in x coordinate mod0 <- mppm(X~1, data=H, Poisson()) modx <- mppm(X~x, data=H, Poisson()) anova(mod0, modx, test="Chi") # not significant anova(modx, test="Chi") # not significant #' test for inhibition mod0S <- mppm(X~1, data=H, Strauss(2)) anova(mod0, mod0S, test="Chi") # significant! #' test for trend after accounting for inhibition modxS <- mppm(X~x, data=H, Strauss(2)) anova(mod0S, modxS, test="Chi") # not significant
H <- hyperframe(X=waterstriders) #' test for loglinear trend in x coordinate mod0 <- mppm(X~1, data=H, Poisson()) modx <- mppm(X~x, data=H, Poisson()) anova(mod0, modx, test="Chi") # not significant anova(modx, test="Chi") # not significant #' test for inhibition mod0S <- mppm(X~1, data=H, Strauss(2)) anova(mod0, mod0S, test="Chi") # significant! #' test for trend after accounting for inhibition modxS <- mppm(X~x, data=H, Strauss(2)) anova(mod0S, modxS, test="Chi") # not significant
Performs analysis of deviance for one or more fitted point process models.
## S3 method for class 'ppm' anova(object, ..., test=NULL, adjust=TRUE, warn=TRUE, fine=FALSE)
## S3 method for class 'ppm' anova(object, ..., test=NULL, adjust=TRUE, warn=TRUE, fine=FALSE)
object |
A fitted point process model
(object of class |
... |
Optional. Additional objects of class |
test |
Character string, partially matching one of
|
adjust |
Logical value indicating whether to correct the pseudolikelihood ratio when some of the models are not Poisson processes. |
warn |
Logical value indicating whether to issue warnings if problems arise. |
fine |
Logical value, passed to |
This is a method for anova
for
fitted point process models (objects of class "ppm"
,
usually generated by the model-fitting function ppm
).
If the fitted models are all Poisson point processes,
then by default, this function performs an Analysis of Deviance of
the fitted models. The output shows the deviance differences
(i.e. 2 times log likelihood ratio),
the difference in degrees of freedom, and (if test="Chi"
or test="LRT"
)
the two-sided p-values for the chi-squared tests. Their interpretation
is very similar to that in anova.glm
.
If test="Rao"
or test="score"
,
the score test (Rao, 1948) is performed instead.
If some of the fitted models are not Poisson point processes,
the ‘deviance’ differences in this table are
'pseudo-deviances' equal to 2 times the differences
in the maximised values of the log pseudolikelihood (see
ppm
). It is not valid to compare these
values to the chi-squared distribution. In this case,
if adjust=TRUE
(the default), the
pseudo-deviances will be adjusted using the method of Pace et al
(2011) and Baddeley et al (2015) so that the chi-squared test is valid.
It is strongly advisable to perform this adjustment.
An object of class "anova"
, or NULL
.
There may be an error message that the models are not “nested”.
For an Analysis of Deviance the models must be nested, i.e. one model
must be a special case of the other. For example the point process
model with formula ~x
is a special case of the model with
formula ~x+y
, so these models are nested. However
the two point process
models with formulae ~x
and ~y
are not nested.
If you get this error message and you believe that the models should be nested, the problem may be the inability of R to recognise that the two formulae are nested. Try modifying the formulae to make their relationship more obvious.
There may be an error message from anova.glmlist
that
“models were not all fitted to the same size of dataset”.
This implies that the models were fitted using different
quadrature schemes (see quadscheme
) and/or
with different edge corrections or different values of the
border edge correction distance rbord
.
To ensure that models are comparable, check the following:
the models must all have been fitted to the same point pattern dataset, in the same window.
all models must have been fitted by the same
fitting method as specified by the argument method
in
ppm
.
If some of the models depend on covariates, then
they should all have been fitted using the same list of
covariates, and using allcovar=TRUE
to ensure that the
same quadrature scheme is used.
all models must have been fitted using the same edge
correction as specified by the arguments correction
and rbord
. If you did not specify the value of
rbord
, then it may have
taken a different value for different models. The default value of
rbord
is equal to zero for a Poisson model,
and otherwise equals the reach (interaction distance) of the
interaction term (see reach
).
To ensure that the models are comparable, set rbord
to
equal the maximum reach of the interactions that you
are fitting.
An error message that reports
system is computationally singular indicates that the
determinant of the Fisher information matrix of one of the models
was either too large or too small for reliable numerical calculation.
See vcov.ppm
for suggestions on how to handle this.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Turner, R. and Rubak, E. (2015) Adjusted composite likelihood ratio test for Gibbs point processes. Journal of Statistical Computation and Simulation 86 (5) 922–941. DOI: 10.1080/00949655.2015.1044530.
Pace, L., Salvan, A. and Sartori, N. (2011) Adjusting composite likelihood ratio statistics. Statistica Sinica 21, 129–148.
Rao, C.R. (1948) Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Proceedings of the Cambridge Philosophical Society 44, 50–57.
mod0 <- ppm(swedishpines ~1) modx <- ppm(swedishpines ~x) # Likelihood ratio test anova(mod0, modx, test="Chi") # Score test anova(mod0, modx, test="Rao") # Single argument modxy <- ppm(swedishpines ~x + y) anova(modxy, test="Chi") # Adjusted composite likelihood ratio test modP <- ppm(swedishpines ~1, rbord=9) modS <- ppm(swedishpines ~1, Strauss(9)) anova(modP, modS, test="Chi")
mod0 <- ppm(swedishpines ~1) modx <- ppm(swedishpines ~x) # Likelihood ratio test anova(mod0, modx, test="Chi") # Score test anova(mod0, modx, test="Rao") # Single argument modxy <- ppm(swedishpines ~x + y) anova(modxy, test="Chi") # Adjusted composite likelihood ratio test modP <- ppm(swedishpines ~1, rbord=9) modS <- ppm(swedishpines ~1, Strauss(9)) anova(modP, modS, test="Chi")
Performs Analysis of Deviance for two or more fitted Spatial Logistic Regression models.
## S3 method for class 'slrm' anova(object, ..., test = NULL)
## S3 method for class 'slrm' anova(object, ..., test = NULL)
object |
a fitted spatial logistic regression model.
An object of class |
... |
additional objects of the same type (optional). |
test |
a character string, (partially) matching one of
|
This is a method for anova
for fitted spatial logistic
regression models (objects of class "slrm"
, usually obtained
from the function slrm
).
The output shows the deviance differences (i.e. 2 times log
likelihood ratio), the difference in degrees of freedom, and (if
test="Chi"
) the two-sided -values for the chi-squared tests.
Their interpretation is very similar to that
in
anova.glm
.
An object of class "anova"
, inheriting from
class "data.frame"
, representing the analysis of deviance table.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
X <- rpoispp(42) fit0 <- slrm(X ~ 1) fit1 <- slrm(X ~ x+y) anova(fit0, fit1, test="Chi")
X <- rpoispp(42) fit0 <- slrm(X ~ 1) fit1 <- slrm(X ~ x+y) anova(fit0, fit1, test="Chi")
Creates an instance of the Area Interaction point process model (Widom-Rowlinson penetrable spheres model) which can then be fitted to point pattern data.
AreaInter(r)
AreaInter(r)
r |
The radius of the discs in the area interaction process |
This function defines the interpoint interaction structure of a point process called the Widom-Rowlinson penetrable sphere model or area-interaction process. It can be used to fit this model to point pattern data.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the area interaction structure is
yielded by the function AreaInter()
. See the examples below.
In standard form, the area-interaction process
(Widom and Rowlinson, 1970;
Baddeley and Van Lieshout, 1995) with disc radius ,
intensity parameter
and interaction parameter
is a point process with probability density
for a point pattern , where
represent the
points of the pattern,
is the number of points in the
pattern, and
is the area of the region formed by
the union of discs of radius
centred at the points
.
Here
is a normalising constant.
The interaction parameter can be any positive number.
If
then the model reduces to a Poisson
process with intensity
.
If
then the process is regular,
while if
the process is clustered.
Thus, an area interaction process can be used to model either
clustered or regular point patterns. Two points interact if the
distance between them is less than
.
The standard form of the model, shown above, is a little
complicated to interpret in practical applications.
For example, each isolated point of the pattern contributes a factor
to the probability density.
In spatstat, the model is parametrised in a different form, which is easier to interpret. In canonical scale-free form, the probability density is rewritten as
where is the new intensity parameter,
is the new interaction parameter, and
is the interaction potential. Here
is the normalised area (so that the discs have unit area).
In this formulation, each isolated point of the pattern contributes a
factor to the probability density (so the
first order trend is
). The quantity
is a true interaction potential, in the sense that
if the point pattern
does not contain any
points that lie close together (closer than
units
apart).
When a new point is added to an existing point pattern
, the rescaled potential
increases by
a value between 0 and 1.
The increase is zero if
is not close to any point of
.
The increase is 1 if the disc of radius
centred at
is completely contained in the union of discs of radius
centred at the data points
. Thus, the increase in
potential is a measure of how close the new point
is to the
existing pattern
. Addition of the point
contributes a factor
to the probability density, where
is the
increase in potential.
The old parameters of the
standard form are related to
the new parameters
of the canonical
scale-free form, by
and
provided and
are positive and finite.
In the canonical scale-free form, the parameter
can take any nonnegative value. The value
again corresponds to a Poisson process, with intensity
.
If
then the process is regular,
while if
the process is clustered.
The value
corresponds to a hard core process
with hard core radius
(interaction distance
).
The nonstationary area interaction process is similar except that
the contribution of each individual point
is a function
of location, rather than a constant beta.
Note the only argument of AreaInter()
is the disc radius r
.
When r
is fixed, the model becomes an exponential family.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
AreaInter()
.
An object of class "interact"
describing the interpoint interaction
structure of the area-interaction process with disc radius .
The interaction distance of this process is equal to
2 * r
. Two discs of radius r
overlap if their centres
are closer than 2 * r
units apart.
The estimate of the interaction parameter
is unreliable if the interaction radius
r
is too small
or too large. In these situations the model is approximately Poisson
so that is unidentifiable.
As a rule of thumb, one can inspect the empty space function
of the data, computed by
Fest
. The value
of the empty space function at the interaction radius
r
should
be between 0.2 and 0.8.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A.J. and Van Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of the Institute of Statistical Mathematics 47 (1995) 601–619.
Widom, B. and Rowlinson, J.S. (1970). New model for the study of liquid-vapor phase transitions. The Journal of Chemical Physics 52 (1970) 1670–1684.
ppm
,
pairwise.family
,
ppm.object
ragsAreaInter
and rmh
for simulation
of area-interaction models.
# prints a sensible description of itself AreaInter(r=0.1) # Note the reach is twice the radius reach(AreaInter(r=1)) # Fit the stationary area interaction process to Swedish Pines data ppm(swedishpines ~1, AreaInter(r=7)) # Fit the stationary area interaction process to `cells' ppm(cells ~1, AreaInter(r=0.06)) # eta=0 indicates hard core process. # Fit a nonstationary area interaction with log-cubic polynomial trend ppm(swedishpines ~polynom(x/10,y/10,3), AreaInter(r=7))
# prints a sensible description of itself AreaInter(r=0.1) # Note the reach is twice the radius reach(AreaInter(r=1)) # Fit the stationary area interaction process to Swedish Pines data ppm(swedishpines ~1, AreaInter(r=7)) # Fit the stationary area interaction process to `cells' ppm(cells ~1, AreaInter(r=0.06)) # eta=0 indicates hard core process. # Fit a nonstationary area interaction with log-cubic polynomial trend ppm(swedishpines ~polynom(x/10,y/10,3), AreaInter(r=7))
Converts an object of class "leverage.ppm"
to a function of the
and
coordinates.
## S3 method for class 'leverage.ppm' as.function(x, ...)
## S3 method for class 'leverage.ppm' as.function(x, ...)
x |
Object of class |
... |
Ignored. |
An object of class "leverage.ppm"
represents the leverage
function of a fitted point process model. This command converts the object
to a function(x,y)
where the arguments x
and y
are (vectors of) spatial
coordinates. This function returns the leverage values at the
specified locations (calculated by referring to the nearest location
where the leverage has been computed).
A function in the R language, also belonging to the
class "funxy"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) lev <- leverage(fit) f <- as.function(lev) f(0.2, 0.3) # evaluate at (x,y) coordinates y <- f(X) # evaluate at a point pattern
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) lev <- leverage(fit) f <- as.function(lev) f(0.2, 0.3) # evaluate at (x,y) coordinates y <- f(X) # evaluate at a point pattern
Converts fitted model into a function table (an object of class "fv"
).
## S3 method for class 'kppm' as.fv(x) ## S3 method for class 'dppm' as.fv(x) ## S3 method for class 'minconfit' as.fv(x)
## S3 method for class 'kppm' as.fv(x) ## S3 method for class 'dppm' as.fv(x) ## S3 method for class 'minconfit' as.fv(x)
x |
A fitted model which will be converted into a function table |
The generic command as.fv
converts data x
, that
could be interpreted as the values of a function,
into a function value table (object of the class "fv"
as described in fv.object
). This object can then
be plotted easily using plot.fv
.
Objects of class "kppm"
(and related classes) represent
a model that has been fitted to a dataset
by computing a summary function of the dataset and
matching it to the corresponding summary function of the model.
The methods for as.fv
for classes "kppm"
,
"dppm"
and "minconfit"
extract this information:
the result is a function table containing the
observed summary function and the best fit summary function.
An object of class "fv"
(see fv.object
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
as.fv(kppm(redwood))
as.fv(kppm(redwood))
Extracts the interpoint interaction structure from a point pattern model.
as.interact(object) ## S3 method for class 'fii' as.interact(object) ## S3 method for class 'interact' as.interact(object) ## S3 method for class 'ppm' as.interact(object)
as.interact(object) ## S3 method for class 'fii' as.interact(object) ## S3 method for class 'interact' as.interact(object) ## S3 method for class 'ppm' as.interact(object)
object |
A fitted point process model (object of class
|
The function as.interact
extracts
the interpoint interaction structure from a suitable object.
An object of class "interact"
describes an interpoint
interaction structure, before it has been fitted to point pattern
data. The irregular parameters of the interaction (such as the
interaction range) are fixed, but the regular parameters
(such as interaction strength) are undetermined.
Objects of this class are created by the functions
Poisson
, Strauss
and so on.
The main use of such objects is in a call to ppm
.
The function as.interact
is generic, with methods for the classes
"ppm"
, "fii"
and "interact"
.
The result is an object of class "interact"
which can be printed.
An object of class "interact"
representing the
interpoint interaction. This object can be printed and plotted.
This function does not extract the fitted coefficients
of the interaction. To extract the fitted interaction including
the fitted coefficients, use fitin
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
model <- ppm(cells ~1, Strauss(0.07)) f <- as.interact(model) f
model <- ppm(cells ~1, Strauss(0.07)) f <- as.interact(model) f
Converts a measure into a layered object.
## S3 method for class 'msr' as.layered(X)
## S3 method for class 'msr' as.layered(X)
X |
A measure (object of class |
This function converts the object X
into an object of class "layered"
.
It is a method for the generic as.layered
for the class of measures.
If X
is a vector-valued measure,
then as.layered(X)
consists of several layers,
each containing a scalar-valued measure.
An object of class "layered"
(see layered
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
P <- rpoispp(100) fit <- ppm(P ~ x+y) rs <- residuals(fit, type="score") as.layered(rs)
P <- rpoispp(100) fit <- ppm(P ~ x+y) rs <- residuals(fit, type="score") as.layered(rs)
Converts data specifying an observation window
in any of several formats, into an object of class "owin"
.
## S3 method for class 'ppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'kppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'dppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'slrm' as.owin(W, ..., from=c("points", "covariates")) ## S3 method for class 'msr' as.owin(W, ..., fatal=TRUE)
## S3 method for class 'ppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'kppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'dppm' as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE) ## S3 method for class 'slrm' as.owin(W, ..., from=c("points", "covariates")) ## S3 method for class 'msr' as.owin(W, ..., fatal=TRUE)
W |
Data specifying an observation window, in any of several formats described under Details below. |
fatal |
Logical value determining what to do if the data cannot be converted to an observation window. See Details. |
... |
Ignored. |
from |
Character string. See Details. |
The class "owin"
is a way of specifying the observation window
for a point pattern. See owin.object
for an overview.
The generic function as.owin
converts data in any of several formats
into an object of class "owin"
for use by the spatstat
package. The function as.owin
is generic, with methods
for different classes of objects, and a default method.
The argument W
may be
an object of class "owin"
a structure with entries xrange
, yrange
specifying the
and
dimensions of a rectangle
a structure with entries named xmin
, xmax
, ymin
,
ymax
(in any order)
specifying the and
dimensions of a rectangle.
This will accept objects of class
bbox
in the sf
package.
a numeric vector of length 4
(interpreted as (xmin, xmax, ymin, ymax)
in that order)
specifying the and
dimensions of a rectangle
a structure with entries named xl
, xu
, yl
, yu
(in any order)
specifying the and
dimensions of a rectangle
as
(xmin, xmax) = (xl, xu)
and
(ymin, ymax) = (yl, yu)
. This will accept objects of
class spp
used in the Venables and Ripley spatial
package.
an object of class "ppp"
representing a point pattern.
In this case, the object's window
structure will be
extracted.
an object of class "psp"
representing a line segment pattern.
In this case, the object's window
structure will be
extracted.
an object of class "tess"
representing a tessellation.
In this case, the object's window
structure will be
extracted.
an object of class "quad"
representing a quadrature scheme.
In this case, the window of the data
component will be
extracted.
an object of class "im"
representing a pixel image.
In this case, a window of type "mask"
will be returned,
with the same pixel raster coordinates as the image.
An image pixel value of NA
, signifying that the pixel
lies outside the window, is transformed into the logical value
FALSE
, which is the corresponding convention for window masks.
an object of class "ppm"
, "kppm"
, "slrm"
or "dppm"
representing a fitted point process
model. In this case, if from="data"
(the default),
as.owin
extracts the original point
pattern data to which the model was fitted, and returns the
observation window of this point pattern. If
from="covariates"
then as.owin
extracts the
covariate images to which the model was fitted,
and returns a binary mask window that specifies the pixel locations.
an object of class "lpp"
representing a point pattern on a linear network.
In this case, as.owin
extracts the linear network
and returns a window containing this network.
an object of class "lppm"
representing a fitted point process model on a linear network.
In this case, as.owin
extracts the linear network
and returns a window containing this network.
A data.frame
with exactly three columns. Each row of the
data frame corresponds to one pixel. Each row contains the
and
coordinates of a pixel, and a logical value
indicating whether the pixel lies inside the window.
A data.frame
with exactly two columns. Each row of the
data frame contains the and
coordinates of a pixel
that lies inside the window.
an object of class "distfun"
, "nnfun"
or "funxy"
representing a function of spatial location,
defined on a spatial domain. The spatial domain of the function will be
extracted.
an object of class "rmhmodel"
representing a
point process model that can be simulated using rmh
.
The window (spatial domain) of the model will be extracted.
The window may be NULL
in some circumstances (indicating that the
simulation window has not yet been determined). This is not treated
as an error, because the argument fatal
defaults to
FALSE
for this method.
an object of class "layered"
representing a
list of spatial objects. See layered
.
In this case, as.owin
will be applied to each
of the objects in the list, and the union of these windows
will be returned.
an object of some other suitable class from another package.
For full details, see vignette('shapefiles')
.
If the argument W
is not in one of these formats
and cannot be converted to a window, then an error will
be generated (if fatal=TRUE
) or a value of NULL
will be returned (if fatal=FALSE
).
When W
is a data frame, the argument step
can be used to specify the pixel grid spacing; otherwise, the spacing
will be guessed from the data.
An object of class "owin"
(see owin.object
)
specifying an observation window.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
as.owin
,
as.owin.rmhmodel
,
as.owin.lpp
.
Additional methods for as.owin
may be provided
by other packages outside the spatstat family.
fit <- ppm(cells ~ 1) as.owin(fit)
fit <- ppm(cells ~ 1) as.owin(fit)
Extracts the fitted point process model from some kind of fitted model.
as.ppm(object) ## S3 method for class 'ppm' as.ppm(object) ## S3 method for class 'profilepl' as.ppm(object) ## S3 method for class 'kppm' as.ppm(object) ## S3 method for class 'dppm' as.ppm(object) ## S3 method for class 'rppm' as.ppm(object)
as.ppm(object) ## S3 method for class 'ppm' as.ppm(object) ## S3 method for class 'profilepl' as.ppm(object) ## S3 method for class 'kppm' as.ppm(object) ## S3 method for class 'dppm' as.ppm(object) ## S3 method for class 'rppm' as.ppm(object)
object |
An object that includes a
fitted Poisson or Gibbs point process model.
An object of class |
The function as.ppm
extracts
the fitted point process model (of class "ppm"
)
from a suitable object.
The function as.ppm
is generic, with methods for the classes
"ppm"
, "profilepl"
, "kppm"
, "dppm"
and "rppm"
, and possibly for other classes.
For the class "profilepl"
of models fitted by maximum profile
pseudolikelihood, the method as.ppm.profilepl
extracts the
fitted point process model (with the optimal values of the
irregular parameters).
For the class "kppm"
of models fitted by minimum contrast (or Palm or composite likelihood)
using Waagepetersen's two-step estimation procedure
(see kppm
), the method as.ppm.kppm
extracts the Poisson point process model that is fitted in the
first stage of the procedure.
The behaviour for the class "dppm"
is analogous to the
"kppm"
case above.
For the class "rppm"
of models fitted by recursive partitioning
(regression trees), the method as.ppm.rppm
extracts the
corresponding loglinear model that is fitted in the first stage of the
procedure (whose purpose is merely to identify and evaluate the
explanatory variables).
An object of class "ppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
# fit a model by profile maximum pseudolikelihood rvals <- data.frame(r=(1:10)/100) pfit <- profilepl(rvals, Strauss, cells, ~1) # extract the fitted model fit <- as.ppm(pfit)
# fit a model by profile maximum pseudolikelihood rvals <- data.frame(r=(1:10)/100) pfit <- profilepl(rvals, Strauss, cells, ~1) # extract the fitted model fit <- as.ppm(pfit)
Compute the AUC (area under the Receiver Operating Characteristic curve) for a fitted point process model.
## S3 method for class 'ppm' auc(X, ...) ## S3 method for class 'kppm' auc(X, ...) ## S3 method for class 'slrm' auc(X, ...)
## S3 method for class 'ppm' auc(X, ...) ## S3 method for class 'kppm' auc(X, ...) ## S3 method for class 'slrm' auc(X, ...)
X |
Point pattern (object of class |
... |
Arguments passed to |
This command computes the AUC, the area under the Receiver Operating
Characteristic curve. The ROC itself is computed by roc
.
For a fitted point process model X
,
the AUC measures the ability of the
fitted model intensity to separate the spatial domain
into areas of high and low density of points.
Suppose is the intensity function of the model.
The AUC is the probability that
.
That is, AUC is the probability that a randomly-selected data point
has higher predicted intensity than does a randomly-selected spatial
location.
The AUC is not a measure of the goodness-of-fit of the model
(Lobo et al, 2007).
(For spatial logistic regression models (class "slrm"
)
replace “intensity” by “probability of presence”
in the text above.)
Numeric.
For auc.ppm
, auc.kppm
and auc.lppm
, the result is a
numeric vector of length 2 giving the AUC value
and the theoretically expected AUC value for this model.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Lobo, J.M., Jimenez-Valverde, A. and Real, R. (2007) AUC: a misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography 17(2) 145–151.
Nam, B.-H. and D'Agostino, R. (2002) Discrimination index, the area under the ROC curve. Pages 267–279 in Huber-Carol, C., Balakrishnan, N., Nikulin, M.S. and Mesbah, M., Goodness-of-fit tests and model validity, Birkhauser, Basel.
fit <- ppm(swedishpines ~ x+y) auc(fit)
fit <- ppm(swedishpines ~ x+y) auc(fit)
Creates an instance of the Baddeley-Geyer point process model, defined as a hybrid of several Geyer interactions. The model can then be fitted to point pattern data.
BadGey(r, sat)
BadGey(r, sat)
r |
vector of interaction radii |
sat |
vector of saturation parameters, or a single common value of saturation parameter |
This is Baddeley's generalisation of the
Geyer saturation point process model,
described in Geyer
, to a process with multiple interaction
distances.
The BadGey point process with interaction radii
,
saturation thresholds
,
intensity parameter
and
interaction parameters
,
is the point process
in which each point
in the pattern
contributes a factor
to the probability density of the point pattern, where
where denotes the
number of points in the pattern
which lie
within a distance
from the point
.
BadGey
is used to fit this model to data.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the piecewise constant Saturated pairwise
interaction is yielded by the function BadGey()
.
See the examples below.
The argument r
specifies the vector of interaction distances.
The entries of r
must be strictly increasing, positive numbers.
The argument sat
specifies the vector of saturation parameters
that are applied to the point counts .
It should be a vector of the same length as
r
, and its entries
should be nonnegative numbers. Thus sat[1]
is applied to the
count of points within a distance r[1]
, and sat[2]
to the
count of points within a distance r[2]
, etc.
Alternatively sat
may be a single number, and this saturation
value will be applied to every count.
Infinite values of the
saturation parameters are also permitted; in this case
and there is effectively no ‘saturation’ for the distance range in
question. If all the saturation parameters are set to
Inf
then
the model is effectively a pairwise interaction process, equivalent to
PairPiece
(however the interaction parameters
obtained from
BadGey
have a complicated relationship to the interaction
parameters obtained from
PairPiece
).
If r
is a single number, this model is virtually equivalent to the
Geyer process, see Geyer
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
A ‘hybrid’ interaction is one which is built by combining
several different interactions (Baddeley et al, 2013).
The BadGey
interaction can be described as a
hybrid of several Geyer
interactions.
The Hybrid
command can be used to build
hybrids of any interactions. If the Hybrid
operator
is applied to several Geyer
models, the result is
equivalent to a BadGey
model.
This can be useful for incremental model selection.
Adrian Baddeley [email protected] and Rolf Turner [email protected] in collaboration with Hao Wang and Jeff Picka
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013)
Hybrids of Gibbs point process models and their implementation.
Journal of Statistical Software 55:11, 1–43.
DOI: 10.18637/jss.v055.i11
ppm
,
pairsat.family
,
Geyer
,
PairPiece
,
SatPiece
,
Hybrid
BadGey(c(0.1,0.2), c(1,1)) # prints a sensible description of itself BadGey(c(0.1,0.2), 1) # fit a stationary Baddeley-Geyer model ppm(cells ~1, BadGey(c(0.07, 0.1, 0.13), 2)) # nonstationary process with log-cubic polynomial trend ppm(cells ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
BadGey(c(0.1,0.2), c(1,1)) # prints a sensible description of itself BadGey(c(0.1,0.2), 1) # fit a stationary Baddeley-Geyer model ppm(cells ~1, BadGey(c(0.07, 0.1, 0.13), 2)) # nonstationary process with log-cubic polynomial trend ppm(cells ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
Applies a first-order bias correction to a fitted model.
bc(fit, ...) ## S3 method for class 'ppm' bc(fit, ..., nfine = 256)
bc(fit, ...) ## S3 method for class 'ppm' bc(fit, ..., nfine = 256)
fit |
A fitted point process model (object of class |
... |
Additional arguments are currently ignored. |
nfine |
Grid dimensions for fine grid of locations. An integer, or a pair of integers. See Details. |
This command applies the first order Newton-Raphson bias correction method of
Baddeley and Turner (2014, sec 4.2) to a fitted model.
The function bc
is generic, with a method for fitted point
process models of class "ppm"
.
A fine grid of locations, of dimensions nfine * nfine
or
nfine[2] * nfine[1]
, is created over the original window of the
data, and the intensity or conditional intensity of the fitted model is
calculated on this grid. The result is used to update the fitted
model parameters once by a Newton-Raphson update.
This is only useful if the quadrature points used to fit the original
model fit
are coarser than the grid of points specified by
nfine
.
A numeric vector, of the same length as coef(fit)
, giving updated
values for the fitted model coefficients.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Baddeley, A. and Turner, R. (2014) Bias correction for parameter estimates of spatial point process models. Journal of Statistical Computation and Simulation 84, 1621–1643. DOI: 10.1080/00949655.2012.755976
fit <- ppm(cells ~ x, Strauss(0.07)) coef(fit) if(!interactive()) { bc(fit, nfine=64) } else { bc(fit) }
fit <- ppm(cells ~ x, Strauss(0.07)) coef(fit) if(!interactive()) { bc(fit, nfine=64) } else { bc(fit) }
Tests the goodness-of-fit of a Poisson point process model using methods of Berman (1986).
## S3 method for class 'ppm' berman.test(model, covariate, which = c("Z1", "Z2"), alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'ppm' berman.test(model, covariate, which = c("Z1", "Z2"), alternative = c("two.sided", "less", "greater"), ...)
model |
A fitted point process model (object of class |
covariate |
The spatial covariate on which the test will be based.
An image (object of class |
which |
Character string specifying the choice of test. |
alternative |
Character string specifying the alternative hypothesis. |
... |
Additional arguments controlling the pixel resolution
(arguments |
These functions perform a goodness-of-fit test of a Poisson point
process model fitted to point pattern data. The observed distribution
of the values of a spatial covariate at the data points,
and the predicted distribution of the same values under the model,
are compared using either of two test statistics
and
proposed by Berman (1986).
The
test is also known as the
Lawson-Waller test.
The function berman.test
is generic, with methods for
point patterns ("ppp"
or "lpp"
)
and point process models ("ppm"
or "lppm"
).
If X
is a point pattern dataset (object of class
"ppp"
or "lpp"
), then
berman.test(X, ...)
performs a goodness-of-fit test of the
uniform Poisson point process (Complete Spatial Randomness, CSR)
for this dataset.
If model
is a fitted point process model
(object of class "ppm"
or "lppm"
)
then berman.test(model, ...)
performs
a test of goodness-of-fit for this fitted model. In this case,
model
should be a Poisson point process.
The test is performed by comparing the observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same covariate under the model. Thus, you must nominate a spatial covariate for this test.
The argument covariate
should be either a function(x,y)
or a pixel image (object of class "im"
containing the values
of a spatial function.
If covariate
is an image, it should have numeric values,
and its domain should cover the observation window of the
model
. If covariate
is a function, it should expect
two arguments x
and y
which are vectors of coordinates,
and it should return a numeric vector of the same length
as x
and y
.
First the original data point pattern is extracted from model
.
The values of the covariate
at these data points are
collected.
Next the values of the covariate
at all locations in the
observation window are evaluated. The point process intensity
of the fitted model is also evaluated at all locations in the window.
If which="Z1"
,
the test statistic is computed as follows.
The sum
of the covariate values at all data
points is evaluated. The predicted mean
and variance
of
are computed
from the values of the covariate at all locations in the window.
Then we compute
.
Closely-related tests were proposed independently
by Waller et al (1993) and Lawson (1993)
so this test is often termed the
Lawson-Waller test in epidemiological literature.
If which="Z2"
,
the test statistic is computed as follows.
The values of the
covariate
at all locations in the
observation window, weighted by the point process intensity,
are compiled into a cumulative distribution function .
The probability integral transformation is then applied:
the values of the
covariate
at the original data points
are transformed by the predicted cumulative distribution function
into numbers between 0 and 1. If the model is correct,
these numbers are i.i.d. uniform random numbers.
The standardised sample mean of these numbers is the
statistic
.
In both cases the null distribution of the test statistic is the standard normal distribution, approximately.
The return value is an object of class "htest"
containing the
results of the hypothesis test. The print method for this class
gives an informative summary of the test outcome.
An object of class "htest"
(hypothesis test)
and also of class "bermantest"
,
containing the results of the test. The return value can be
plotted (by plot.bermantest
) or printed
to give an informative summary of the test.
The meaning of a one-sided test must be carefully scrutinised: see the printed output.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54–62.
Lawson, A.B. (1993) On the analysis of mortality events around a prespecified fixed point. Journal of the Royal Statistical Society, Series A 156 (3) 363–377.
Waller, L., Turnbull, B., Clark, L.C. and Nasca, P. (1992) Chronic Disease Surveillance and testing of clustering of disease and exposure: Application to leukaemia incidence and TCE-contaminated dumpsites in upstate New York. Environmetrics 3, 281–300.
# Berman's data X <- copper$SouthPoints L <- copper$SouthLines D <- distmap(L, eps=1) # test of fitted model fit <- ppm(X ~ x+y) berman.test(fit, D)
# Berman's data X <- copper$SouthPoints L <- copper$SouthLines D <- distmap(L, eps=1) # test of fitted model fit <- ppm(X ~ x+y) berman.test(fit, D)
Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast.
cauchy.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
cauchy.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the model. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
This algorithm fits the Neyman-Scott cluster point process model
with Cauchy kernel to a point pattern dataset
by the Method of Minimum Contrast, using the function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The function of the point pattern will be computed
using
Kest
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits the Neyman-Scott cluster point process
with Cauchy kernel to X
,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical function of the Matern Cluster process
and the observed
function.
For a more detailed explanation of the Method of Minimum Contrast,
see
mincontrast
.
The model is described in Jalilian et al (2013).
It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
follow a common distribution described in Jalilian et al (2013).
If the argument lambda
is provided, then this is used
as the value of the point process intensity .
Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The corresponding model can be simulated using rCauchy
.
For computational reasons, the optimisation procedure uses the parameter
eta2
, which is equivalent to 4 * scale^2
where scale
is the scale parameter for the model
as used in rCauchy
.
Homogeneous or inhomogeneous Neyman-Scott/Cauchy models can also be
fitted using the function kppm
and the fitted models
can be simulated using simulate.kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley [email protected]
Ghorbani, M. (2013) Cauchy cluster process. Metrika 76, 697–706.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
cauchy.estpcf
,
lgcp.estK
,
thomas.estK
,
vargamma.estK
,
mincontrast
,
Kest
,
Kmodel
.
rCauchy
to simulate the model.
u <- cauchy.estK(redwood) u plot(u)
u <- cauchy.estK(redwood) u plot(u)
Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
cauchy.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
cauchy.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the model. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
This algorithm fits the Neyman-Scott cluster point process model with Cauchy kernel to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf
or one of its relatives.
The algorithm fits the Neyman-Scott cluster point process
with Cauchy kernel to X
,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical pair correlation function of the Matern Cluster process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The model is described in Jalilian et al (2013).
It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
follow a common distribution described in Jalilian et al (2013).
If the argument lambda
is provided, then this is used
as the value of the point process intensity .
Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The corresponding model can be simulated using rCauchy
.
For computational reasons, the optimisation procedure internally uses
the parameter eta2
, which is equivalent to 4 * scale^2
where scale
is the scale parameter for the model as used in
rCauchy
.
Homogeneous or inhomogeneous Neyman-Scott/Cauchy models can also be
fitted using the function kppm
and the fitted models
can be simulated using simulate.kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley [email protected]
Ghorbani, M. (2013) Cauchy cluster process. Metrika 76, 697–706.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
cauchy.estK
,
lgcp.estpcf
,
thomas.estpcf
,
vargamma.estpcf
,
mincontrast
,
pcf
,
pcfmodel
.
rCauchy
to simulate the model.
u <- cauchy.estpcf(redwood) u plot(u, legendpos="topright")
u <- cauchy.estpcf(redwood) u plot(u, legendpos="topright")
Performs a spatial distribution test of a point process model fitted to multiple spatial point patterns. The test compares the observed and predicted distributions of the values of a spatial covariate, using either the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling test of goodness-of-fit.
## S3 method for class 'mppm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., nsim=19, verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)
## S3 method for class 'mppm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., nsim=19, verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)
model |
An object of class |
covariate |
The spatial covariate on which the test will be based.
A function, a pixel image, a list of functions, a list of pixel
images, a hyperframe, a character string containing the name
of one of the covariates in |
test |
Character string identifying the test to be performed:
|
... |
Arguments passed to |
nsim |
Number of simulated realisations which should be generated, if a Monte Carlo test is required. |
verbose |
Logical flag indicating whether to print progress reports. |
interpolate |
Logical flag indicating whether to interpolate between
pixel values when |
fast |
Logical flag. If |
jitter |
Logical flag. If |
This function is a method for the generic function
cdf.test
for the class mppm
.
This function performs a goodness-of-fit test of a point process model that has been fitted to multiple point patterns. The observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same values under the model, are compared using the Kolmogorov-Smirnov, Cramer-von Mises or Anderson-Darling test of goodness-of-fit. These are exact tests if the model is Poisson; otherwise, for a Gibbs model, a Monte Carlo p-value is computed by generating simulated realisations of the model and applying the selected goodness-of-fit test to each simulation.
The argument model
should be a fitted point process model
fitted to multiple point patterns
(object of class "mppm"
).
The argument covariate
contains the values of a spatial
function. It can be
a function(x,y)
a pixel image (object of class "im"
a list of function(x,y)
, one for each point pattern
a list of pixel images, one for each point pattern
a hyperframe (see hyperframe
)
of which the first
column will be taken as containing the covariate
a character string giving the name of one of the covariates
in model
one of the character strings "x"
or "y"
,
indicating the spatial coordinates.
If covariate
is an image, it should have numeric values,
and its domain should cover the observation window of the
model
. If covariate
is a function, it should expect
two arguments x
and y
which are vectors of coordinates,
and it should return a numeric vector of the same length
as x
and y
.
First the original data point pattern is extracted from model
.
The values of the covariate
at these data points are
collected.
The predicted distribution of the values of the covariate
under the fitted model
is computed as follows.
The values of the covariate
at all locations in the
observation window are evaluated,
weighted according to the point process intensity of the fitted model,
and compiled into a cumulative distribution function using
ewcdf
.
The probability integral transformation is then applied:
the values of the covariate
at the original data points
are transformed by the predicted cumulative distribution function
into numbers between 0 and 1. If the model is correct,
these numbers are i.i.d. uniform random numbers.
A goodness-of-fit test of the uniform distribution is applied
to these numbers using
ks.test
,
cvm.test
or ad.test
.
The argument interpolate
determines
how pixel values will be handled when covariate
is a pixel image.
The value of the covariate at a data point is obtained
by looking up the value of the nearest pixel if
interpolate=FALSE
, or by linearly interpolating
between the values of the four nearest pixels
if interpolate=TRUE
. Linear interpolation is slower,
but is sometimes necessary to avoid tied values of the covariate
arising when the pixel grid is coarse.
If model
is a Poisson point process, then the
Kolmogorov-Smirnov,
Cramer-von Mises
and Anderson-Darling tests are theoretically exact.
This test was apparently first described (in the context of
spatial data, and for Kolmogorov-Smirnov) by Berman (1986).
See also Baddeley et al (2005).
If model
is not a Poisson point process, then the
Kolmogorov-Smirnov,
Cramer-von Mises
and Anderson-Darling tests are biased.
Instead they are used as the basis of a Monte Carlo test.
First nsim
simulated realisations of the model will be generated.
Each simulated realisation consists of a list of simulated point
patterns, one for each of the original data patterns. This
can take a very long time. The model is then re-fitted to each
simulation, and the refitted model is subjected to the goodness-of-fit
test described above. A Monte Carlo p-value is then computed by
comparing the p-value of the original test with the
p-values obtained from the simulations.
An object of class "cdftest"
and "htest"
containing the results of the
test. See cdf.test
for details.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54–62.
# three i.i.d. realisations of nonuniform Poisson process lambda <- as.im(function(x,y) { 200 * exp(x) }, square(1)) dat <- hyperframe(X=list(rpoispp(lambda), rpoispp(lambda), rpoispp(lambda))) # fit uniform Poisson process fit0 <- mppm(X~1, dat) # fit correct nonuniform Poisson process fit1 <- mppm(X~x, dat) # test wrong model cdf.test(fit0, "x") # test right model cdf.test(fit1, "x") # Gibbs model fitGibbs <- update(fit0, interaction=Strauss(0.05)) ns <- if(interactive()) 19 else 2 cdf.test(fitGibbs, "x", nsim=ns)
# three i.i.d. realisations of nonuniform Poisson process lambda <- as.im(function(x,y) { 200 * exp(x) }, square(1)) dat <- hyperframe(X=list(rpoispp(lambda), rpoispp(lambda), rpoispp(lambda))) # fit uniform Poisson process fit0 <- mppm(X~1, dat) # fit correct nonuniform Poisson process fit1 <- mppm(X~x, dat) # test wrong model cdf.test(fit0, "x") # test right model cdf.test(fit1, "x") # Gibbs model fitGibbs <- update(fit0, interaction=Strauss(0.05)) ns <- if(interactive()) 19 else 2 cdf.test(fitGibbs, "x", nsim=ns)
Performs a test of goodness-of-fit of a point process model. The observed and predicted distributions of the values of a spatial covariate are compared using either the Kolmogorov-Smirnov test, Cramer-von Mises test or Anderson-Darling test. For non-Poisson models, a Monte Carlo test is used.
## S3 method for class 'ppm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., interpolate=TRUE, jitter=TRUE, nsim=99, verbose=TRUE) ## S3 method for class 'slrm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., modelname=NULL, covname=NULL)
## S3 method for class 'ppm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., interpolate=TRUE, jitter=TRUE, nsim=99, verbose=TRUE) ## S3 method for class 'slrm' cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., modelname=NULL, covname=NULL)
model |
A fitted point process model (object of class |
covariate |
The spatial covariate on which the test will be based.
A function, a pixel image (object of class |
test |
Character string identifying the test to be performed:
|
... |
Arguments passed to |
interpolate |
Logical flag indicating whether to interpolate pixel images.
If |
jitter |
Logical flag. If |
modelname , covname
|
Character strings giving alternative names for |
nsim |
Number of simulated realisations from the |
verbose |
Logical value indicating whether to print progress reports when performing a Monte Carlo test. |
These functions perform a goodness-of-fit test of a Poisson or Gibbs point process model fitted to point pattern data. The observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same values under the model, are compared using the Kolmogorov-Smirnov test, the Cramer-von Mises test or the Anderson-Darling test. For Gibbs models, a Monte Carlo test is performed using these test statistics.
The function cdf.test
is generic, with methods for
point patterns ("ppp"
or "lpp"
),
point process models ("ppm"
or "lppm"
)
and spatial logistic regression models ("slrm"
).
If X
is a point pattern dataset (object of class
"ppp"
), then cdf.test(X, ...)
performs a goodness-of-fit test of the
uniform Poisson point process (Complete Spatial Randomness, CSR)
for this dataset.
For a multitype point pattern, the uniform intensity
is assumed to depend on the type of point (sometimes called
Complete Spatial Randomness and Independence, CSRI).
If model
is a fitted point process model
(object of class "ppm"
or "lppm"
)
then cdf.test(model, ...)
performs
a test of goodness-of-fit for this fitted model.
If model
is a fitted spatial logistic regression
(object of class "slrm"
) then cdf.test(model, ...)
performs
a test of goodness-of-fit for this fitted model.
The test is performed by comparing the observed distribution of the values of a spatial covariate at the data points, and the predicted distribution of the same covariate under the model, using a classical goodness-of-fit test. Thus, you must nominate a spatial covariate for this test.
If X
is a point pattern that does not have marks,
the argument covariate
should be either a function(x,y)
or a pixel image (object of class "im"
containing the values
of a spatial function, or one of the characters "x"
or
"y"
indicating the Cartesian coordinates.
If covariate
is an image, it should have numeric values,
and its domain should cover the observation window of the
model
. If covariate
is a function, it should expect
two arguments x
and y
which are vectors of coordinates,
and it should return a numeric vector of the same length
as x
and y
.
If X
is a multitype point pattern, the argument covariate
can be either a function(x,y,marks)
,
or a pixel image, or a list of pixel images corresponding to
each possible mark value, or one of the characters "x"
or
"y"
indicating the Cartesian coordinates.
First the original data point pattern is extracted from model
.
The values of the covariate
at these data points are
collected.
The predicted distribution of the values of the covariate
under the fitted model
is computed as follows.
The values of the covariate
at all locations in the
observation window are evaluated,
weighted according to the point process intensity of the fitted model,
and compiled into a cumulative distribution function using
ewcdf
.
The probability integral transformation is then applied:
the values of the covariate
at the original data points
are transformed by the predicted cumulative distribution function
into numbers between 0 and 1. If the model is correct,
these numbers are i.i.d. uniform random numbers. The
A goodness-of-fit test of the uniform distribution is applied
to these numbers using
stats::ks.test
,
goftest::cvm.test
or
goftest::ad.test
.
This test was apparently first described (in the context of spatial data, and using Kolmogorov-Smirnov) by Berman (1986). See also Baddeley et al (2005).
If model
is not a Poisson process, then
a Monte Carlo test is performed, by generating nsim
point patterns which are simulated realisations of the model
,
re-fitting the model to each simulated point pattern,
and calculating the test statistic for each fitted model.
The Monte Carlo value is determined by comparing
the simulated values of the test statistic
with the value for the original data.
The return value is an object of class "htest"
containing the
results of the hypothesis test. The print method for this class
gives an informative summary of the test outcome.
The return value also belongs to the class "cdftest"
for which there is a plot method plot.cdftest
.
The plot method displays the empirical cumulative distribution
function of the covariate at the data points, and the predicted
cumulative distribution function of the covariate under the model,
plotted against the value of the covariate.
The argument jitter
controls whether covariate values are
randomly perturbed, in order to avoid ties.
If the original data contains any ties in the covariate (i.e. points
with equal values of the covariate), and if jitter=FALSE
, then
the Kolmogorov-Smirnov test implemented in ks.test
will issue a warning that it cannot calculate the exact -value.
To avoid this, if
jitter=TRUE
each value of the covariate will
be perturbed by adding a small random value. The perturbations are
normally distributed with standard deviation equal to one hundredth of
the range of values of the covariate. This prevents ties,
and the -value is still correct. There is
a very slight loss of power.
An object of class "htest"
containing the results of the
test. See ks.test
for details. The return value can be
printed to give an informative summary of the test.
The value also belongs to the class "cdftest"
for which there is
a plot method.
The outcome of the test involves a small amount of random variability,
because (by default) the coordinates are randomly perturbed to
avoid tied values. Hence, if cdf.test
is executed twice, the
-values will not be exactly the same. To avoid this behaviour,
set
jitter=FALSE
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic process. Applied Statistics 35, 54–62.
plot.cdftest
,
quadrat.test
,
berman.test
,
ks.test
,
cvm.test
,
ad.test
,
ppm
op <- options(useFancyQuotes=FALSE) # fit inhomogeneous Poisson model and test model <- ppm(nztrees ~x) cdf.test(model, "x") if(interactive()) { # synthetic data: nonuniform Poisson process X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1)) # fit uniform Poisson process fit0 <- ppm(X ~1) # fit correct nonuniform Poisson process fit1 <- ppm(X ~x) # test wrong model cdf.test(fit0, "x") # test right model cdf.test(fit1, "x") } # multitype point pattern yimage <- as.im(function(x,y){y}, W=Window(amacrine)) cdf.test(ppm(amacrine ~marks+y), yimage) options(op)
op <- options(useFancyQuotes=FALSE) # fit inhomogeneous Poisson model and test model <- ppm(nztrees ~x) cdf.test(model, "x") if(interactive()) { # synthetic data: nonuniform Poisson process X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1)) # fit uniform Poisson process fit0 <- ppm(X ~1) # fit correct nonuniform Poisson process fit1 <- ppm(X ~x) # test wrong model cdf.test(fit0, "x") # test right model cdf.test(fit1, "x") } # multitype point pattern yimage <- as.im(function(x,y){y}, W=Window(amacrine)) cdf.test(ppm(amacrine ~marks+y), yimage) options(op)
Low-level functions to count the number of close pairs of points.
closepaircounts(X, r) crosspaircounts(X, Y, r)
closepaircounts(X, r) crosspaircounts(X, Y, r)
X , Y
|
Point patterns (objects of class |
r |
Maximum distance between pairs of points to be counted as close pairs. |
These are the efficient low-level functions used by spatstat to count close pairs of points in a point pattern or between two point patterns.
closepaircounts(X,r)
counts the number of neighbours for
each point in the pattern X
. That is, for each point
X[i]
, it counts the number of other points X[j]
with j != i
such that d(X[i],X[j]) <= r
where
d
denotes Euclidean distance. The result is an integer vector
v
such that v[i]
is the number of neighbours of
X[i]
.
crosspaircounts(X,Y,r)
counts, for each point
in the pattern X
, the number of neighbours in the pattern
Y
. That is, for each point
X[i]
, it counts the number of points Y[j]
such that d(X[i],X[j]) <= r
. The result is an integer vector
v
such that v[i]
is the number of neighbours of
X[i]
in the pattern Y
.
An integer vector of length equal to the number of points in X
.
The results of these functions may not agree exactly with
the correct answer (as calculated by a human) and may not
be consistent between different computers and different installations
of R. The discrepancies arise in marginal cases where the interpoint
distance is equal to, or very close to, the threshold rmax
.
Floating-point numbers in a computer
are not mathematical Real Numbers: they are approximations using
finite-precision binary arithmetic.
The approximation is accurate to a tolerance of about
.Machine$double.eps
.
If the true interpoint distance and the threshold
rmax
are equal, or if their difference is no more than .Machine$double.eps
,
the result may be incorrect.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
closepairs
to identify all close pairs of points.
a <- closepaircounts(cells, 0.1) sum(a) Y <- split(amacrine) b <- crosspaircounts(Y$on, Y$off, 0.1)
a <- closepaircounts(cells, 0.1) sum(a) Y <- split(amacrine) b <- crosspaircounts(Y$on, Y$off, 0.1)
Calculate the superposition of cluster kernels at the location of a point pattern.
## S3 method for class 'kppm' clusterfield(model, locations = NULL, ...)
## S3 method for class 'kppm' clusterfield(model, locations = NULL, ...)
model |
Cluster model. Either a fitted cluster model (object of class
|
locations |
A point pattern giving the locations of the kernels. Defaults to the
centroid of the observation window for the |
... |
Additional arguments passed to |
The function clusterfield
is generic,
with a method for "kppm"
(described here)
and methods for "character"
and "function"
.
The method clusterfield.kppm
extracts the relevant information
from the fitted model and calls clusterfield.function
.
The calculations are performed by density.ppp
and
...
arguments are passed thereto for control over the pixel
resolution etc.
(These arguments are then passed on to pixellate.ppp
and as.mask
.)
A pixel image (object of class "im"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
kppm
,
fit <- kppm(redwood~1, "Thomas") Z <- clusterfield(fit, eps = 0.01)
fit <- kppm(redwood~1, "Thomas") Z <- clusterfield(fit, eps = 0.01)
Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern by the Method of Minimum Contrast.
clusterfit(X, clusters, lambda = NULL, startpar = NULL, ..., q = 1/4, p = 2, rmin = NULL, rmax = NULL, ctrl=list(q=q, p=p, rmin=rmin, rmax=rmax), statistic = NULL, statargs = NULL, algorithm="Nelder-Mead", verbose=FALSE, pspace=NULL)
clusterfit(X, clusters, lambda = NULL, startpar = NULL, ..., q = 1/4, p = 2, rmin = NULL, rmax = NULL, ctrl=list(q=q, p=p, rmin=rmin, rmax=rmax), statistic = NULL, statargs = NULL, algorithm="Nelder-Mead", verbose=FALSE, pspace=NULL)
X |
Data to which the cluster or Cox model will be fitted. Either a point pattern or a summary statistic. See Details. |
clusters |
Character string determining the cluster or Cox model.
Partially matched.
Options are |
lambda |
Optional. An estimate of the intensity of the point process.
Either a single numeric specifying a constant intensity,
a pixel image (object of class |
startpar |
Vector of initial values of the parameters of the
point process mode. If |
q , p
|
Optional. Exponents for the contrast criterion.
See |
rmin , rmax
|
Optional. The interval of |
ctrl |
Optional. Named list containing values of the parameters
|
... |
Additional arguments passed to |
statistic |
Optional. Name of the summary statistic to be used
for minimum contrast estimation: either |
statargs |
Optional list of arguments to be used when calculating
the |
algorithm |
Character string determining the mathematical optimisation algorithm
to be used by |
verbose |
Logical value indicating whether to print detailed progress reports for debugging purposes. |
pspace |
For internal use by package code only. |
This function fits the clustering parameters of a cluster or Cox point
process model by the Method of Minimum Contrast, that is, by
matching the theoretical -function of the model to the
empirical
-function of the data, as explained in
mincontrast
.
If statistic="pcf"
(or X
appears to be an
estimated pair correlation function) then instead of using the
-function, the algorithm will use the pair correlation
function.
If X
is a point pattern of class "ppp"
an estimate of
the summary statistic specfied by statistic
(defaults to
"K"
) is first computed before minimum contrast estimation is
carried out as described above. In this case the argument
statargs
can be used for controlling the summary statistic
estimation. The precise algorithm for computing the summary statistic
depends on whether the intensity specification (lambda
) is:
If lambda
is NUll
or a single numeric the pattern is
considered homogeneous and either Kest
or
pcf
is invoked. In this case lambda
is
not used for anything when estimating the summary statistic.
If lambda
is a pixel image (object of class "im"
),
a fitted point process model (object of class "ppm"
or
"kppm"
) or a function(x,y)
the pattern is considered
inhomogeneous. In this case either Kinhom
or
pcfinhom
is invoked with lambda
as an
argument.
After the clustering parameters of the model have been estimated by
minimum contrast lambda
(if non-null) is used to compute the
additional model parameter .
The algorithm parameters q,p,rmax,rmin
are described in the
help for mincontrast
. They may be provided either
as individually-named arguments, or as entries in the list
ctrl
. The individually-named arguments q,p,rmax,rmin
override the entries in the list ctrl
.
An object of class "minconfit"
. There are methods for printing
and plotting this object. See mincontrast
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63 (2007) 252–258.
fit <- clusterfit(redwood, "Thomas") fit if(interactive()){ plot(fit) } K <- Kest(redwood) fit2 <- clusterfit(K, "MatClust")
fit <- clusterfit(redwood, "Thomas") fit if(interactive()){ plot(fit) } K <- Kest(redwood) fit2 <- clusterfit(K, "MatClust")
Given a fitted cluster point process model, this command returns the probability density of the cluster offspring.
## S3 method for class 'kppm' clusterkernel(model, ...)
## S3 method for class 'kppm' clusterkernel(model, ...)
model |
Cluster model. Either a fitted cluster or Cox model
(object of class |
... |
Parameter values for the model,
when |
Given a cluster point process model, this command
returns a function(x,y)
giving the two-dimensional
probability density of the cluster offspring points assuming a cluster parent
located at the origin.
The function clusterkernel
is generic,
with methods for class "kppm"
(described here)
and "character"
(described in clusterkernel.character
).
A function in the R language with arguments x,y,...
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
clusterkernel.character
,
clusterfield
, kppm
fit <- kppm(redwood ~ x, "MatClust") f <- clusterkernel(fit) f(0.05, 0.02)
fit <- kppm(redwood ~ x, "MatClust") f <- clusterkernel(fit) f(0.05, 0.02)
Given a cluster point process model, this command returns a value beyond which the the probability density of the cluster offspring is neglible.
## S3 method for class 'kppm' clusterradius(model, ..., thresh = NULL, precision = FALSE)
## S3 method for class 'kppm' clusterradius(model, ..., thresh = NULL, precision = FALSE)
model |
Cluster model. Either a fitted cluster or Cox model
(object of class |
... |
Parameter values for the model,
when |
thresh |
Numerical threshold relative to the cluster kernel value at the origin (parent location) determining when the cluster kernel will be considered neglible. A sensible default is provided. |
precision |
Logical. If |
Given a cluster model this function by default returns the effective
range of the model with the given parameters as used in spatstat. For
the Matern cluster model (see e.g. rMatClust
) this is
simply the finite radius of the offsring density given by the paramter
scale
irrespective of other options given to this function. The
remaining models in spatstat have infinite theoretical range, and an
effective finite value is given as follows: For the Thomas model (see
e.g. rThomas
the default is 4*scale
where scale
is the scale or standard deviation parameter of the model. If
thresh
is given the value is instead found as described for the
other models below.
For the Cauchy model (see e.g. rCauchy
) and the Variance
Gamma (Bessel) model (see e.g. rVarGamma
) the value of
thresh
defaults to 0.001, and then this is used to compute the
range numerically as follows. If
with
denotes the isotropic cluster kernel then
is the
density function of the offspring distance from the parent. The range
is determined as the value of
where
falls below
thresh
times .
If precision=TRUE
the precision related to the chosen range is
returned as an attribute. Here the precision is defined as the polar
integral of the kernel from distance 0 to the calculated
range. Ideally this should be close to the value 1 which would be
obtained for the true theretical infinite range.
A positive numeric.
Additionally, the precision related to this range value is returned as
an attribute "prec"
, if precision=TRUE
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
clusterkernel
, kppm
,
rMatClust
, rThomas
,
rCauchy
,
rVarGamma
, rNeymanScott
.
fit <- kppm(redwood ~ x, "MatClust") clusterradius(fit)
fit <- kppm(redwood ~ x, "MatClust") clusterradius(fit)
Given a point process model fitted to a list of point patterns,
extract the coefficients of the fitted model.
A method for coef
.
## S3 method for class 'mppm' coef(object, ...)
## S3 method for class 'mppm' coef(object, ...)
object |
The fitted point process model (an object of class |
... |
Ignored. |
This function is a method for the generic function coef
.
The argument object
must be a fitted point process model
(object of class "mppm"
) produced by the
fitting algorithm mppm
). This represents a
point process model that has been fitted
to a list of several point pattern datasets. See mppm
for information.
This function extracts the vector of coefficients of the fitted model.
This is the estimate of the parameter vector
such that the conditional intensity of the model
is of the form
where is a (vector-valued) statistic.
For example, if the model object
is the uniform Poisson process,
then coef(object)
will yield a single value
(named "(Intercept)"
) which is the logarithm of the
fitted intensity of the Poisson process.
If the fitted model includes random effects (i.e. if the argument
random
was specified in the call to mppm
),
then the fitted coefficients are different for each point pattern
in the original data, so coef(object)
is a data frame
with one row for each point pattern, and one column for each
parameter. Use fixef.mppm
to extract the vector of fixed effect
coefficients, and ranef.mppm
to extract the random
effect coefficients at each level.
Use print.mppm
to print a more useful
description of the fitted model.
Either a vector containing the fitted coefficients, or a data frame containing the fitted coefficients for each point pattern.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
fixef.mppm
and ranef.mppm
for the fixed and random effect coefficients in a model that includes
random effects.
H <- hyperframe(X=waterstriders) fit.Poisson <- mppm(X ~ 1, H) coef(fit.Poisson) # The single entry "(Intercept)" # is the log of the fitted intensity of the Poisson process fit.Strauss <- mppm(X~1, H, Strauss(7)) coef(fit.Strauss) # The two entries "(Intercept)" and "Interaction" # are respectively log(beta) and log(gamma) # in the usual notation for Strauss(beta, gamma, r) # Tweak data to exaggerate differences H$X[[1]] <- rthin(H$X[[1]], 0.3) # Model with random effects fitran <- mppm(X ~ 1, H, random=~1|id) coef(fitran)
H <- hyperframe(X=waterstriders) fit.Poisson <- mppm(X ~ 1, H) coef(fit.Poisson) # The single entry "(Intercept)" # is the log of the fitted intensity of the Poisson process fit.Strauss <- mppm(X~1, H, Strauss(7)) coef(fit.Strauss) # The two entries "(Intercept)" and "Interaction" # are respectively log(beta) and log(gamma) # in the usual notation for Strauss(beta, gamma, r) # Tweak data to exaggerate differences H$X[[1]] <- rthin(H$X[[1]], 0.3) # Model with random effects fitran <- mppm(X ~ 1, H, random=~1|id) coef(fitran)
Given a point process model fitted to a point pattern,
extract the coefficients of the fitted model.
A method for coef
.
## S3 method for class 'ppm' coef(object, ...)
## S3 method for class 'ppm' coef(object, ...)
object |
The fitted point process model (an object of class |
... |
Ignored. |
This function is a method for the generic function coef
.
The argument object
must be a fitted point process model
(object of class "ppm"
). Such objects are produced by the maximum
pseudolikelihood fitting algorithm ppm
).
This function extracts the vector of coefficients of the fitted model.
This is the estimate of the parameter vector
such that the conditional intensity of the model
is of the form
where is a (vector-valued) statistic.
For example, if the model object
is the uniform Poisson process,
then coef(object)
will yield a single value
(named "(Intercept)"
) which is the logarithm of the
fitted intensity of the Poisson process.
Use print.ppm
to print a more useful
description of the fitted model.
A vector containing the fitted coefficients.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
poi <- ppm(cells, ~1, Poisson()) coef(poi) # This is the log of the fitted intensity of the Poisson process stra <- ppm(cells, ~1, Strauss(r=0.07)) coef(stra) # The two entries "(Intercept)" and "Interaction" # are respectively log(beta) and log(gamma) # in the usual notation for Strauss(beta, gamma, r)
poi <- ppm(cells, ~1, Poisson()) coef(poi) # This is the log of the fitted intensity of the Poisson process stra <- ppm(cells, ~1, Strauss(r=0.07)) coef(stra) # The two entries "(Intercept)" and "Interaction" # are respectively log(beta) and log(gamma) # in the usual notation for Strauss(beta, gamma, r)
Extracts the coefficients (parameters) from a fitted Spatial Logistic Regression model.
## S3 method for class 'slrm' coef(object, ...)
## S3 method for class 'slrm' coef(object, ...)
object |
a fitted spatial logistic regression model.
An object of class |
... |
Ignored. |
This is a method for coef
for fitted spatial logistic
regression models (objects of class "slrm"
, usually obtained
from the function slrm
).
It extracts the fitted canonical parameters, i.e.\ the coefficients in the linear predictor of the spatial logistic regression.
Numeric vector of coefficients.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
X <- rpoispp(42) fit <- slrm(X ~ x+y) coef(fit)
X <- rpoispp(42) fit <- slrm(X ~ x+y) coef(fit)
Compares several fitted point process models using the same residual diagnostic.
compareFit(object, Fun, r = NULL, breaks = NULL, ..., trend = ~1, interaction = Poisson(), rbord = NULL, modelnames = NULL, same = NULL, different = NULL)
compareFit(object, Fun, r = NULL, breaks = NULL, ..., trend = ~1, interaction = Poisson(), rbord = NULL, modelnames = NULL, same = NULL, different = NULL)
object |
Object or objects to be analysed.
Either a fitted point process model (object of class |
Fun |
Diagnostic function to be computed for each model.
One of the functions |
r |
Optional.
Vector of values of the argument |
breaks |
Optional alternative to |
... |
Extra arguments passed to |
trend , interaction , rbord
|
Optional. Arguments passed to |
modelnames |
Character vector. Short descriptive names for the different models. |
same , different
|
Character strings or character vectors passed to
|
This is a convenient way to collect diagnostic information for several different point process models fitted to the same point pattern dataset, or for point process models of the same form fitted to several different datasets, etc.
The first argument, object
, is usually a list of
fitted point process models
(objects of class "ppm"
), obtained from the
model-fitting function ppm
.
For convenience, object
can also be a list of point patterns
(objects of class "ppp"
).
In that case, point process models will be fitted to
each of the point pattern datasets,
by calling ppm
using the arguments
trend
(for the first order trend),
interaction
(for the interpoint interaction)
and rbord
(for the erosion distance in the border correction
for the pseudolikelihood). See ppm
for details
of these arguments.
Alternatively object
can be a single point pattern
(object of class "ppp"
) and one or more of the arguments
trend
, interaction
or rbord
can be a list. In this case, point process models will be fitted to
the same point pattern dataset, using each of the model specifications
listed.
The diagnostic function Fun
will be applied to each of the
point process models. The results will be collected into a single
function value table. The modelnames
are used to label the
results from each fitted model.
Function value table (object of class "fv"
).
Ege Rubak [email protected], Adrian Baddeley [email protected] and Jesper Moller.
ppm
,
Kcom
,
Kres
,
Gcom
,
Gres
,
psst
,
psstA
,
psstG
,
collapse.fv
nd <- 40 ilist <- list(Poisson(), Geyer(7, 2), Strauss(7)) iname <- c("Poisson", "Geyer", "Strauss") K <- compareFit(swedishpines, Kcom, interaction=ilist, rbord=9, correction="translate", same="trans", different="tcom", modelnames=iname, nd=nd) K
nd <- 40 ilist <- list(Poisson(), Geyer(7, 2), Strauss(7)) iname <- c("Poisson", "Geyer", "Strauss") K <- compareFit(swedishpines, Kcom, interaction=ilist, rbord=9, correction="translate", same="trans", different="tcom", modelnames=iname, nd=nd) K
Creates an instance of the Connected Component point process model which can then be fitted to point pattern data.
Concom(r)
Concom(r)
r |
Threshold distance |
This function defines the interpoint interaction structure of a point process called the connected component process. It can be used to fit this model to point pattern data.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the connected component interaction is
yielded by the function Concom()
. See the examples below.
In standard form, the connected component process
(Baddeley and Moller, 1989) with disc radius ,
intensity parameter
and interaction parameter
is a point process with probability density
for a point pattern , where
represent the
points of the pattern,
is the number of points in the
pattern, and
is defined below.
Here
is a normalising constant.
To define the term C(x)
, suppose that we construct a planar
graph by drawing an edge between
each pair of points which are less than
units apart. Two points belong to the same connected component
of this graph if they are joined by a path in the graph.
Then
is the number of connected components of the graph.
The interaction parameter can be any positive number.
If
then the model reduces to a Poisson
process with intensity
.
If
then the process is regular,
while if
the process is clustered.
Thus, a connected-component interaction process can be used to model either
clustered or regular point patterns.
In spatstat, the model is parametrised in a different form, which is easier to interpret. In canonical form, the probability density is rewritten as
where is the new intensity parameter and
is the interaction potential.
In this formulation, each isolated point of the pattern contributes a
factor
to the probability density (so the
first order trend is
). The quantity
is a true interaction potential, in the sense that
if the point pattern
does not contain any
points that lie close together.
When a new point is added to an existing point pattern
, the rescaled potential
increases by
zero or a positive integer.
The increase is zero if
is not close to any point of
.
The increase is a positive integer
if there are
different connected components of
that lie close to
.
Addition of the point
contributes a factor
to the probability density, where
is the
increase in potential.
If desired, the original parameter can be recovered from
the canonical parameter by
.
The nonstationary connected component process is similar except that
the contribution of each individual point
is a function
of location, rather than a constant beta.
Note the only argument of Concom()
is the threshold distance r
.
When r
is fixed, the model becomes an exponential family.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
Concom()
.
An object of class "interact"
describing the interpoint interaction
structure of the connected component process with disc radius .
The interaction distance of this process is infinite.
There are no well-established procedures for edge correction
for fitting such models, and accordingly the model-fitting function
ppm
will give an error message saying that the user must
specify an edge correction. A reasonable solution is
to use the border correction at the same distance r
, as shown in the
Examples.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A.J. and Moller, J. (1989) Nearest-neighbour Markov point processes and random sets. International Statistical Review 57, 89–121.
ppm
,
pairwise.family
,
ppm.object
# prints a sensible description of itself Concom(r=0.1) # Fit the stationary connected component process to redwood data ppm(redwood ~1, Concom(r=0.07), rbord=0.07) # Fit the stationary connected component process to `cells' data ppm(cells ~1, Concom(r=0.06), rbord=0.06) # eta=0 indicates hard core process. # Fit a nonstationary connected component model # with log-cubic polynomial trend ppm(swedishpines ~polynom(x/10,y/10,3), Concom(r=7), rbord=7)
# prints a sensible description of itself Concom(r=0.1) # Fit the stationary connected component process to redwood data ppm(redwood ~1, Concom(r=0.07), rbord=0.07) # Fit the stationary connected component process to `cells' data ppm(cells ~1, Concom(r=0.06), rbord=0.06) # eta=0 indicates hard core process. # Fit a nonstationary connected component model # with log-cubic polynomial trend ppm(swedishpines ~polynom(x/10,y/10,3), Concom(r=7), rbord=7)
Given a fitted point process model, this function extracts the original point pattern dataset to which the model was fitted.
data.ppm(object)
data.ppm(object)
object |
fitted point process model (an object of class |
An object of class "ppm"
represents a point process model
that has been fitted to data. It is typically produced by
the model-fitting algorithm ppm
.
The object contains complete information about the original data
point pattern to which the model was fitted.
This function extracts the original data pattern.
See ppm.object
for a list of all operations that can be
performed on objects of class "ppm"
.
A point pattern (object of class "ppp"
).
Adrian Baddeley [email protected] and Rolf Turner [email protected].
fit <- ppm(cells, ~1, Strauss(r=0.1)) X <- data.ppm(fit) # 'X' is identical to 'cells'
fit <- ppm(cells, ~1, Strauss(r=0.1)) X <- data.ppm(fit) # 'X' is identical to 'cells'
Function to ease the implementation of a new determinantal point process model family.
detpointprocfamilyfun(kernel = NULL, specden = NULL, basis = "fourierbasis", convkernel = NULL, Kfun = NULL, valid = NULL, intensity = NULL, dim = 2, name = "User-defined", isotropic = TRUE, range = NULL, parbounds = NULL, specdenrange = NULL, startpar = NULL, ...)
detpointprocfamilyfun(kernel = NULL, specden = NULL, basis = "fourierbasis", convkernel = NULL, Kfun = NULL, valid = NULL, intensity = NULL, dim = 2, name = "User-defined", isotropic = TRUE, range = NULL, parbounds = NULL, specdenrange = NULL, startpar = NULL, ...)
kernel |
function specifying the kernel.
May be set to |
specden |
function specifying the spectral density.
May be set to |
basis |
character string giving the name of the basis. Defaults to the Fourier basis. See Details. |
convkernel |
function specifying the k-fold auto-convolution of the kernel.
May be set to |
Kfun |
function specifying the K-function.
May be set to |
valid |
function determining whether a given set of parameter values
yields a valid model. May be set to |
intensity |
character string specifying which parameter is the intensity in the
model family. Should be |
dim |
character strig specifying which parameter is the dimension of the state space in this model family (if any). Alternatively a positive integer specifying the dimension. |
name |
character string giving the name of the model family used for printing. |
isotropic |
logical value indicating whether or not the model is isotropic. |
range |
function determining the interaction range of the model. May be
set to |
parbounds |
function determining the bounds for each model parameter when all
other parameters are fixed. May be set to |
specdenrange |
function specifying the the range of the spectral density if it
is finite (only the case for very few models). May be set to
|
startpar |
function determining starting values for parameters in any estimation
algorithm. May be set to |
... |
Additional arguments for inclusion in the returned model object. These are not checked in any way. |
A determinantal point process family is specified either
in terms of a kernel (a positive semi-definite function, i.e. a
covariance function) or a spectral density, or preferably both.
One of these can be NULL
if it is unknown, but not both.
When both are supplied they must have the same arguments.
The first argument gives
the values at which the function should be evaluated. In general
the function should accept an by
matrix or
data.frame
specifying
points in dimension
. If the model is isotropic it only needs to
accept a non-negative valued numeric of length
. (In fact
there is currently almost no support for non-isotropic models, so
it is recommended not to specify such a model.) The name
of this argument could be chosen freely, but
is
recommended. The remaining arguments are the parameters of the
model. If one of these is an intensity parameter the name should
be mentioned in the argument
intensity
. If one of these
specifies the dimension of the model it should be mentioned in the
argument dim
.
The kernel and spectral density is with respect to a
specific set of basis functions, which is typically the Fourier
basis. However this can be changed to any user-supplied basis in
the argument basis
. If such an alternative is supplied it
must be the name of a function expecting the same arguments as
fourierbasis
and returning the results in the same
form as fourierbasis
.
If supplied, the arguments of convkernel must obey the
following: first argument should be like the first argument of
kernel and/or specden (see above). The second argument (preferably
called k
) should be the positive integer specifying how many times
the auto-convolution is done (i.e. the in
-fold
auto-convolution). The remaining arguments must agree with the
arguments of
kernel
and/or specden
(see above).
If supplied, the arguments of Kfun
should be like the
arguments of kernel
and specden
(see above).
A function in the R language, belonging to the class
"detpointprocfamilyfun"
. The function has formal arguments
...
and returns a determinantal point process family
(object of class "detpointprocfamily"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
## Example of how to define the Gauss family exGauss <- detpointprocfamilyfun( name="Gaussian", kernel=function(x, lambda, alpha, d){ lambda*exp(-(x/alpha)^2) }, specden=function(x, lambda, alpha, d){ lambda * (sqrt(pi)*alpha)^d * exp(-(x*alpha*pi)^2) }, convkernel=function(x, k, lambda, alpha, d){ logres <- k*log(lambda*pi*alpha^2) - log(pi*k*alpha^2) - x^2/(k*alpha^2) return(exp(logres)) }, Kfun = function(x, lambda, alpha, d){ pi*x^2 - pi*alpha^2/2*(1-exp(-2*x^2/alpha^2)) }, valid=function(lambda, alpha, d){ lambda>0 && alpha>0 && d>=1 && lambda <= (sqrt(pi)*alpha)^(-d) }, isotropic=TRUE, intensity="lambda", dim="d", range=function(alpha, bound = .99){ if(missing(alpha)) stop("The parameter alpha is missing.") if(!(is.numeric(bound)&&bound>0&&bound<1)) stop("Argument bound must be a numeric between 0 and 1.") return(alpha*sqrt(-log(sqrt(1-bound)))) }, parbounds=function(name, lambda, alpha, d){ switch(name, lambda = c(0, (sqrt(pi)*alpha)^(-d)), alpha = c(0, lambda^(-1/d)/sqrt(pi)), stop("Parameter name misspecified") ) }, startpar=function(model, X){ rslt <- NULL if("lambda" %in% model$freepar){ lambda <- intensity(X) rslt <- c(rslt, "lambda" = lambda) model <- update(model, lambda=lambda) } if("alpha" %in% model$freepar){ alpha <- .8*dppparbounds(model, "alpha")[2] rslt <- c(rslt, "alpha" = alpha) } return(rslt) } ) exGauss m <- exGauss(lambda=100, alpha=.05, d=2) m
## Example of how to define the Gauss family exGauss <- detpointprocfamilyfun( name="Gaussian", kernel=function(x, lambda, alpha, d){ lambda*exp(-(x/alpha)^2) }, specden=function(x, lambda, alpha, d){ lambda * (sqrt(pi)*alpha)^d * exp(-(x*alpha*pi)^2) }, convkernel=function(x, k, lambda, alpha, d){ logres <- k*log(lambda*pi*alpha^2) - log(pi*k*alpha^2) - x^2/(k*alpha^2) return(exp(logres)) }, Kfun = function(x, lambda, alpha, d){ pi*x^2 - pi*alpha^2/2*(1-exp(-2*x^2/alpha^2)) }, valid=function(lambda, alpha, d){ lambda>0 && alpha>0 && d>=1 && lambda <= (sqrt(pi)*alpha)^(-d) }, isotropic=TRUE, intensity="lambda", dim="d", range=function(alpha, bound = .99){ if(missing(alpha)) stop("The parameter alpha is missing.") if(!(is.numeric(bound)&&bound>0&&bound<1)) stop("Argument bound must be a numeric between 0 and 1.") return(alpha*sqrt(-log(sqrt(1-bound)))) }, parbounds=function(name, lambda, alpha, d){ switch(name, lambda = c(0, (sqrt(pi)*alpha)^(-d)), alpha = c(0, lambda^(-1/d)/sqrt(pi)), stop("Parameter name misspecified") ) }, startpar=function(model, X){ rslt <- NULL if("lambda" %in% model$freepar){ lambda <- intensity(X) rslt <- c(rslt, "lambda" = lambda) model <- update(model, lambda=lambda) } if("alpha" %in% model$freepar){ alpha <- .8*dppparbounds(model, "alpha")[2] rslt <- c(rslt, "alpha" = alpha) } return(rslt) } ) exGauss m <- exGauss(lambda=100, alpha=.05, d=2) m
Computes the deletion influence measure for each parameter in a fitted point process model.
## S3 method for class 'ppm' dfbetas(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
## S3 method for class 'ppm' dfbetas(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
model |
Fitted point process model (object of class |
... |
Ignored, except for the arguments |
drop |
Logical. Whether to include ( |
iScore , iHessian
|
Components of the score vector and Hessian matrix for the irregular parameters, if required. See Details. |
iArgs |
List of extra arguments for the functions |
Given a fitted spatial point process model
,
this function computes the influence measure for each parameter,
as described in Baddeley, Chang and Song (2013)
and Baddeley, Rubak and Turner (2019).
This is a method for the generic function dfbetas
.
The influence measure for each parameter is a
signed measure in two-dimensional space. It consists of a discrete
mass on each data point (i.e. each point in the point pattern to which
the
model
was originally fitted) and a continuous density at
all locations. The mass at a data point represents the change in the
fitted value of the parameter that would occur
if this data point were to be deleted.
The density at other non-data locations represents the
effect (on the fitted value of
)
of deleting these locations (and their associated covariate values)
from the input to the fitting procedure.
If the point process model trend has irregular parameters that were
fitted (using ippm
)
then the influence calculation requires the first and second
derivatives of the log trend with respect to the irregular parameters.
The argument iScore
should be a list,
with one entry for each irregular parameter, of R functions that compute the
partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular
parameter. The argument iHessian
should be a list,
with entries where
is the number of irregular
parameters, of R functions that compute the second order
partial derivatives of the
log trend with respect to each pair of irregular parameters.
An object of class "msr"
representing a signed or vector-valued
measure. This object can be printed and plotted.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
Baddeley, A., Rubak, E. and Turner, R. (2019) Leverage and influence diagnostics for Gibbs spatial point processes. Spatial Statistics 29, 15–48.
leverage.ppm
,
influence.ppm
,
ppmInfluence
.
See msr
for information on how to use a measure.
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(dfbetas(fit)) plot(Smooth(dfbetas(fit)))
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(dfbetas(fit)) plot(Smooth(dfbetas(fit)))
Computes the case deletion effect measure DFFIT
for a fitted model.
dffit(object, ...) ## S3 method for class 'ppm' dffit(object, ..., collapse = FALSE, dfb = NULL)
dffit(object, ...) ## S3 method for class 'ppm' dffit(object, ..., collapse = FALSE, dfb = NULL)
object |
A fitted model, such as a point process model
(object of class |
... |
Additional arguments passed to |
collapse |
Logical value specifying whether to collapse the vector-valued measure to a scalar-valued measure by adding all the components. |
dfb |
Optional. The result of |
The case deletion effect measure DFFIT
is a model diagnostic
traditionally used for regression models. In that context,
DFFIT[i,j]
is the negative change, in the value of the
j
th term in the linear predictor, that would occur if the i
th
data value was deleted. It is closely related to the
diagnostic DFBETA
.
For a spatial point process model, dffit
computes
the analogous spatial case deletion diagnostic, described in
Baddeley, Rubak and Turner (2019).
A measure (object of class "msr"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2019) Leverage and influence diagnostics for Gibbs spatial point processes. Spatial Statistics 29, 15–48.
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(dffit(fit)) plot(dffit(fit, collapse=TRUE))
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(dffit(fit)) plot(dffit(fit, collapse=TRUE))
Given a point process model fitted to a point pattern, produce diagnostic plots based on residuals.
diagnose.ppm(object, ..., type="raw", which="all", sigma=NULL, rbord=reach(object), cumulative=TRUE, plot.it=TRUE, rv = NULL, compute.sd=is.poisson(object), compute.cts=TRUE, envelope=FALSE, nsim=39, nrank=1, typename, check=TRUE, repair=TRUE, oldstyle=FALSE, splineargs=list(spar=0.5)) ## S3 method for class 'diagppm' plot(x, ..., which, plot.neg=c("image", "discrete", "contour", "imagecontour"), plot.smooth=c("imagecontour", "image", "contour", "persp"), plot.sd, spacing=0.1, outer=3, srange=NULL, monochrome=FALSE, main=NULL)
diagnose.ppm(object, ..., type="raw", which="all", sigma=NULL, rbord=reach(object), cumulative=TRUE, plot.it=TRUE, rv = NULL, compute.sd=is.poisson(object), compute.cts=TRUE, envelope=FALSE, nsim=39, nrank=1, typename, check=TRUE, repair=TRUE, oldstyle=FALSE, splineargs=list(spar=0.5)) ## S3 method for class 'diagppm' plot(x, ..., which, plot.neg=c("image", "discrete", "contour", "imagecontour"), plot.smooth=c("imagecontour", "image", "contour", "persp"), plot.sd, spacing=0.1, outer=3, srange=NULL, monochrome=FALSE, main=NULL)
object |
The fitted point process model (an object of class |
type |
String indicating the type of residuals or weights to be used.
Current options are |
which |
Character string or vector indicating the choice(s) of
plots to be generated. Options are
|
sigma |
Bandwidth for kernel smoother in |
rbord |
Width of border to avoid edge effects.
The diagnostic calculations
will be confined to those points of the data pattern which are
at least |
cumulative |
Logical flag indicating whether the lurking variable plots
for the |
plot.it |
Logical value indicating whether
plots should be shown. If |
plot.neg |
String indicating how the density part of the residual measure should be plotted. |
plot.smooth |
String indicating how the smoothed residual field should be plotted. |
compute.sd , plot.sd
|
Logical values indicating whether
error bounds should be computed and added to the |
envelope , nsim , nrank
|
Arguments passed to |
rv |
Usually absent. Advanced use only.
If this argument is present, the values of the residuals will not be
calculated from the fitted model |
spacing |
The spacing between plot panels (when a four-panel plot is generated) expressed as a fraction of the width of the window of the point pattern. |
outer |
The distance from the outermost line of text to the nearest plot panel, expressed as a multiple of the spacing between plot panels. |
srange |
Vector of length 2 that will be taken as giving the range of values of the smoothed residual field, when generating an image plot of this field. This is useful if you want to generate diagnostic plots for two different fitted models using the same colour map. |
monochrome |
Flag indicating whether images should be displayed in greyscale (suitable for publication) or in colour (suitable for the screen). The default is to display in colour. |
check |
Logical value indicating whether to check the internal format
of |
repair |
Logical value indicating whether to repair the internal format
of |
oldstyle |
Logical flag indicating whether error bounds should be plotted
using the approximation given in the original paper
( |
splineargs |
Argument passed to |
x |
The value returned from a previous call to
|
typename |
String to be used as the name of the residuals. |
main |
Main title for the plot. |
... |
Extra arguments, controlling either the resolution of the smoothed image
(passed from |
compute.cts |
Advanced use only. |
The function diagnose.ppm
generates several diagnostic plots for a
fitted point process model.
The plots display the residuals from the fitted model
(Baddeley et al, 2005)
or alternatively the ‘exponential energy marks’ (Stoyan and Grabarnik, 1991).
These plots can be used to
assess goodness-of-fit, to identify outliers in the data,
and to reveal departures from the fitted model.
See also the companion function qqplot.ppm
.
The argument object
must be a fitted point process model
(object of class "ppm"
) typically produced by the maximum
pseudolikelihood fitting algorithm ppm
).
The argument type
selects the type of residual or weight
that will be computed. Current options are:
"eem"
:exponential energy marks (Stoyan and Grabarnik, 1991)
computed by eem
.
These are positive weights attached to the data points
(i.e. the points of the point pattern dataset
to which the model was fitted).
If the fitted model is correct, then the sum of these weights
for all data points in a spatial region
has expected value equal to the
area of
. See
eem
for further explanation.
"raw"
, "inverse"
or "pearson"
:point process residuals (Baddeley et al, 2005)
computed by the function residuals.ppm
.
These are residuals attached both to the data points and to some
other points in the window of observation (namely, to the dummy
points of the quadrature scheme used to fit the model).
If the fitted model is correct, then the sum of the
residuals in a spatial region has mean zero.
The options are
"raw"
: the raw residuals;
"inverse"
: the ‘inverse-lambda’ residuals,
a counterpart of the exponential energy weights;
"pearson"
: the Pearson residuals.
See residuals.ppm
for further explanation.
The argument which
selects the type of plot that is
produced. Options are:
"marks"
:plot the residual measure.
For the exponential energy weights (type="eem"
)
this displays circles centred at the points of the data pattern,
with radii proportional to the exponential energy weights.
For the residuals (type="raw"
, type="inverse"
or type="pearson"
) this again displays circles centred at
the points of the data pattern with radii proportional to the
(positive) residuals, while the plotting of the negative residuals
depends on the argument plot.neg
. If
plot.neg="image"
then the negative part of the residual
measure, which is a density, is plotted as a colour image.
If plot.neg="discrete"
then the discretised negative
residuals (obtained by approximately integrating the negative
density using the quadrature scheme of the fitted model)
are plotted as squares centred at the dummy points
with side lengths proportional to the (negative) residuals.
[To control the size of the circles and squares, use the argument
maxsize
.]
"smooth"
:plot a kernel-smoothed version of the residual measure.
Each data or dummy point is taken to have a ‘mass’ equal to its
residual or exponential energy weight.
(Note that residuals can be negative).
This point mass is then replaced by
a bivariate isotropic Gaussian density
with standard deviation sigma
.
The value of the smoothed residual field at
any point in the window is the sum of these weighted densities.
If the fitted model is correct, this smoothed field
should be flat, and its height should be close to 0
(for the residuals) or 1 (for the exponential energy weights).
The field is plotted either as an image, contour plot or
perspective view of a surface, according to the
argument plot.smooth
.
The range of values of the smoothed field is printed
if the option which="sum"
is also selected.
"x"
:produce a ‘lurking variable’ plot for the coordinate.
This is a plot of
against
(solid lines)
and of
against
(dashed lines),
where
is defined below, and
denotes the
expectation of
assuming the fitted model is true.
if cumulative=TRUE
then is the cumulative sum of
the weights or residuals for all points
which have
coordinate less than or equal to
.
For the residuals
,
and for the exponential energy weights
area of the subset of the window to the left of
the line
.
if cumulative=FALSE
then
is the marginal integral of
the smoothed residual field (see the case
which="smooth"
described above) on the axis.
This is approximately the derivative
of the plot for
cumulative=TRUE
.
The value of is computed by summing the values of
the smoothed residual field over all pixels with
the given
coordinate.
For the residuals
,
and for the exponential energy weights
length of the intersection between the
observation window and the line
.
If plot.sd = TRUE
, then superimposed on the lurking variable
plot are the pointwise
two-standard-deviation error limits for calculated for the
inhomogeneous Poisson process. The default is
plot.sd = TRUE
for Poisson models and plot.sd = FALSE
for non-Poisson
models.
"y"
:produce a similar lurking variable plot for the coordinate.
"sum"
:print the sum of the weights or residuals for all points
in the window (clipped by a margin rbord
if required)
and the area of the same window. If the fitted model is correct
the sum of the exponential energy weights should equal the area of
the window, while the sum of the residuals should equal zero.
Also print the range of values of the smoothed field
displayed in the "smooth"
case.
"all"
:All four of the diagnostic plots listed above are plotted together
in a two-by-two display. Top left panel is "marks"
plot.
Bottom right panel is "smooth"
plot. Bottom left panel is
"x"
plot. Top right panel is "y"
plot, rotated 90 degrees.
The argument rbord
ensures there are no edge
effects in the computation of the residuals. The diagnostic calculations
will be confined to those points of the data pattern which are
at least rbord
units away from the edge of the window.
The value of rbord
should be greater than or equal to
the range of interaction permitted in the model.
By default, the two-standard-deviation limits are calculated
from the exact formula for the asymptotic variance
of the residuals under the asymptotic normal approximation,
equation (37) of Baddeley et al (2006).
However, for compatibility with the original paper
of Baddeley et al (2005), if oldstyle=TRUE
,
the two-standard-deviation limits are calculated
using the innovation variance, an over-estimate of the true
variance of the residuals. (However, see the section about
Replicated Data).
The argument rv
would normally be used only by experts.
It enables the user to substitute arbitrary values for the
residuals or marks, overriding the usual calculations.
If rv
is present, then instead of calculating the residuals from
the fitted model, the algorithm takes the residuals from the object
rv
, and plots them in the manner appropriate to the type of residual
or mark selected by type
. If type ="eem"
then
rv
should be similar to the return value of eem
,
namely, a numeric vector of length equal to
the number of points in the original data point pattern.
Otherwise, rv
should be similar to the return value of
residuals.ppm
, that is, it should be an object of
class "msr"
(see msr
) representing a signed
measure.
The return value of diagnose.ppm
is an object of class "diagppm"
.
The plot
method for this class is documented here.
There is also a print
method. See the Examples.
In plot.diagppm
,
if a four-panel diagnostic plot is produced (the default), then
the extra arguments xlab
, ylab
, rlab
determine the
text labels for the and
coordinates
and the residuals, respectively.
The undocumented arguments
col.neg
and col.smooth
control the colour maps used in the top left and bottom right
panels respectively.
See also the companion functions qqplot.ppm
, which produces a
Q-Q plot of the residuals, and lurking
, which produces
lurking variable plots for any spatial covariate.
An object of class "diagppm"
which contains
the coordinates needed to reproduce the selected plots.
This object can be plotted using plot.diagppm
and printed using print.diagppm
.
Note that if object
is a model that was obtained by
first fitting a model to replicated point pattern data using
mppm
and then using subfits
to extract
a model for one of the individual point patterns, then the
variance calculations are only implemented for the
innovation variance (oldstyle=TRUE
) and this is the default
in such cases.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
residuals.ppm
,
eem
,
ppm.object
,
qqplot.ppm
,
lurking
,
ppm
fit <- ppm(cells ~x, Strauss(r=0.15)) diagnose.ppm(fit) diagnose.ppm(fit, type="pearson") diagnose.ppm(fit, which="marks") diagnose.ppm(fit, type="raw", plot.neg="discrete") diagnose.ppm(fit, type="pearson", which="smooth") # save the diagnostics and plot them later u <- diagnose.ppm(fit, rbord=0.15, plot.it=FALSE) if(interactive()) { plot(u) plot(u, which="marks") }
fit <- ppm(cells ~x, Strauss(r=0.15)) diagnose.ppm(fit) diagnose.ppm(fit, type="pearson") diagnose.ppm(fit, which="marks") diagnose.ppm(fit, type="raw", plot.neg="discrete") diagnose.ppm(fit, type="pearson", which="smooth") # save the diagnostics and plot them later u <- diagnose.ppm(fit, rbord=0.15, plot.it=FALSE) if(interactive()) { plot(u) plot(u, which="marks") }
Creates an instance of the Diggle-Gates-Stibbard point process model which can then be fitted to point pattern data.
DiggleGatesStibbard(rho)
DiggleGatesStibbard(rho)
rho |
Interaction range |
Diggle, Gates and Stibbard (1987) proposed a pairwise interaction
point process in which each pair of points separated by
a distance contributes a factor
to the
probability density, where
for , and
is equal to 1
for
.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Diggle-Gates-Stibbard
pairwise interaction is
yielded by the function DiggleGatesStibbard()
.
See the examples below.
Note that this model does not have any regular parameters
(as explained in the section on Interaction Parameters
in the help file for ppm
).
The parameter is not estimated by
ppm
.
An object of class "interact"
describing the interpoint interaction
structure of the Diggle-Gates-Stibbard
process with interaction range rho
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.
ppm
,
pairwise.family
,
DiggleGratton
,
rDGS
,
ppm.object
DiggleGatesStibbard(0.02) # prints a sensible description of itself ppm(cells ~1, DiggleGatesStibbard(0.05)) # fit the stationary D-G-S process to `cells' ppm(cells ~ polynom(x,y,3), DiggleGatesStibbard(0.05)) # fit a nonstationary D-G-S process # with log-cubic polynomial trend
DiggleGatesStibbard(0.02) # prints a sensible description of itself ppm(cells ~1, DiggleGatesStibbard(0.05)) # fit the stationary D-G-S process to `cells' ppm(cells ~ polynom(x,y,3), DiggleGatesStibbard(0.05)) # fit a nonstationary D-G-S process # with log-cubic polynomial trend
Creates an instance of the Diggle-Gratton pairwise interaction point process model, which can then be fitted to point pattern data.
DiggleGratton(delta=NA, rho)
DiggleGratton(delta=NA, rho)
delta |
lower threshold |
rho |
upper threshold |
Diggle and Gratton (1984, pages 208-210)
introduced the pairwise interaction point
process with pair potential of the form
with for
and
for
.
Here
,
and
are parameters.
Note that we use the symbol
where Diggle and Gratton (1984) and Diggle, Gates and Stibbard (1987)
use
, since in spatstat we reserve the symbol
for an intensity parameter.
The parameters must all be nonnegative,
and must satisfy .
The potential is inhibitory, i.e.\ this model is only appropriate for
regular point patterns. The strength of inhibition increases with
. For
the model is
a hard core process with hard core radius
.
For
the model is a hard core
process with hard core radius
.
The irregular parameters
must be given in the call to
DiggleGratton
, while the
regular parameter will be estimated.
If the lower threshold delta
is missing or NA
,
it will be estimated from the data when ppm
is called.
The estimated value of delta
is the minimum nearest neighbour distance
multiplied by , where
is the
number of data points.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Diggle, P.J., Gates, D.J. and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction point processes. Biometrika 74, 763 – 770.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
ppm(cells ~1, DiggleGratton(0.05, 0.1))
ppm(cells ~1, DiggleGratton(0.05, 0.1))
Extracts the dimension of a determinantal point process model.
## S3 method for class 'detpointprocfamily' dim(x)
## S3 method for class 'detpointprocfamily' dim(x)
x |
object of class |
A numeric (or NULL if the dimension of the model is unspecified).
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Given a spatial object such as a point pattern, in any number of dimensions, this function extracts the spatial domain in which the object is defined.
## S3 method for class 'ppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'kppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'dppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'slrm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'msr' domain(X, ...)
## S3 method for class 'ppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'kppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'dppm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'slrm' domain(X, ..., from=c("points", "covariates")) ## S3 method for class 'msr' domain(X, ...)
X |
A spatial object such as a point pattern (in any number of dimensions), line segment pattern or pixel image. |
... |
Extra arguments. They are ignored by all the methods listed here. |
from |
Character string. See Details. |
The function domain
is generic.
For a spatial object X
in any number of dimensions,
domain(X)
extracts the spatial domain in which X
is
defined.
For a two-dimensional object X
, typically domain(X)
is the same as Window(X)
.
Exceptions occur for methods related to linear networks.
The argument from
applies when X
is a fitted
point process model
(object of class "ppm"
, "kppm"
or "dppm"
).
If from="data"
(the default),
domain
extracts the window of the original point
pattern data to which the model was fitted.
If from="covariates"
then domain
returns the
window in which the spatial covariates of the model were provided.
A spatial object representing the domain of X
.
Typically a window (object of class "owin"
),
a three-dimensional box ("box3"
), a multidimensional
box ("boxx"
) or a linear network ("linnet"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
domain
,
domain.quadratcount
,
domain.quadrattest
,
domain.rmhmodel
,
domain.lpp
.
Window
,
Frame
.
domain(ppm(redwood ~ 1))
domain(ppm(redwood ~ 1))
Returns an approximation to the kernel of a determinantal
point process, as a function of one argument .
dppapproxkernel(model, trunc = 0.99, W = NULL)
dppapproxkernel(model, trunc = 0.99, W = NULL)
model |
Object of class |
trunc |
Numeric specifying how the model truncation is
performed. See Details section of |
W |
Optional window – undocumented at the moment. |
A function
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Returns an approximation to the
theoretical pair correlation function of a
determinantal point process model, as a
function of one argument .
dppapproxpcf(model, trunc = 0.99, W = NULL)
dppapproxpcf(model, trunc = 0.99, W = NULL)
model |
Object of class |
trunc |
Numeric value specifying how the model truncation is
performed.
See Details section of |
W |
Optional window – undocumented at the moment. |
This function is usually NOT needed for anything. It only exists for investigative purposes.
A function in the R language, with one numeric argument x
,
that returns the value of the approximate pair correlation
at distances x
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
f <- dppapproxpcf(dppMatern(lambda = 100, alpha=.028, nu=1, d=2)) plot(f, xlim = c(0,0.1))
f <- dppapproxpcf(dppMatern(lambda = 100, alpha=.028, nu=1, d=2)) plot(f, xlim = c(0,0.1))
Function generating an instance of the Bessel-type determinantal point process model.
dppBessel(...)
dppBessel(...)
... |
arguments of the form |
The possible parameters are:
the intensity lambda
as a positive numeric
the scale parameter alpha
as a positive numeric
the shape parameter sigma
as a non-negative numeric
the dimension d
as a positive integer
An object of class "detpointprocfamily"
.
Frederic Lavancier and Christophe Biscio. Modified by Ege Rubak [email protected] , Adrian Baddeley [email protected]
and Rolf Turner [email protected]
dppCauchy
,
dppGauss
,
dppMatern
,
dppPowerExp
m <- dppBessel(lambda=100, alpha=.05, sigma=0, d=2)
m <- dppBessel(lambda=100, alpha=.05, sigma=0, d=2)
Function generating an instance of the (generalized) Cauchy determinantal point process model.
dppCauchy(...)
dppCauchy(...)
... |
arguments of the form |
The (generalized) Cauchy DPP is defined in (Lavancier, Moller and Rubak, 2015) The possible parameters are:
the intensity lambda
as a positive numeric
the scale parameter alpha
as a positive numeric
the shape parameter nu
as a positive numeric
(artificially required to be less than 20 in the code for numerical
stability)
the dimension d
as a positive integer
An object of class "detpointprocfamily"
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853–977.
dppBessel
,
dppGauss
,
dppMatern
,
dppPowerExp
m <- dppCauchy(lambda=100, alpha=.05, nu=1, d=2)
m <- dppCauchy(lambda=100, alpha=.05, nu=1, d=2)
This function is mainly for internal package use and is usually not called by the user.
dppeigen(model, trunc, Wscale, stationary = FALSE)
dppeigen(model, trunc, Wscale, stationary = FALSE)
model |
object of class |
trunc |
numeric giving the truncation |
Wscale |
numeric giving the scale of the window relative to a unit box |
stationary |
logical indicating whether the stationarity of the model should be used (only works in dimension 2). |
A list
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Function generating an instance of the Gaussian determinantal point process model.
dppGauss(...)
dppGauss(...)
... |
arguments of the form |
The Gaussian DPP is defined in (Lavancier, Moller and Rubak, 2015) The possible parameters are:
the intensity lambda
as a positive numeric
the scale parameter alpha
as a positive numeric
the dimension d
as a positive integer
An object of class "detpointprocfamily"
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853–977.
dppBessel
,
dppCauchy
,
dppMatern
,
dppPowerExp
m <- dppGauss(lambda=100, alpha=.05, d=2)
m <- dppGauss(lambda=100, alpha=.05, d=2)
Returns the kernel of a determinantal point process model as a
function of one argument x
.
dppkernel(model, ...)
dppkernel(model, ...)
model |
Model of class |
... |
Arguments passed to |
A function
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
kernelMatern <- dppkernel(dppMatern(lambda = 100, alpha=.01, nu=1, d=2)) plot(kernelMatern, xlim = c(0,0.1))
kernelMatern <- dppkernel(dppMatern(lambda = 100, alpha=.01, nu=1, d=2)) plot(kernelMatern, xlim = c(0,0.1))
Fit a determinantal point process model to a point pattern.
dppm(formula, family, data=NULL, ..., startpar = NULL, method = c("mincon", "clik2", "palm", "adapcl"), weightfun = NULL, control = list(), algorithm, statistic = "K", statargs = list(), rmax = NULL, epsilon = 0.01, covfunargs = NULL, use.gam = FALSE, nd = NULL, eps = NULL)
dppm(formula, family, data=NULL, ..., startpar = NULL, method = c("mincon", "clik2", "palm", "adapcl"), weightfun = NULL, control = list(), algorithm, statistic = "K", statargs = list(), rmax = NULL, epsilon = 0.01, covfunargs = NULL, use.gam = FALSE, nd = NULL, eps = NULL)
formula |
A |
family |
Information specifying the family of point processes
to be used in the model.
Typically one of the family functions
|
data |
The values of spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows, tessellations or numeric constants. |
... |
Additional arguments. See Details. |
startpar |
Named vector of starting parameter values for the optimization. |
method |
The fitting method. Either
|
weightfun |
Optional weighting function |
control |
List of control parameters passed to the optimization function
|
algorithm |
Character string determining the mathematical algorithm
to be used to solve the fitting problem.
If |
statistic |
Name of the summary statistic to be used
for minimum contrast estimation: either |
statargs |
Optional list of arguments to be used when calculating
the |
rmax |
Maximum value of interpoint distance to use in the composite likelihood. |
epsilon |
Tuning parameter for the adaptive composite likelihood. |
covfunargs , use.gam , nd , eps
|
Arguments passed to |
This function fits a determinantal point process model to a point pattern dataset as described in Lavancier et al. (2015).
The model to be fitted is specified by the arguments
formula
and family
.
The argument formula
should normally be a formula
in the
R language. The left hand side of the formula
specifies the point pattern dataset to which the model should be fitted.
This should be a single argument which may be a point pattern
(object of class "ppp"
) or a quadrature scheme
(object of class "quad"
). The right hand side of the formula is called
the trend
and specifies the form of the
logarithm of the intensity of the process.
Alternatively the argument formula
may be a point pattern or quadrature
scheme, and the trend formula is taken to be ~1
.
The argument family
specifies the family of point processes
to be used in the model.
It is typically one of the family functions
dppGauss
, dppMatern
,
dppCauchy
, dppBessel
or dppPowerExp
.
Alternatively it may be a character string giving the name
of a family function, or the result of calling one of the
family functions. A family function belongs to class
"detpointprocfamilyfun"
. The result of calling a family
function is a point process family, which belongs to class
"detpointprocfamily"
.
The algorithm first estimates the intensity function
of the point process using ppm
.
If the trend formula is ~1
(the default if a point pattern or quadrature
scheme is given rather than a "formula"
)
then the model is homogeneous. The algorithm begins by
estimating the intensity as the number of points divided by
the area of the window.
Otherwise, the model is inhomogeneous.
The algorithm begins by fitting a Poisson process with log intensity
of the form specified by the formula trend
.
(See ppm
for further explanation).
The interaction parameters of the model are then fitted either by minimum contrast estimation, or by a composite likelihood method (maximum composite likelihood, maximum Palm likelihood, or by solving the adaptive composite likelihood estimating equation).
If method = "mincon"
(the default) interaction parameters of
the model will be fitted
by minimum contrast estimation, that is, by matching the theoretical
-function of the model to the empirical
-function
of the data, as explained in
mincontrast
.
For a homogeneous model ( trend = ~1
)
the empirical -function of the data is computed
using
Kest
,
and the interaction parameters of the model are estimated by
the method of minimum contrast.
For an inhomogeneous model,
the inhomogeneous function is estimated
by
Kinhom
using the fitted intensity.
Then the interaction parameters of the model
are estimated by the method of minimum contrast using the
inhomogeneous function. This two-step estimation
procedure is heavily inspired by Waagepetersen (2007).
If statistic="pcf"
then instead of using the
-function, the algorithm will use
the pair correlation function
pcf
for homogeneous
models and the inhomogeneous pair correlation function
pcfinhom
for inhomogeneous models.
In this case, the smoothing parameters of the pair correlation
can be controlled using the argument statargs
,
as shown in the Examples.
Additional arguments ...
will be passed to
clusterfit
to control the minimum contrast fitting
algorithm.
If method = "clik2"
the interaction parameters of the
model will be fitted by maximising the second-order composite likelihood
(Guan, 2006). The log composite likelihood is
where the sums are taken over all pairs of data points
separated by a distance
less than
rmax
,
and the double integral is taken over all pairs of locations
in the spatial window of the data.
Here
is the
pair correlation function of the model with
interaction parameters
.
The function in the composite likelihood
is a weighting function and may be chosen arbitrarily.
It is specified by the argument
weightfun
.
If this is missing or NULL
then the default is
a threshold weight function,
, where
is
rmax/2
.
If method = "palm"
the interaction parameters of the
model will be fitted by maximising the Palm loglikelihood
(Tanaka et al, 2008)
with the same notation as above. Here
is the Palm intensity of
the model at location
given there is a point at
.
If method = "cladap"
the clustering parameters of the
model will be fitted by solving the adaptive second order
composite likelihood estimating equation (Lavancier et al, 2021).
The estimating function is
where the sum is taken over all distinct pairs of points.
Here is the
pair correlation function with
parameters
.
The partial derivative with respect to
is
, and
denotes
the fitted intensity function of the model.
The tuning parameter is
independent of the data. It can be specified by the
argument
epsilon
and has default value .
The function in the estimating function
is a weighting function of bounded support
.
It is specified by the argument
weightfun
.
If this is missing or NULL
then the default is
.
The estimating equation is solved using the
nonlinear equation solver
nleqslv
from the package nleqslv. The package nleqslv
must be installed in order to use this option.
It is also possible to fix any parameters desired before the
optimisation by specifying them as name=value
in the call to the family function. See Examples.
An object of class "dppm"
representing the fitted model.
There are methods for printing, plotting, predicting and simulating
objects of this class.
The following details allow greater control over the fitting procedure.
For the first three fitting methods
(method="mincon", "clik2"
and "palm"
),
the optimisation is performed by the generic
optimisation algorithm optim
.
The behaviour of this algorithm can be modified using the
arguments control
and algorithm
.
Useful control arguments include
trace
, maxit
and abstol
(documented in the help for optim
).
For method="adapcl"
, the estimating equation is solved
using the nonlinear equation solver nleqslv
from the package nleqslv.
Arguments available for controlling the solver are
documented in the help for
nleqslv
; they include control
,
globStrat
, startparm
for the initial estimates and
algorithm
for the method.
The package nleqslv must be installed in order to use this option.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]. Adaptive composite likelihood method contributed by Chiara Fend and modified by Adrian Baddeley.
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502–1512.
Lavancier, F., Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference. Journal of the Royal Statistical Society, Series B 77, 853–977.
Lavancier, F., Poinas, A., and Waagepetersen, R. (2021) Adaptive estimating function inference for nonstationary determinantal point processes. Scandinavian Journal of Statistics, 48 (1), 87–107.
Tanaka, U., Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43–57.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
methods for dppm
objects:
plot.dppm
,
fitted.dppm
,
predict.dppm
,
simulate.dppm
,
methods.dppm
,
as.ppm.dppm
,
Kmodel.dppm
,
pcfmodel.dppm
.
Minimum contrast fitting algorithm:
higher level interface clusterfit
;
low-level algorithm mincontrast
.
Deterimantal point process models:
dppGauss
,
dppMatern
,
dppCauchy
,
dppBessel
,
dppPowerExp
,
Summary statistics:
Kest
,
Kinhom
,
pcf
,
pcfinhom
.
See also ppm
jpines <- residualspaper$Fig1 dppm(jpines ~ 1, dppGauss) dppm(jpines ~ 1, dppGauss, method="c") dppm(jpines ~ 1, dppGauss, method="p") dppm(jpines ~ 1, dppGauss, method="a") if(interactive()) { # Fixing the intensity at lambda=2 rather than the Poisson MLE 2.04: dppm(jpines ~ 1, dppGauss(lambda=2)) # The following is quite slow (using K-function) dppm(jpines ~ x, dppMatern) } # much faster using pair correlation function dppm(jpines ~ x, dppMatern, statistic="pcf", statargs=list(stoyan=0.2)) # Fixing the Matern shape parameter at nu=2 rather than estimating it: dppm(jpines ~ x, dppMatern(nu=2))
jpines <- residualspaper$Fig1 dppm(jpines ~ 1, dppGauss) dppm(jpines ~ 1, dppGauss, method="c") dppm(jpines ~ 1, dppGauss, method="p") dppm(jpines ~ 1, dppGauss, method="a") if(interactive()) { # Fixing the intensity at lambda=2 rather than the Poisson MLE 2.04: dppm(jpines ~ 1, dppGauss(lambda=2)) # The following is quite slow (using K-function) dppm(jpines ~ x, dppMatern) } # much faster using pair correlation function dppm(jpines ~ x, dppMatern, statistic="pcf", statargs=list(stoyan=0.2)) # Fixing the Matern shape parameter at nu=2 rather than estimating it: dppm(jpines ~ x, dppMatern(nu=2))
Function generating an instance of the Whittle-Matern determinantal point process model
dppMatern(...)
dppMatern(...)
... |
arguments of the form |
The Whittle-Matern DPP is defined in (Lavancier, Moller and Rubak, 2015) The possible parameters are:
the intensity lambda
as a positive numeric
the scale parameter alpha
as a positive numeric
the shape parameter nu
as a positive numeric
(artificially required to be less than 20 in the code for numerical
stability)
the dimension d
as a positive integer
An object of class "detpointprocfamily"
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853–977.
dppBessel
,
dppCauchy
,
dppGauss
,
dppPowerExp
m <- dppMatern(lambda=100, alpha=.02, nu=1, d=2)
m <- dppMatern(lambda=100, alpha=.02, nu=1, d=2)
Returns the lower and upper bound for a specific parameter of a determinantal point process model when all other parameters are fixed.
dppparbounds(model, name, ...)
dppparbounds(model, name, ...)
model |
Model of class |
name |
name of the parameter for which the bound should be computed. |
... |
Additional arguments passed to the |
A data.frame
containing lower and upper bounds.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) dppparbounds(model, "lambda")
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) dppparbounds(model, "lambda")
Function generating an instance of the Power Exponential Spectral determinantal point process model.
dppPowerExp(...)
dppPowerExp(...)
... |
arguments of the form |
The Power Exponential Spectral DPP is defined in (Lavancier, Moller and Rubak, 2015) The possible parameters are:
the intensity lambda
as a positive numeric
the scale parameter alpha
as a positive numeric
the shape parameter nu
as a positive numeric
(artificially required to be less than 20 in the code for numerical
stability)
the dimension d
as a positive integer
An object of class "detpointprocfamily"
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853–977.
dppBessel
,
dppCauchy
,
dppGauss
,
dppMatern
m <- dppPowerExp(lambda=100, alpha=.01, nu=1, d=2)
m <- dppPowerExp(lambda=100, alpha=.01, nu=1, d=2)
Returns the spectral density of a determinantal point process
model as a function of one argument x
.
dppspecden(model)
dppspecden(model)
model |
Model of class |
A function
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
model <- dppMatern(lambda = 100, alpha=.01, nu=1, d=2) dppspecden(model)
model <- dppMatern(lambda = 100, alpha=.01, nu=1, d=2) dppspecden(model)
Computes the range of the spectral density of a determinantal point process model.
dppspecdenrange(model)
dppspecdenrange(model)
model |
Model of class |
Numeric value (possibly Inf
).
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
m <- dppBessel(lambda=100, alpha=0.05, sigma=1, d=2) dppspecdenrange(m)
m <- dppBessel(lambda=100, alpha=0.05, sigma=1, d=2) dppspecdenrange(m)
Converts data of any kind to numeric values. A factor is expanded to a set of dummy variables.
dummify(x)
dummify(x)
x |
Vector, factor, matrix or data frame to be converted. |
This function converts data (such as a factor) to numeric values in order that the user may calculate, for example, the mean, variance, covariance and correlation of the data.
If x
is a numeric vector or integer vector, it is returned
unchanged.
If x
is a logical vector, it is converted to a 0-1 matrix with
2 columns. The first column contains a 1 if the logical value is
FALSE
, and the second column contains a 1 if the logical
value is TRUE
.
If x
is a complex vector, it is converted to a matrix with 2
columns, containing the real and imaginary parts.
If x
is a factor, the result is a matrix of 0-1 dummy
variables. The matrix has one column for each possible level of the
factor. The (i,j)
entry is
equal to 1 when the i
th factor value equals the
j
th level, and is equal to 0 otherwise.
If x
is a matrix or data frame, the appropriate conversion is
applied to each column of x
.
Note that, unlike model.matrix
, this command converts a
factor into a full set of dummy variables (one column for each level of
the factor).
A numeric matrix.
Adrian Baddeley [email protected]
chara <- sample(letters[1:3], 8, replace=TRUE) logi <- (runif(8) < 0.3) comp <- round(4*runif(8) + 3*runif(8) * 1i, 1) nume <- 8:1 + 0.1 df <- data.frame(nume, chara, logi, comp) df dummify(df)
chara <- sample(letters[1:3], 8, replace=TRUE) logi <- (runif(8) < 0.3) comp <- round(4*runif(8) + 3*runif(8) * 1i, 1) nume <- 8:1 + 0.1 df <- data.frame(nume, chara, logi, comp) df dummify(df)
Given a fitted point process model, this function extracts the ‘dummy points’ of the quadrature scheme used to fit the model.
dummy.ppm(object, drop=FALSE)
dummy.ppm(object, drop=FALSE)
object |
fitted point process model (an object of class |
drop |
Logical value determining whether to delete dummy points that were not used to fit the model. |
An object of class "ppm"
represents a point process model
that has been fitted to data. It is typically produced by
the model-fitting algorithm ppm
.
The maximum pseudolikelihood algorithm in ppm
approximates the pseudolikelihood
integral by a sum over a finite set of quadrature points,
which is constructed by augmenting
the original data point pattern by a set of “dummy” points.
The fitted model object returned by ppm
contains complete information about this quadrature scheme.
See ppm
or ppm.object
for further
information.
This function dummy.ppm
extracts the dummy points of the quadrature scheme.
A typical use of this function would be to count the number of dummy
points, to gauge the accuracy of the approximation to the
exact pseudolikelihood.
It may happen that some dummy points are not actually used in
fitting the model (typically because the value of a covariate is NA
at these points). The argument drop
specifies whether these
unused dummy points shall be deleted (drop=TRUE
) or
retained (drop=FALSE
) in the return value.
See ppm.object
for a list of all operations that can be
performed on objects of class "ppm"
.
A point pattern (object of class "ppp"
).
Adrian Baddeley [email protected] and Rolf Turner [email protected]
fit <- ppm(cells, ~1, Strauss(r=0.1)) X <- dummy.ppm(fit) npoints(X) # this is the number of dummy points in the quadrature scheme
fit <- ppm(cells, ~1, Strauss(r=0.1)) X <- dummy.ppm(fit) npoints(X) # this is the number of dummy points in the quadrature scheme
Given a point process model fitted to a point pattern, compute the Stoyan-Grabarnik diagnostic “exponential energy marks” for the data points.
eem(fit, ...) ## S3 method for class 'ppm' eem(fit, check=TRUE, ...) ## S3 method for class 'slrm' eem(fit, check=TRUE, ...)
eem(fit, ...) ## S3 method for class 'ppm' eem(fit, check=TRUE, ...) ## S3 method for class 'slrm' eem(fit, check=TRUE, ...)
fit |
The fitted point process model. An object of class |
check |
Logical value indicating whether to check the internal format
of |
... |
Ignored. |
Stoyan and Grabarnik (1991) proposed a diagnostic
tool for point process models fitted to spatial point pattern data.
Each point of the data pattern
is given a ‘mark’ or ‘weight’
where
is the conditional intensity of the fitted model.
If the fitted model is correct, then the sum of these marks
for all points in a region
has expected value equal to the
area of
.
The argument fit
must be a fitted point process model
(object of class "ppm"
or "slrm"
).
Such objects are produced by the fitting algorithms ppm
)
and slrm
.
This fitted model object contains complete
information about the original data pattern and the model that was
fitted to it.
The value returned by eem
is the vector
of weights associated with the points
of the original data pattern. The original data pattern
(in corresponding order) can be
extracted from
fit
using response
.
The function diagnose.ppm
produces a set of sensible diagnostic plots based on these weights.
A vector containing the values of the exponential energy mark for each point in the pattern.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
diagnose.ppm
,
ppm.object
,
data.ppm
,
residuals.ppm
,
ppm
fit <- ppm(cells ~x, Strauss(r=0.15)) ee <- eem(fit) sum(ee)/area(Window(cells)) # should be about 1 if model is correct Y <- setmarks(cells, ee) plot(Y, main="Cells data\n Exponential energy marks")
fit <- ppm(cells ~x, Strauss(r=0.15)) ee <- eem(fit) sum(ee)/area(Window(cells)) # should be about 1 if model is correct Y <- setmarks(cells, ee) plot(Y, main="Cells data\n Exponential energy marks")
Compute the trend or intensity of a fitted point process model as a function of one of its covariates.
effectfun(model, covname, ..., se.fit=FALSE, nvalues=256)
effectfun(model, covname, ..., se.fit=FALSE, nvalues=256)
model |
A fitted point process model (object of class
|
covname |
The name of the covariate. A character string. (Needed only if the model has more than one covariate.) |
... |
The fixed values of other covariates (in the form
|
se.fit |
Logical. If |
nvalues |
Integer. The number of values of the covariate (if it is numeric) for which the effect function should be evaluated. We recommend at least 256. |
The object model
should be an object of class
"ppm"
, "kppm"
, "lppm"
, "dppm"
, "rppm"
or "profilepl"
representing a point process model fitted to point pattern data.
The model's trend formula should involve a spatial covariate
named covname
. This could be "x"
or "y"
representing one of the Cartesian coordinates.
More commonly the covariate
is another, external variable that was supplied when fitting the model.
The command effectfun
computes the fitted trend
of the point process model
as a function of the covariate
named covname
.
The return value can be plotted immediately, giving a
plot of the fitted trend against the value of the covariate.
If the model also involves covariates other than covname
,
then these covariates will be held fixed. Values for
these other covariates must be provided as arguments
to effectfun
in the form name=value
.
If se.fit=TRUE
, the algorithm also calculates
the asymptotic standard error of the fitted trend,
and a (pointwise) asymptotic 95% confidence interval for the
true trend.
This command is just a wrapper for the prediction method
predict.ppm
. For more complicated computations
about the fitted intensity, use predict.ppm
.
A data frame containing a column of values of the covariate and a column
of values of the fitted trend.
If se.fit=TRUE
, there are 3 additional columns containing the
standard error and the upper and lower limits of a confidence interval.
If the covariate named covname
is numeric (rather than a factor
or logical variable), the return value is
also of class "fv"
so that it can be plotted immediately.
For a Poisson point process model, the trend is the same as the
intensity of the point process. For a more general Gibbs model, the trend
is the first order potential in the model (the first order term in the
Gibbs representation). In Poisson or Gibbs models fitted by
ppm
, the trend is the only part of the model that
depends on the covariates.
The function dppm
which fits
a determinantal point process model allows the user to specify the
intensity lambda
. In such cases the effect function is
undefined, and effectfun
stops with an error message.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
X <- copper$SouthPoints D <- distfun(copper$SouthLines) fit <- ppm(X ~ polynom(D, 5)) effectfun(fit) plot(effectfun(fit, se.fit=TRUE)) fitx <- ppm(X ~ x + polynom(D, 5)) plot(effectfun(fitx, "D", x=20))
X <- copper$SouthPoints D <- distfun(copper$SouthLines) fit <- ppm(X ~ polynom(D, 5)) effectfun(fit) plot(effectfun(fit, se.fit=TRUE)) fitx <- ppm(X ~ x + polynom(D, 5)) plot(effectfun(fitx, "D", x=20))
Check whether a model is valid, and if not, find the nearest model which is valid.
emend(object, ...)
emend(object, ...)
object |
A statistical model, belonging to some class. |
... |
Arguments passed to methods. |
The function emend
is generic,
and has methods for several classes of statistical models
in the spatstat package (mostly point process models).
Its purpose is to check whether a given model is valid
(for example, that none of the model parameters are NA
)
and, if not, to find the nearest model which is valid.
See the methods for more information.
Another model of the same kind.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Ensures that a fitted point process model satisfies the integrability conditions for existence of the point process.
project.ppm(object, ..., fatal=FALSE, trace=FALSE) ## S3 method for class 'ppm' emend(object, ..., fatal=FALSE, trace=FALSE)
project.ppm(object, ..., fatal=FALSE, trace=FALSE) ## S3 method for class 'ppm' emend(object, ..., fatal=FALSE, trace=FALSE)
object |
Fitted point process model (object of class |
... |
Ignored. |
fatal |
Logical value indicating whether to generate an error if the model cannot be projected to a valid model. |
trace |
Logical value indicating whether to print a trace of the decision process. |
The functions emend.ppm
and project.ppm
are identical:
emend.ppm
is a method for the generic emend
,
while project.ppm
is an older name for the same function.
The purpose of the function is to ensure that a fitted model is valid.
The model-fitting function ppm
fits Gibbs point process models to point pattern data.
By default, the fitted model returned by ppm
may not
actually exist as a point process.
First, some of the fitted coefficients of the model
may be NA
or infinite values.
This usually occurs when the data are insufficient to estimate
all the parameters. The model is said to be
unidentifiable or confounded.
Second, unlike a regression model, which is well-defined for any finite values
of the fitted regression coefficients, a Gibbs point process model
is only well-defined if the fitted interaction parameters
satisfy some constraints.
A famous example is the Strauss process (see Strauss
)
which exists only when the interaction parameter
is less than or equal to 1. For values
,
the probability density is not integrable and the process does not
exist (and cannot be simulated).
By default, ppm
does not enforce the constraint that
a fitted Strauss process (for example) must satisfy
.
This is because a fitted parameter value of
could be useful information for data analysis, as it indicates that
the Strauss model is not appropriate, and suggests a clustered model should be
fitted.
The function emend.ppm
or project.ppm
modifies the model object
so that the model is valid. It
identifies the terms in the model object
that are associated with illegal parameter values (i.e. parameter
values which are either NA
, infinite, or outside their permitted
range). It considers all possible sub-models of object
obtained by deleting one or more
of these terms. It identifies which of these submodels are valid,
and chooses the valid submodel with the largest pseudolikelihood. The result
of emend.ppm
or project.ppm
is the
true maximum pseudolikelihood fit to the data.
For large datasets or complex models, the algorithm used in
emend.ppm
or
project.ppm
may be time-consuming, because it takes time to
compute all the sub-models. A faster, approximate
algorithm can be applied by setting
spatstat.options(project.fast=TRUE)
. This produces a
valid submodel, which may not be the maximum pseudolikelihood submodel.
Use the function valid.ppm
to check whether a fitted model
object specifies a well-defined point process.
Use the expression all(is.finite(coef(object)))
to determine
whether all parameters are identifiable.
Another point process model (object of class "ppm"
).
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
ppm
,
valid.ppm
,
emend
,
spatstat.options
fit <- ppm(redwood ~1, Strauss(0.1)) coef(fit) fit2 <- emend(fit) coef(fit2)
fit <- ppm(redwood ~1, Strauss(0.1)) coef(fit) fit2 <- emend(fit) coef(fit2)
Ensures that a fitted spatial logistic regression specifies a well-defined model.
## S3 method for class 'slrm' emend(object, ..., fatal=FALSE, trace=FALSE)
## S3 method for class 'slrm' emend(object, ..., fatal=FALSE, trace=FALSE)
object |
Fitted point process model (object of class |
... |
Ignored. |
fatal |
Logical value indicating whether to generate an error if the model cannot be projected to a valid model. |
trace |
Logical value indicating whether to print a trace of the decision process. |
emend.slrm
is a method for the generic emend
,
The purpose of the function is to ensure that a fitted model is valid.
The model-fitting function slrm
fits spatial logistic regression models to point pattern data.
In some circumstances, the fitted model returned by slrm
may not
specify a well-defined model, because
some of the fitted coefficients of the model
may be NA
or infinite values.
This usually occurs when the data are insufficient to estimate
all the parameters. The model is said to be
unidentifiable or confounded.
The function emend.slrm
modifies the model object
so that the model is valid. It
identifies the terms in the model object
that are associated with illegal parameter values (i.e. parameter
values which are either NA
, infinite, or outside their permitted
range). It considers all possible sub-models of object
obtained by deleting one or more
of these terms. It identifies which of these submodels are valid,
and chooses the valid submodel with the largest pseudolikelihood. The result
of emend.slrm
or project.slrm
is the
true maximum pseudolikelihood fit to the data.
For large datasets or complex models, the algorithm used in
emend.slrm
may be time-consuming, because it takes time to
compute all the sub-models. A faster, approximate
algorithm can be applied by setting
spatstat.options(project.fast=TRUE)
. This produces a
valid submodel, which may not be the maximum likelihood submodel.
Use the function valid.slrm
to check whether a fitted model
object specifies a well-defined model.
Another point process model (object of class "slrm"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
slrm
,
valid.slrm
,
emend
,
spatstat.options
fit <- slrm(redwood ~ x + I(x)) coef(fit) fit2 <- emend(fit) coef(fit2)
fit <- slrm(redwood ~ x + I(x)) coef(fit) fit2 <- emend(fit) coef(fit2)
Computes simulation envelopes of a summary function.
## S3 method for class 'ppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, fix.n=FALSE, fix.marks=FALSE, verbose=TRUE, clipdata=TRUE, start=NULL, control=update(default.rmhcontrol(Y), nrep=nrep), nrep=1e5, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL) ## S3 method for class 'kppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, verbose=TRUE, clipdata=TRUE, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL) ## S3 method for class 'slrm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, verbose=TRUE, clipdata=TRUE, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL)
## S3 method for class 'ppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, fix.n=FALSE, fix.marks=FALSE, verbose=TRUE, clipdata=TRUE, start=NULL, control=update(default.rmhcontrol(Y), nrep=nrep), nrep=1e5, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL) ## S3 method for class 'kppm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, verbose=TRUE, clipdata=TRUE, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL) ## S3 method for class 'slrm' envelope(Y, fun=Kest, nsim=99, nrank=1, ..., funargs=list(), funYargs=funargs, simulate=NULL, verbose=TRUE, clipdata=TRUE, transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL, alternative=c("two.sided", "less", "greater"), scale=NULL, clamp=FALSE, savefuns=FALSE, savepatterns=FALSE, nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim, rejectNA=FALSE, silent=FALSE, do.pwrong=FALSE, envir.simul=NULL)
Y |
Object containing point pattern data.
A point pattern (object of class
|
fun |
Function that computes the desired summary statistic for a point pattern. |
nsim |
Number of simulated point patterns to be generated when computing the envelopes. |
nrank |
Integer. Rank of the envelope value amongst the |
... |
Extra arguments passed to |
funargs |
A list, containing extra arguments to be passed to |
funYargs |
Optional. A list, containing extra arguments to be passed to
|
simulate |
Optional. Specifies how to generate the simulated point patterns.
If |
fix.n |
Logical. If |
fix.marks |
Logical. If |
verbose |
Logical flag indicating whether to print progress reports during the simulations. |
clipdata |
Logical flag indicating whether the data point pattern should be
clipped to the same window as the simulated patterns,
before the summary function for the data is computed.
This should usually be |
start , control
|
Optional. These specify the arguments |
nrep |
Number of iterations in the Metropolis-Hastings simulation
algorithm. Applicable only when |
transform |
Optional. A transformation to be applied to the function values, before the envelopes are computed. An expression object (see Details). |
global |
Logical flag indicating whether envelopes should be pointwise
( |
ginterval |
Optional.
A vector of length 2 specifying
the interval of |
use.theory |
Logical value indicating whether to use the theoretical value,
computed by |
alternative |
Character string determining whether the envelope corresponds
to a two-sided test ( |
scale |
Optional. Scaling function for global envelopes.
A function in the R language which determines the
relative scale of deviations, as a function of
distance |
clamp |
Logical value indicating how to compute envelopes when
|
savefuns |
Logical flag indicating whether to save all the simulated function values. |
savepatterns |
Logical flag indicating whether to save all the simulated point patterns. |
nsim2 |
Number of extra simulated point patterns to be generated
if it is necessary to use simulation to estimate the theoretical
mean of the summary function. Only relevant when |
VARIANCE |
Logical. If |
nSD |
Number of estimated standard deviations used to determine
the critical envelopes, if |
Yname |
Character string that should be used as the name of the
data point pattern |
maxnerr |
Maximum number of rejected patterns.
If |
rejectNA |
Logical value specifying whether to reject a simulated pattern
if the resulting values of |
silent |
Logical value specifying whether to print a report each time a simulated pattern is rejected. |
do.pwrong |
Logical. If |
envir.simul |
Environment in which to evaluate the expression |
The envelope
command performs simulations and
computes envelopes of a summary statistic based on the simulations.
The result is an object that can be plotted to display the envelopes.
The envelopes can be used to assess the goodness-of-fit of
a point process model to point pattern data.
For the most basic use, if you have a point pattern X
and
you want to test Complete Spatial Randomness (CSR), type
plot(envelope(X, Kest,nsim=39))
to see the function
for
X
plotted together with the envelopes of the
function for 39 simulations of CSR.
The envelope
function is generic, with methods for
the classes "ppp"
, "ppm"
, "kppm"
and "slrm"
described here. There are also methods for the classes "pp3"
,
"lpp"
and "lppm"
which are described separately
under envelope.pp3
and envelope.lpp
.
Envelopes can also be computed from other envelopes, using
envelope.envelope
.
To create simulation envelopes, the command envelope(Y, ...)
first generates nsim
random point patterns
in one of the following ways.
If Y
is a point pattern (an object of class "ppp"
)
and simulate=NULL
,
then we generate nsim
simulations of
Complete Spatial Randomness (i.e. nsim
simulated point patterns
each being a realisation of the uniform Poisson point process)
with the same intensity as the pattern Y
.
(If Y
is a multitype point pattern, then the simulated patterns
are also given independent random marks; the probability
distribution of the random marks is determined by the
relative frequencies of marks in Y
.)
If Y
is a fitted point process model (an object of class
"ppm"
or "kppm"
or "slrm"
) and simulate=NULL
,
then this routine generates nsim
simulated
realisations of that model.
If simulate
is supplied, then it determines how the
simulated point patterns are generated. It may be either
an expression in the R language, typically containing a call
to a random generator. This expression will be evaluated
nsim
times to yield nsim
point patterns. For example
if simulate=expression(runifpoint(100))
then each simulated
pattern consists of exactly 100 independent uniform random points.
a function in the R language, typically containing a call to a
random generator. This function will be applied repeatedly
to the original data pattern Y
to yield nsim
point
patterns. For example if simulate=rlabel
then each
simulated pattern was generated by evaluating rlabel(Y)
and consists of a randomly-relabelled version of Y
.
a list of point patterns. The entries in this list will be taken as the simulated patterns.
an object of class "envelope"
. This should have been
produced by calling envelope
with the
argument savepatterns=TRUE
.
The simulated point patterns that were saved in this object
will be extracted and used as the simulated patterns for the
new envelope computation. This makes it possible to plot envelopes
for two different summary functions based on exactly the same set of
simulated point patterns.
The summary statistic fun
is applied to each of these simulated
patterns. Typically fun
is one of the functions
Kest
, Gest
, Fest
, Jest
, pcf
,
Kcross
, Kdot
, Gcross
, Gdot
,
Jcross
, Jdot
, Kmulti
, Gmulti
,
Jmulti
or Kinhom
. It may also be a character string
containing the name of one of these functions.
The statistic fun
can also be a user-supplied function;
if so, then it must have arguments X
and r
like those in the functions listed above, and it must return an object
of class "fv"
.
Upper and lower critical envelopes are computed in one of the following ways:
by default, envelopes are calculated pointwise
(i.e. for each value of the distance argument ), by sorting the
nsim
simulated values, and taking the m
-th lowest
and m
-th highest values, where m = nrank
.
For example if nrank=1
, the upper and lower envelopes
are the pointwise maximum and minimum of the simulated values.
The pointwise envelopes are not “confidence bands”
for the true value of the function! Rather,
they specify the critical points for a Monte Carlo test
(Ripley, 1981). The test is constructed by choosing a
fixed value of , and rejecting the null hypothesis if the
observed function value
lies outside the envelope at this value of
.
This test has exact significance level
alpha = 2 * nrank/(1 + nsim)
.
if global=TRUE
, then the envelopes are
determined as follows. First we calculate the theoretical mean value of
the summary statistic (if we are testing CSR, the theoretical
value is supplied by fun
; otherwise we perform a separate
set of nsim2
simulations, compute the
average of all these simulated values, and take this average
as an estimate of the theoretical mean value). Then, for each simulation,
we compare the simulated curve to the theoretical curve, and compute the
maximum absolute difference between them (over the interval
of values specified by
ginterval
). This gives a
deviation value for each of the
nsim
simulations. Finally we take the m
-th largest of the
deviation values, where m=nrank
, and call this
dcrit
. Then the simultaneous envelopes are of the form
lo = expected - dcrit
and hi = expected + dcrit
where
expected
is either the theoretical mean value theo
(if we are testing CSR) or the estimated theoretical value
mmean
(if we are testing another model). The simultaneous critical
envelopes have constant width 2 * dcrit
.
The simultaneous critical envelopes allow us to perform a different
Monte Carlo test (Ripley, 1981). The test rejects the null
hypothesis if the graph of the observed function
lies outside the envelope at any value of .
This test has exact significance level
alpha = nrank/(1 + nsim)
.
This test can also be performed using mad.test
.
if VARIANCE=TRUE
,
the algorithm calculates the
(pointwise) sample mean and sample variance of
the simulated functions. Then the envelopes are computed
as mean plus or minus nSD
standard deviations.
These envelopes do not have an exact significance interpretation.
They are a naive approximation to
the critical points of the Neyman-Pearson test
assuming the summary statistic is approximately Normally
distributed.
The return value is an object of class "fv"
containing
the summary function for the data point pattern,
the upper and lower simulation envelopes, and
the theoretical expected value (exact or estimated) of the summary function
for the model being tested. It can be plotted
using plot.envelope
.
If VARIANCE=TRUE
then the return value also includes the
sample mean, sample variance and other quantities.
Arguments can be passed to the function fun
through
...
. This means that you simply specify these arguments in the call to
envelope
, and they will be passed to fun
.
In particular, the argument correction
determines the edge correction to be used to calculate the summary
statistic. See the section on Edge Corrections, and the Examples.
Arguments can also be passed to the function fun
through the list funargs
. This mechanism is typically used if
an argument of fun
has the same name as an argument of
envelope
. The list funargs
should contain
entries of the form name=value
, where each name
is the name
of an argument of fun
.
There is also an option, rarely used, in which different function
arguments are used when computing the summary function
for the data Y
and for the simulated patterns.
If funYargs
is given, it will be used
when the summary function for the data Y
is computed,
while funargs
will be used when computing the summary function
for the simulated patterns.
This option is only needed in rare cases: usually the basic principle
requires that the data and simulated patterns must be treated
equally, so that funargs
and funYargs
should be identical.
If Y
is a fitted cluster point process model (object of
class "kppm"
), and simulate=NULL
,
then the model is simulated directly
using simulate.kppm
.
If Y
is a fitted Gibbs point process model (object of
class "ppm"
), and simulate=NULL
,
then the model is simulated
by running the Metropolis-Hastings algorithm rmh
.
Complete control over this algorithm is provided by the
arguments start
and control
which are passed
to rmh
.
For simultaneous critical envelopes (global=TRUE
)
the following options are also useful:
ginterval
determines the interval of values
over which the deviation between curves is calculated.
It should be a numeric vector of length 2.
There is a sensible default (namely, the recommended plotting
interval for
fun(X)
, or the range of r
values if
r
is explicitly specified).
transform
specifies a transformation of the
summary function fun
that will be carried out before the
deviations are computed.
Such transforms are useful if global=TRUE
or
VARIANCE=TRUE
.
The transform
must be an expression object
using the symbol .
to represent the function value
(and possibly other symbols recognised by with.fv
).
For example,
the conventional way to normalise the function
(Ripley, 1981) is to transform it to the
function
and this is implemented by setting
transform=expression(sqrt(./pi))
.
It is also possible to extract the summary functions for each of the
individual simulated point patterns, by setting savefuns=TRUE
.
Then the return value also
has an attribute "simfuns"
containing all the
summary functions for the individual simulated patterns.
It is an "fv"
object containing
functions named sim1, sim2, ...
representing the nsim
summary functions.
It is also possible to save the simulated point patterns themselves,
by setting savepatterns=TRUE
. Then the return value also has
an attribute "simpatterns"
which is a list of length
nsim
containing all the simulated point patterns.
See plot.envelope
and plot.fv
for information about how to plot the envelopes.
Different envelopes can be recomputed from the same data
using envelope.envelope
.
Envelopes can be combined using pool.envelope
.
An object of class "envelope"
and "fv"
, see fv.object
,
which can be printed and plotted directly.
Essentially a data frame containing columns
r |
the vector of values of the argument |
obs |
values of the summary function for the data point pattern |
lo |
lower envelope of simulations |
hi |
upper envelope of simulations |
and either
theo |
theoretical value of the summary function under CSR (Complete Spatial Randomness, a uniform Poisson point process) if the simulations were generated according to CSR |
mmean |
estimated theoretical value of the summary function, computed by averaging simulated values, if the simulations were not generated according to CSR. |
Additionally, if savepatterns=TRUE
, the return value has an attribute
"simpatterns"
which is a list containing the nsim
simulated patterns. If savefuns=TRUE
, the return value
has an attribute "simfuns"
which is an object of class
"fv"
containing the summary functions
computed for each of the nsim
simulated patterns.
An error may be generated if one of the simulations produces a
point pattern that is empty, or is otherwise unacceptable to the
function fun
.
The upper envelope may be NA
(plotted as plus or minus
infinity) if some of the function values
computed for the simulated point patterns are NA
.
Whether this occurs will depend on the function fun
,
but it usually happens when the simulated point pattern does not contain
enough points to compute a meaningful value.
Simulation envelopes do not compute confidence intervals;
they generate significance bands.
If you really need a confidence interval for the true summary function
of the point process, use lohboot
.
See also varblock
.
It is common to apply a correction for edge effects when
calculating a summary function such as the function.
Typically the user has a choice between several possible edge
corrections.
In a call to
envelope
, the user can specify the edge correction
to be applied in fun
, using the argument correction
.
See the Examples below.
Summary functions that are available in spatstat, such as
Kest
, Gest
and pcf
,
have a standard argument called correction
which specifies
the name of one or more edge corrections.
The list of available edge
corrections is different for each summary function,
and may also depend on the kind of window in which the point pattern is
recorded.
In the
case of Kest
(the default and most frequently used value of
fun
) the best edge correction is Ripley's isotropic
correction if the window is rectangular or polygonal,
and the translation correction if the window is a binary mask.
See the help files for the individual
functions for more information.
All the summary functions in spatstat
recognise the option correction="best"
which gives the “best” (most accurate) available edge correction
for that function.
In a call to envelope
, if fun
is one of the
summary functions provided in spatstat, then the default
is correction="best"
. This means that
by default, the envelope will be computed
using the “best” available edge correction.
The user can override this default by specifying the argument
correction
. For example the computation can be accelerated
by choosing another edge correction which is less accurate
than the “best” one, but faster to compute.
If fun
is a function written by the user,
then envelope
has to guess what to do.
If fun
has an argument
called correction
, or has ...
arguments,
then envelope
assumes that the function
can handle a correction argument. To compute the envelope,
fun
will be called with a correction
argument.
The default is correction="best"
, unless
overridden in the call to envelope
.
Otherwise, if fun
does not have an argument
called correction
and does not have ...
arguments,
then envelope
assumes that the function
cannot handle a correction argument. To compute the
envelope, fun
is called without a correction argument.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Arnold, 2003.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
dclf.test
,
mad.test
for envelope-based tests.
fv.object
,
plot.envelope
,
plot.fv
,
envelope.envelope
,
pool.envelope
for handling envelopes.
There are also methods for print
and summary
.
Kest
,
Gest
,
Fest
,
Jest
,
pcf
,
ppp
,
ppm
,
default.expand
X <- simdat online <- interactive() Nsim <- if(online) 19 else 3 # Envelope of K function for simulations from Gibbs model if(online) { fit <- ppm(cells ~1, Strauss(0.05)) plot(envelope(fit)) plot(envelope(fit, global=TRUE)) } else { fit <- ppm(cells ~1, Strauss(0.05), nd=20) E <- envelope(fit, nsim=Nsim, correction="border", nrep=100) E <- envelope(fit, nsim=Nsim, correction="border", global=TRUE, nrep=100) } # Envelope of K function for simulations from cluster model fit <- kppm(redwood ~1, "Thomas") if(online) { plot(envelope(fit, Gest)) plot(envelope(fit, Gest, global=TRUE)) } else { E <- envelope(fit, Gest, correction="rs", nsim=Nsim, global=TRUE, nrep=100) } # Envelope of INHOMOGENEOUS K-function with fitted trend # The following is valid. # Setting lambda=fit means that the fitted model is re-fitted to # each simulated pattern to obtain the intensity estimates for Kinhom. # (lambda=NULL would also be valid) fit <- kppm(redwood ~1, clusters="MatClust") if(online) { plot(envelope(fit, Kinhom, lambda=fit, nsim=19)) } else { envelope(fit, Kinhom, lambda=fit, nsim=Nsim) }
X <- simdat online <- interactive() Nsim <- if(online) 19 else 3 # Envelope of K function for simulations from Gibbs model if(online) { fit <- ppm(cells ~1, Strauss(0.05)) plot(envelope(fit)) plot(envelope(fit, global=TRUE)) } else { fit <- ppm(cells ~1, Strauss(0.05), nd=20) E <- envelope(fit, nsim=Nsim, correction="border", nrep=100) E <- envelope(fit, nsim=Nsim, correction="border", global=TRUE, nrep=100) } # Envelope of K function for simulations from cluster model fit <- kppm(redwood ~1, "Thomas") if(online) { plot(envelope(fit, Gest)) plot(envelope(fit, Gest, global=TRUE)) } else { E <- envelope(fit, Gest, correction="rs", nsim=Nsim, global=TRUE, nrep=100) } # Envelope of INHOMOGENEOUS K-function with fitted trend # The following is valid. # Setting lambda=fit means that the fitted model is re-fitted to # each simulated pattern to obtain the intensity estimates for Kinhom. # (lambda=NULL would also be valid) fit <- kppm(redwood ~1, clusters="MatClust") if(online) { plot(envelope(fit, Kinhom, lambda=fit, nsim=19)) } else { envelope(fit, Kinhom, lambda=fit, nsim=Nsim) }
Computes, to very high accuracy, the Maximum Pseudolikelihood Estimates of the parameters of a stationary Strauss point process.
exactMPLEstrauss(X, R, ngrid = 2048, plotit = FALSE, project=TRUE)
exactMPLEstrauss(X, R, ngrid = 2048, plotit = FALSE, project=TRUE)
X |
Data to which the Strauss process will be fitted.
A point pattern dataset (object of class |
R |
Interaction radius of the Strauss process. A non-negative number. |
ngrid |
Grid size for calculation of integrals. An integer, giving the
number of grid points in the |
plotit |
Logical. If |
project |
Logical. If |
This function is intended mainly for technical investigation of algorithm performance. Its practical use is quite limited.
It fits the stationary Strauss point process model
to the point pattern dataset X
by maximum pseudolikelihood
(with the border edge correction) using an algorithm with very high accuracy.
This algorithm is more accurate than the
default behaviour of the model-fitting function
ppm
because the discretisation is much finer.
Ripley (1988) and Baddeley and Turner (2000) derived the
log pseudolikelihood for the stationary Strauss
process, and eliminated the parameter ,
obtaining an exact formula for the partial log pseudolikelihood
as a function of the interaction parameter
only.
The algorithm evaluates this expression to a high degree of accuracy,
using numerical integration on a
ngrid * ngrid
lattice,
uses optim
to maximise the log pseudolikelihood
with respect to , and finally recovers
.
The result is a vector of length 2, containing the fitted coefficients
and
.
These values correspond to the entries that would be obtained with
coef(ppm(X, ~1, Strauss(R)))
.
The fitted coefficients are typically accurate to
within as shown in Baddeley and Turner (2013).
Note however that (by default) exactMPLEstrauss
constrains the parameter to lie in the
interval
in which the point process is well defined
(Kelly and Ripley, 1976)
whereas
ppm
does not constrain
the value of (by default). This behaviour is controlled by
the argument
project
to ppm
and
exactMPLEstrauss
. The default for ppm
is project=FALSE
, while the default for exactMPLEstrauss
is project=TRUE
.
Vector of length 2.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Baddeley, A. and Turner, R. (2013)
Bias correction for parameter estimates of spatial point process models.
Journal of Statistical Computation and Simulation
2012. DOI: 10.1080/00949655.2012.755976
Kelly, F.P. and Ripley, B.D. (1976) On Strauss's model for clustering. Biometrika 63, 357–360.
Ripley, B.D. (1988) Statistical inference for spatial processes. Cambridge University Press.
if(interactive()) { rc <- 0.09 exactMPLEstrauss(cells, rc, plotit=TRUE) coef(ppm(cells ~1, Strauss(rc))) coef(ppm(cells ~1, Strauss(rc), nd=128)) rr <- 0.04 exactMPLEstrauss(redwood, rr) exactMPLEstrauss(redwood, rr, project=FALSE) coef(ppm(redwood ~1, Strauss(rr))) } else { rc <- 0.09 exactMPLEstrauss(cells, rc, ngrid=64, plotit=TRUE) exactMPLEstrauss(cells, rc, ngrid=64, project=FALSE) }
if(interactive()) { rc <- 0.09 exactMPLEstrauss(cells, rc, plotit=TRUE) coef(ppm(cells ~1, Strauss(rc))) coef(ppm(cells ~1, Strauss(rc), nd=128)) rr <- 0.04 exactMPLEstrauss(redwood, rr) exactMPLEstrauss(redwood, rr, project=FALSE) coef(ppm(redwood ~1, Strauss(rr))) } else { rc <- 0.09 exactMPLEstrauss(cells, rc, ngrid=64, plotit=TRUE) exactMPLEstrauss(cells, rc, ngrid=64, project=FALSE) }
Extract a subset of an influence object, or extract the influence values at specified locations.
## S3 method for class 'influence.ppm' x[i, ...]
## S3 method for class 'influence.ppm' x[i, ...]
x |
A influence object (of class |
i |
Subset index (passed to |
... |
Ignored. |
An object of class "influence.ppm"
contains the values of the
likelihood influence for a point process model, computed by
influence.ppm
. This is effectively a marked point
pattern obtained by marking each of the original data points
with its likelihood influence.
This function extracts a designated subset of the influence values, either as another influence object, or as a vector of numeric values.
The function [.influence.ppm
is a method for [
for the
class "influence.ppm"
. The argument i
should be
an index applicable to a point pattern. It may be either
a spatial window (object of class "owin"
)
or a sequence index.
The result will be another influence object
(of class influence.ppm
).
To extract the influence values as a numeric vector,
use marks(as.ppp(x))
.
Another object of class "influence.ppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
fit <- ppm(cells, ~x) infl <- influence(fit) b <- owin(c(0.1, 0.3), c(0.2, 0.4)) infl[b] infl[1:5] marks(as.ppp(infl))[1:3]
fit <- ppm(cells, ~x) infl <- influence(fit) b <- owin(c(0.1, 0.3), c(0.2, 0.4)) infl[b] infl[1:5] marks(as.ppp(infl))[1:3]
Extract a subset of a leverage map, or extract the leverage values at specified locations.
## S3 method for class 'leverage.ppm' x[i, ..., update=TRUE]
## S3 method for class 'leverage.ppm' x[i, ..., update=TRUE]
x |
A leverage object (of class |
i |
Subset index (passed to |
... |
Further arguments passed to |
update |
Logical value indicating whether to update the internally-stored value of the mean leverage, by averaging over the specified subset. |
An object of class "leverage.ppm"
contains the values of the
leverage function for a point process model, computed by
leverage.ppm
.
This function extracts a designated subset of the leverage values, either as another leverage object, or as a vector of numeric values.
The function [.leverage.ppm
is a method for [
for the
class "leverage.ppm"
. The argument i
should be either
a spatial window (object of class "owin"
)
determining a region where the leverage map is required.
The result will typically be another leverage map
(object of class leverage.ppm
).
a spatial point pattern (object of class "ppp"
)
specifying locations at which the leverage values are required.
The result will be a numeric vector.
The subset operator for images, [.im
, is applied to
the leverage map. If this yields a pixel image, then the result of
[.leverage.ppm
is another leverage object. Otherwise,
a vector containing the numeric values of leverage is returned.
Another object of class "leverage.ppm"
,
or a vector of numeric values of leverage.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
fit <- ppm(cells ~x) lev <- leverage(fit) b <- owin(c(0.1, 0.3), c(0.2, 0.4)) lev[b] lev[cells]
fit <- ppm(cells ~x) lev <- leverage(fit) b <- owin(c(0.1, 0.3), c(0.2, 0.4)) lev[b] lev[cells]
Extract a subset of a signed measure or vector-valued measure.
## S3 method for class 'msr' x[i, j, ...]
## S3 method for class 'msr' x[i, j, ...]
x |
A signed or vector measure.
An object of class |
i |
Object defining the subregion or subset to be extracted.
Either a spatial window (an object of class |
j |
Subset index selecting the vector coordinates to be extracted,
if |
... |
Ignored. |
This operator extracts a subset of
the data which determines the signed measure
or vector-valued measure x
. The result is another measure.
An object of class "msr"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") rp[square(0.5)] rs[ , 2:3]
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") rp[square(0.5)] rs[ , 2:3]
Creates an instance of Fiksel's double exponential pairwise interaction point process model, which can then be fitted to point pattern data.
Fiksel(r, hc=NA, kappa)
Fiksel(r, hc=NA, kappa)
r |
The interaction radius of the Fiksel model |
hc |
The hard core distance |
kappa |
The rate parameter |
Fiksel (1984) introduced a pairwise interaction point process
with the following interaction function .
For two points
and
separated by a distance
, the interaction
is equal to
if
,
equal to
if
, and
equal to
if , where
are parameters.
A graph of this interaction function is shown in the Examples. The interpretation of the parameters is as follows.
is the hard core distance: distinct points are
not permitted to come closer than a distance
apart.
is the interaction range: points further than
this distance do not interact.
is the rate or slope parameter,
controlling the decay of the interaction as distance increases.
is the interaction strength parameter,
controlling the strength and type of interaction.
If
is zero, the process is Poisson. If
a
is positive,
the process is clustered. If a
is negative, the process is
inhibited (regular).
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Fiksel
pairwise interaction is
yielded by the function Fiksel()
. See the examples below.
The parameters ,
and
must be
fixed and given in the call to
Fiksel
, while the canonical
parameter is estimated by
ppm()
.
To estimate ,
and
it is possible to use
profilepl
. The maximum likelihood
estimator of is the minimum interpoint distance.
If the hard core distance argument hc
is missing or NA
,
it will be estimated from the data when ppm
is called.
The estimated value of hc
is the minimum nearest neighbour distance
multiplied by , where
is the
number of data points.
See also Stoyan, Kendall and Mecke (1987) page 161.
An object of class "interact"
describing the interpoint interaction
structure of the Fiksel
process with interaction radius ,
hard core distance
hc
and
rate parameter kappa
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Fiksel, T. (1984) Estimation of parameterized pair potentials of marked and non-marked Gibbsian point processes. Electronische Informationsverabeitung und Kybernetika 20, 270–278.
Stoyan, D, Kendall, W.S. and Mecke, J. (1987) Stochastic geometry and its applications. Wiley.
ppm
,
pairwise.family
,
ppm.object
,
StraussHard
Fiksel(r=1,hc=0.02, kappa=2) # prints a sensible description of itself X <- unmark(spruces) fit <- ppm(X ~ 1, Fiksel(r=3.5, kappa=1)) plot(fitin(fit))
Fiksel(r=1,hc=0.02, kappa=2) # prints a sensible description of itself X <- unmark(spruces) fit <- ppm(X ~ 1, Fiksel(r=3.5, kappa=1)) plot(fitin(fit))
Given a point process model that has been fitted to point pattern data, this function extracts the interpoint interaction part of the model as a separate object.
fitin(object) ## S3 method for class 'ppm' fitin(object) ## S3 method for class 'profilepl' fitin(object)
fitin(object) ## S3 method for class 'ppm' fitin(object) ## S3 method for class 'profilepl' fitin(object)
object |
A fitted point process model (object of class
|
An object of class "ppm"
describes a fitted point process
model. It contains information about the original data to which the
model was fitted, the spatial trend that was fitted, the
interpoint interaction that was fitted, and other data.
See ppm.object
) for details of this class.
The function fitin
extracts from this model the information about the
fitted interpoint interaction only.
The information is organised as an object of class "fii"
(fitted interpoint interaction).
This object can be printed or plotted.
Users may find this a convenient way to plot the fitted interpoint interaction term, as shown in the Examples.
For a pairwise interaction, the plot of the fitted interaction shows the pair interaction function (the contribution to the probability density from a pair of points as a function of the distance between them). For a higher-order interaction, the plot shows the strongest interaction (the value most different from 1) that could ever arise at the given distance.
The fitted interaction coefficients can also be extracted
from this object using coef
.
An object of class "fii"
representing the fitted
interpoint interaction. This object can be printed and plotted.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Methods for handling fitted interactions:
methods.fii
, reach.fii
,
as.interact.fii
.
Background:
ppm
,
ppm.object
.
# unmarked model <- ppm(swedishpines ~1, PairPiece(seq(3,19,by=4))) f <- fitin(model) f plot(f) # extract fitted interaction coefficients coef(f) # multitype # fit the stationary multitype Strauss process to `amacrine' r <- 0.02 * matrix(c(1,2,2,1), nrow=2,ncol=2) model <- ppm(amacrine ~1, MultiStrauss(r)) f <- fitin(model) f plot(f)
# unmarked model <- ppm(swedishpines ~1, PairPiece(seq(3,19,by=4))) f <- fitin(model) f plot(f) # extract fitted interaction coefficients coef(f) # multitype # fit the stationary multitype Strauss process to `amacrine' r <- 0.02 * matrix(c(1,2,2,1), nrow=2,ncol=2) model <- ppm(amacrine ~1, MultiStrauss(r)) f <- fitin(model) f plot(f)
Given a point process model fitted to multiple point patterns, compute the fitted conditional intensity of the model at the points of each data pattern, or at the points of the quadrature schemes used to fit the model.
## S3 method for class 'mppm' fitted(object, ..., type = "lambda", dataonly = FALSE)
## S3 method for class 'mppm' fitted(object, ..., type = "lambda", dataonly = FALSE)
object |
The fitted model. An object of class |
... |
Ignored. |
type |
Type of fitted values: either |
dataonly |
If |
This function evaluates the conditional intensity
or spatial trend
of the fitted point process model for
certain locations
, for each of the original point patterns
to which the model was fitted.
The locations at which the fitted conditional intensity/trend is
evaluated, are the points of the quadrature schemes used to fit the
model in
mppm
. They include the data points (the points of the
original point pattern datasets) and other “dummy” points in
the window of observation.
Use predict.mppm
to compute the fitted conditional intensity at
other locations or with other values of the explanatory variables.
A list of vectors (one for each row of the original hyperframe,
i.e. one vector for each of the original point patterns)
containing the values of the fitted conditional intensity
or (if type="trend"
) the fitted spatial trend.
Entries in these vector correspond to the quadrature points (data
or dummy points) used to fit the model. The quadrature points can
be extracted from object
by quad.mppm(object)
.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
model <- mppm(Bugs ~ x, data=hyperframe(Bugs=waterstriders), interaction=Strauss(7)) cifs <- fitted(model)
model <- mppm(Bugs ~ x, data=hyperframe(Bugs=waterstriders), interaction=Strauss(7)) cifs <- fitted(model)
Given a point process model fitted to a point pattern, compute the fitted conditional intensity or fitted trend of the model at the points of the pattern, or at the points of the quadrature scheme used to fit the model.
## S3 method for class 'ppm' fitted(object, ..., type="lambda", dataonly=FALSE, new.coef=NULL, leaveoneout=FALSE, drop=FALSE, check=TRUE, repair=TRUE, ignore.hardcore=FALSE, dropcoef=FALSE)
## S3 method for class 'ppm' fitted(object, ..., type="lambda", dataonly=FALSE, new.coef=NULL, leaveoneout=FALSE, drop=FALSE, check=TRUE, repair=TRUE, ignore.hardcore=FALSE, dropcoef=FALSE)
object |
The fitted point process model (an object of class |
... |
Ignored. |
type |
String (partially matched) indicating whether the fitted value is the
conditional intensity ( |
dataonly |
Logical. If |
new.coef |
Numeric vector of parameter values to replace the
fitted model parameters |
leaveoneout |
Logical. If |
drop |
Logical value determining whether to delete quadrature points that were not used to fit the model. |
check |
Logical value indicating whether to check the internal format
of |
repair |
Logical value indicating whether to repair the internal format
of |
ignore.hardcore |
Advanced use only. Logical value specifying whether to compute only the finite part of the interaction potential (effectively removing any hard core interaction terms). |
dropcoef |
Internal use only. |
The argument object
must be a fitted point process model
(object of class "ppm"
). Such objects are produced by the
model-fitting algorithm ppm
).
This function evaluates the conditional intensity
or spatial trend
of the fitted point process
model for certain locations
,
where
x
is the original point pattern dataset to which
the model was fitted.
The locations at which the fitted conditional intensity/trend
is evaluated, are the points of the
quadrature scheme used to fit the model in
ppm
.
They include the data points (the points of the original point pattern
dataset x
) and other “dummy” points
in the window of observation.
If leaveoneout=TRUE
, fitted values will be computed
for the data points only, using a ‘leave-one-out’ rule:
the fitted value at X[i]
is effectively computed by
deleting this point from the data and re-fitting the model to the
reduced pattern X[-i]
, then predicting the value at
X[i]
. (Instead of literally performing this calculation,
we apply a Taylor approximation using the influence function
computed in dfbetas.ppm
.
The argument drop
is explained in quad.ppm
.
Use predict.ppm
to compute the fitted conditional
intensity at other locations or with other values of the
explanatory variables.
A vector containing the values of the fitted conditional intensity, fitted spatial trend, or logarithm of the fitted conditional intensity.
Entries in this vector correspond to the quadrature points (data or
dummy points) used to fit the model. The quadrature points can be
extracted from object
by union.quad(quad.ppm(object))
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005). Residual analysis for spatial point processes (with discussion). Journal of the Royal Statistical Society, Series B 67, 617–666.
str <- ppm(cells ~x, Strauss(r=0.1)) lambda <- fitted(str) # extract quadrature points in corresponding order quadpoints <- union.quad(quad.ppm(str)) # plot conditional intensity values # as circles centred on the quadrature points quadmarked <- setmarks(quadpoints, lambda) plot(quadmarked) if(!interactive()) str <- ppm(cells ~ x) lambdaX <- fitted(str, leaveoneout=TRUE)
str <- ppm(cells ~x, Strauss(r=0.1)) lambda <- fitted(str) # extract quadrature points in corresponding order quadpoints <- union.quad(quad.ppm(str)) # plot conditional intensity values # as circles centred on the quadrature points quadmarked <- setmarks(quadpoints, lambda) plot(quadmarked) if(!interactive()) str <- ppm(cells ~ x) lambdaX <- fitted(str, leaveoneout=TRUE)
Given a fitted Spatial Logistic Regression model, this function computes the fitted probabilities for each pixel, or the fitted probabilities at each original data point.
## S3 method for class 'slrm' fitted(object, ..., type="probabilities", dataonly=FALSE, leaveoneout=FALSE)
## S3 method for class 'slrm' fitted(object, ..., type="probabilities", dataonly=FALSE, leaveoneout=FALSE)
object |
a fitted spatial logistic regression model.
An object of class |
... |
Ignored. |
type |
Character string (partially) matching one of
|
dataonly |
Logical. If |
leaveoneout |
Logical value
specifying whether to perform a leave-one-out calculation
when |
This is a method for the generic function
fitted
for spatial logistic regression models
(objects of class "slrm"
, usually obtained
from the function slrm
).
By default, the algorithm computes the fitted probabilities of the presence of a random point in each pixel, and returns them as an image.
If dataonly=TRUE
, the algorithm computes the fitted
presence probabilities only at the locations of the original data points.
A pixel image (object of class "im"
) containing the
fitted probability for each pixel,
or a numeric vector containing the fitted probability at each data point.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
X <- rpoispp(42) fit <- slrm(X ~ x+y) plot(fitted(fit)) fitted(fit, dataonly=TRUE)
X <- rpoispp(42) fit <- slrm(X ~ x+y) plot(fitted(fit)) fitted(fit, dataonly=TRUE)
Given a point process model fitted to a list of point patterns,
extract the fixed effects of the model.
A method for fixef
.
## S3 method for class 'mppm' fixef(object, ...)
## S3 method for class 'mppm' fixef(object, ...)
object |
A fitted point process model (an object of class |
... |
Ignored. |
This is a method for the generic function fixef
.
The argument object
must be a fitted point process model
(object of class "mppm"
) produced by the
fitting algorithm mppm
). This represents a
point process model that has been fitted
to a list of several point pattern datasets. See mppm
for information.
This function extracts the coefficients of the fixed effects of the model.
A numeric vector of coefficients.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
H <- hyperframe(Y = waterstriders) # Tweak data to exaggerate differences H$Y[[1]] <- rthin(H$Y[[1]], 0.3) m1 <- mppm(Y ~ id, data=H, Strauss(7)) fixef(m1) m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7)) fixef(m2)
H <- hyperframe(Y = waterstriders) # Tweak data to exaggerate differences H$Y[[1]] <- rthin(H$Y[[1]], 0.3) m1 <- mppm(Y ~ id, data=H, Strauss(7)) fixef(m1) m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7)) fixef(m2)
Extract the trend formula, or the terms in the trend formula, in a fitted Gibbs point process model.
## S3 method for class 'ppm' formula(x, ...) ## S3 method for class 'ppm' terms(x, ...)
## S3 method for class 'ppm' formula(x, ...) ## S3 method for class 'ppm' terms(x, ...)
x |
An object of class |
... |
Arguments passed to other methods. |
These functions are methods for the generic commands
formula
and
terms
for the class "ppm"
.
An object of class "ppm"
represents a fitted
Poisson or Gibbs point process model.
It is obtained from the model-fitting function ppm
.
The method formula.ppm
extracts the trend formula from the
fitted model x
(the formula originally specified as the
argument trend
to ppm
).
The method terms.ppm
extracts the individual
terms in the trend formula.
See the help files for the corresponding generic functions.
Adrian Baddeley [email protected]
ppm
,
as.owin
,
coef.ppm
,
extractAIC.ppm
,
fitted.ppm
,
logLik.ppm
,
model.frame.ppm
,
model.matrix.ppm
,
plot.ppm
,
predict.ppm
,
residuals.ppm
,
simulate.ppm
,
summary.ppm
,
update.ppm
,
vcov.ppm
.
fit <- ppm(cells, ~x) formula(fit) terms(fit)
fit <- ppm(cells, ~x) formula(fit) terms(fit)
Given a point process model fitted to a point pattern dataset,
this function computes the compensator
of the nearest neighbour distance distribution function
based on the fitted model
(as well as the usual nonparametric estimates
of
based on the data alone).
Comparison between the nonparametric and model-compensated
functions serves as a diagnostic for the model.
Gcom(object, r = NULL, breaks = NULL, ..., correction = c("border", "Hanisch"), conditional = !is.poisson(object), restrict=FALSE, model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), ppmcorrection="border", truecoef = NULL, hi.res = NULL)
Gcom(object, r = NULL, breaks = NULL, ..., correction = c("border", "Hanisch"), conditional = !is.poisson(object), restrict=FALSE, model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), ppmcorrection="border", truecoef = NULL, hi.res = NULL)
object |
Object to be analysed.
Either a fitted point process model (object of class |
r |
Optional.
Vector of values of the argument |
breaks |
This argument is for internal use only. |
correction |
Edge correction(s) to be employed in calculating the compensator.
Options are |
conditional |
Optional. Logical value indicating whether to compute the estimates for the conditional case. See Details. |
restrict |
Logical value indicating whether to compute
the restriction estimator ( |
model |
Optional. A fitted point process model (object of
class |
trend , interaction , rbord
|
Optional. Arguments passed to |
... |
Extra arguments passed to |
ppmcorrection |
The |
truecoef |
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with |
hi.res |
Optional. List of parameters passed to |
This command provides a diagnostic for the goodness-of-fit of
a point process model fitted to a point pattern dataset.
It computes different estimates of the nearest neighbour distance
distribution function of the
dataset, which should be approximately equal if the model is a good
fit to the data.
The first argument, object
, is usually a fitted point process model
(object of class "ppm"
), obtained from the
model-fitting function ppm
.
For convenience, object
can also be a point pattern
(object of class "ppp"
).
In that case, a point process
model will be fitted to it,
by calling ppm
using the arguments
trend
(for the first order trend),
interaction
(for the interpoint interaction)
and rbord
(for the erosion distance in the border correction
for the pseudolikelihood). See ppm
for details
of these arguments.
The algorithm first extracts the original point pattern dataset
(to which the model was fitted) and computes the
standard nonparametric estimates of the function.
It then also computes the model-compensated
function. The different functions are returned
as columns in a data frame (of class
"fv"
).
The interpretation of the columns is as follows
(ignoring edge corrections):
bord
:the nonparametric border-correction estimate of ,
where is the distance from the
-th data point
to its nearest neighbour, and
is the distance from the
-th data point to the boundary of the window
.
bcom
:the model compensator of the border-correction estimate
where
denotes the conditional intensity
of the model at the location
, and
denotes the
distance from
to the nearest point in
, while
denotes the distance from
to the boundary of the
window
.
han
:the nonparametric Hanisch estimate of
where
in which denotes the erosion of the window
by a distance
.
hcom
:the corresponding model-compensated function
where is the (‘empty space’)
distance from location
to the nearest point of
.
If the fitted model is a Poisson point process, then the formulae above are exactly what is computed. If the fitted model is not Poisson, the formulae above are modified slightly to handle edge effects.
The modification is determined by the arguments
conditional
and restrict
.
The value of conditional
defaults to FALSE
for Poisson models
and TRUE
for non-Poisson models.
If conditional=FALSE
then the formulae above are not modified.
If conditional=TRUE
, then the algorithm calculates
the restriction estimator if restrict=TRUE
,
and calculates the reweighting estimator if restrict=FALSE
.
See Appendix E of Baddeley, Rubak and Moller (2011).
See also spatstat.options('eroded.intensity')
.
Thus, by default, the reweighting estimator is computed
for non-Poisson models.
The border-corrected and Hanisch-corrected estimates of are
approximately unbiased estimates of the
-function,
assuming the point process is
stationary. The model-compensated functions are unbiased estimates
of the mean value of the corresponding nonparametric estimate,
assuming the model is true. Thus, if the model is a good fit, the mean value
of the difference between the nonparametric and model-compensated
estimates is approximately zero.
To compute the difference between the nonparametric and model-compensated
functions, use Gres
.
A function value table (object of class "fv"
),
essentially a data frame of function values.
There is a plot method for this class. See fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Related functions:
Gest
,
Gres
.
Alternative functions:
Kcom
,
psstA
,
psstG
,
psst
.
Model fitting: ppm
.
fit0 <- ppm(cells ~1) # uniform Poisson G0 <- Gcom(fit0) G0 plot(G0) # uniform Poisson is clearly not correct # Hanisch estimates only plot(Gcom(fit0), cbind(han, hcom) ~ r) fit1 <- ppm(cells, ~1, Strauss(0.08)) plot(Gcom(fit1), cbind(han, hcom) ~ r) # Try adjusting interaction distance fit2 <- update(fit1, Strauss(0.10)) plot(Gcom(fit2), cbind(han, hcom) ~ r) G3 <- Gcom(cells, interaction=Strauss(0.12)) plot(G3, cbind(han, hcom) ~ r)
fit0 <- ppm(cells ~1) # uniform Poisson G0 <- Gcom(fit0) G0 plot(G0) # uniform Poisson is clearly not correct # Hanisch estimates only plot(Gcom(fit0), cbind(han, hcom) ~ r) fit1 <- ppm(cells, ~1, Strauss(0.08)) plot(Gcom(fit1), cbind(han, hcom) ~ r) # Try adjusting interaction distance fit2 <- update(fit1, Strauss(0.10)) plot(Gcom(fit2), cbind(han, hcom) ~ r) G3 <- Gcom(cells, interaction=Strauss(0.12)) plot(G3, cbind(han, hcom) ~ r)
Creates an instance of Geyer's saturation point process model which can then be fitted to point pattern data.
Geyer(r,sat)
Geyer(r,sat)
r |
Interaction radius. A positive real number. |
sat |
Saturation threshold. A non-negative real number. |
Geyer (1999) introduced the “saturation process”,
a modification of the Strauss process (see Strauss
)
in which the total contribution
to the potential from each point (from its pairwise interaction with all
other points) is trimmed to a maximum value .
The interaction structure of this
model is implemented in the function
Geyer()
.
The saturation point process with interaction radius ,
saturation threshold
, and
parameters
and
,
is the point process
in which each point
in the pattern
contributes a factor
to the probability density of the point pattern,
where denotes the
number of ‘close neighbours’ of
in the pattern
. A close neighbour of
is a point
with
such that the distance between
and
is less than or equal to
.
If the saturation threshold is set to infinity,
this model reduces to the Strauss process (see
Strauss
)
with interaction parameter .
If
, the model reduces to the Poisson point process.
If
is a finite positive number, then the interaction parameter
may take any positive value (unlike the case
of the Strauss process), with
values
describing an ‘ordered’ or ‘inhibitive’ pattern,
and
values
describing a ‘clustered’ or ‘attractive’ pattern.
The nonstationary saturation process is similar except that
the value
is replaced by a function
of location.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the saturation process interaction is
yielded by Geyer(r, sat)
where the
arguments r
and sat
specify
the Strauss interaction radius and the saturation threshold
, respectively. See the examples below.
Note the only arguments are the interaction radius r
and the saturation threshold sat
.
When r
and sat
are fixed,
the model becomes an exponential family.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
Geyer()
.
An object of class "interact"
describing the interpoint interaction
structure of Geyer's saturation point process
with interaction radius and saturation threshold
sat
.
The value sat=0
is permitted by Geyer
,
but this is not very useful.
For technical reasons, when ppm
fits a
Geyer model with sat=0
, the default behaviour is to return
an “invalid” fitted model in which the estimate of
is
NA
. In order to get a Poisson
process model returned when sat=0
,
you would need to set emend=TRUE
in
the call to ppm
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79–140.
ppm
,
pairwise.family
,
ppm.object
,
Strauss
.
To make an interaction object like Geyer
but having
multiple interaction radii, see BadGey
or Hybrid
.
ppm(cells, ~1, Geyer(r=0.07, sat=2)) # fit the stationary saturation process to `cells'
ppm(cells, ~1, Geyer(r=0.07, sat=2)) # fit the stationary saturation process to `cells'
Given a point process model fitted to a point pattern dataset,
this function computes the residual function,
which serves as a diagnostic for goodness-of-fit of the model.
Gres(object, ...)
Gres(object, ...)
object |
Object to be analysed.
Either a fitted point process model (object of class |
... |
Arguments passed to |
This command provides a diagnostic for the goodness-of-fit of
a point process model fitted to a point pattern dataset.
It computes a residual version of the function of the
dataset, which should be approximately zero if the model is a good
fit to the data.
In normal use, object
is a fitted point process model
or a point pattern. Then Gres
first calls Gcom
to compute both the nonparametric estimate of the function
and its model compensator. Then
Gres
computes the
difference between them, which is the residual -function.
Alternatively, object
may be a function value table
(object of class "fv"
) that was returned by
a previous call to Gcom
. Then Gres
computes the
residual from this object.
A function value table (object of class "fv"
),
essentially a data frame of function values.
There is a plot method for this class. See fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Related functions:
Gcom
,
Gest
.
Alternative functions:
Kres
,
psstA
,
psstG
,
psst
.
Model-fitting:
ppm
.
fit0 <- ppm(cells, ~1) # uniform Poisson G0 <- Gres(fit0) plot(G0) # Hanisch correction estimate plot(G0, hres ~ r) # uniform Poisson is clearly not correct fit1 <- ppm(cells, ~1, Strauss(0.08)) plot(Gres(fit1), hres ~ r) # fit looks approximately OK; try adjusting interaction distance plot(Gres(cells, interaction=Strauss(0.12))) # How to make envelopes if(interactive()) { E <- envelope(fit1, Gres, model=fit1, nsim=39) plot(E) } # For computational efficiency Gc <- Gcom(fit1) G1 <- Gres(Gc)
fit0 <- ppm(cells, ~1) # uniform Poisson G0 <- Gres(fit0) plot(G0) # Hanisch correction estimate plot(G0, hres ~ r) # uniform Poisson is clearly not correct fit1 <- ppm(cells, ~1, Strauss(0.08)) plot(Gres(fit1), hres ~ r) # fit looks approximately OK; try adjusting interaction distance plot(Gres(cells, interaction=Strauss(0.12))) # How to make envelopes if(interactive()) { E <- envelope(fit1, Gres, model=fit1, nsim=39) plot(E) } # For computational efficiency Gc <- Gcom(fit1) G1 <- Gres(Gc)
Creates an instance of the hard core point process model which can then be fitted to point pattern data.
Hardcore(hc=NA)
Hardcore(hc=NA)
hc |
The hard core distance |
A hard core process with
hard core distance and abundance
parameter
is a pairwise interaction point process
in which distinct points are not allowed to come closer
than a distance
apart.
The probability density is zero if any pair of points
is closer than units apart, and otherwise equals
where represent the
points of the pattern,
is the number of points in the
pattern, and
is the normalising constant.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the hard core process
pairwise interaction is
yielded by the function Hardcore()
. See the examples below.
If the hard core distance argument hc
is missing or NA
,
it will be estimated from the data when ppm
is called.
The estimated value of hc
is the minimum nearest neighbour distance
multiplied by , where
is the
number of data points.
An object of class "interact"
describing the interpoint interaction
structure of the hard core
process with hard core distance hc
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Strauss
,
StraussHard
,
MultiHard
,
ppm
,
pairwise.family
,
ppm.object
Hardcore(0.02) # prints a sensible description of itself ppm(cells ~1, Hardcore(0.05)) # fit the stationary hard core process to `cells' # estimate hard core radius from data ppm(cells ~1, Hardcore()) # equivalent: ppm(cells ~1, Hardcore) # fit a nonstationary hard core process # with log-cubic polynomial trend ppm(cells ~ polynom(x,y,3), Hardcore(0.05))
Hardcore(0.02) # prints a sensible description of itself ppm(cells ~1, Hardcore(0.05)) # fit the stationary hard core process to `cells' # estimate hard core radius from data ppm(cells ~1, Hardcore()) # equivalent: ppm(cells ~1, Hardcore) # fit a nonstationary hard core process # with log-cubic polynomial trend ppm(cells ~ polynom(x,y,3), Hardcore(0.05))
Extract or compute the hard core distance of a point process model.
hardcoredist(x, ...) ## S3 method for class 'fii' hardcoredist(x, ..., epsilon = 0) ## S3 method for class 'ppm' hardcoredist(x, ..., epsilon = 0)
hardcoredist(x, ...) ## S3 method for class 'fii' hardcoredist(x, ..., epsilon = 0) ## S3 method for class 'ppm' hardcoredist(x, ..., epsilon = 0)
x |
An object representing a point process model
(class |
... |
Additional arguments passed to methods. |
epsilon |
Tolerance for defining the hard core. |
A point process model has a hard core distance h
if it is impossible for two random points to lie closer than the
distance h
apart.
The function hardcoredist
is generic, with methods for
objects of class "ppm"
(point process models)
and "fii"
(fitted point process interactions).
It extracts or computes the hard core distance.
If epsilon
is specified, then the code calculates the
largest distance at which the interaction factor is smaller than
epsilon
, implying that points are unlikely to occur closer
than this distance.
The result is zero if the model does not have a hard core distance.
A single numeric value, or for multitype point processes, a numeric matrix giving the hard core distances for each pair of types of points.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
m <- ppm(cells~1, Hardcore()) hardcoredist(m)
m <- ppm(cells~1, Hardcore()) hardcoredist(m)
Evaluates a basis for the harmonic polynomials in and
of degree less than or equal to
.
harmonic(x, y, n)
harmonic(x, y, n)
x |
Vector of |
y |
Vector of |
n |
Maximum degree of polynomial |
This function computes a basis for the harmonic polynomials
in two variables and
up to a given degree
and evaluates them at given
locations.
It can be used in model formulas (for example in
the model-fitting functions
lm,glm,gam
and
ppm
) to specify a
linear predictor which is a harmonic function.
A function is harmonic if
The harmonic polynomials of degree less than or equal to
have a basis consisting of
functions.
This function was implemented on a suggestion of P. McCullagh for fitting nonstationary spatial trend to point process models.
A data frame with 2 * n
columns giving the values of the
basis functions at the coordinates. Each column is labelled by an
algebraic expression for the corresponding basis function.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
# inhomogeneous point pattern X <- unmark(longleaf) # fit Poisson point process with log-cubic intensity fit.3 <- ppm(X ~ polynom(x,y,3), Poisson()) # fit Poisson process with log-cubic-harmonic intensity fit.h <- ppm(X ~ harmonic(x,y,3), Poisson()) # Likelihood ratio test lrts <- 2 * (logLik(fit.3) - logLik(fit.h)) df <- with(coords(X), ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3))) pval <- 1 - pchisq(lrts, df=df)
# inhomogeneous point pattern X <- unmark(longleaf) # fit Poisson point process with log-cubic intensity fit.3 <- ppm(X ~ polynom(x,y,3), Poisson()) # fit Poisson process with log-cubic-harmonic intensity fit.h <- ppm(X ~ harmonic(x,y,3), Poisson()) # Likelihood ratio test lrts <- 2 * (logLik(fit.3) - logLik(fit.h)) df <- with(coords(X), ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3))) pval <- 1 - pchisq(lrts, df=df)
Convert several measures to a common quadrature scheme
## S3 method for class 'msr' harmonise(...)
## S3 method for class 'msr' harmonise(...)
... |
Any number of measures (objects of class |
This function makes any number of measures compatible, by converting them all to a common quadrature scheme.
The command harmonise
is generic. This is the
method for objects of class "msr"
.
A list, of length equal to the number of arguments ...
,
whose entries are measures.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit1 <- ppm(cells ~ x) fit2 <- ppm(rpoispp(ex=cells) ~ x) m1 <- residuals(fit1) m2 <- residuals(fit2) harmonise(m1, m2) s1 <- residuals(fit1, type="score") s2 <- residuals(fit2, type="score") harmonise(s1, s2)
fit1 <- ppm(cells ~ x) fit2 <- ppm(rpoispp(ex=cells) ~ x) m1 <- residuals(fit1) m2 <- residuals(fit2) harmonise(m1, m2) s1 <- residuals(fit1, type="score") s2 <- residuals(fit2, type="score") harmonise(s1, s2)
Creates an instance of the hierarchical hard core point process model which can then be fitted to point pattern data.
HierHard(hradii=NULL, types=NULL, archy=NULL)
HierHard(hradii=NULL, types=NULL, archy=NULL)
hradii |
Optional matrix of hard core distances |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
archy |
Optional: the hierarchical order. See Details. |
This is a hierarchical point process model
for a multitype point pattern
(Hogmander and
Sarkka, 1999;
Grabarnik and Sarkka, 2009).
It is appropriate for analysing multitype point pattern data
in which the types are ordered so that
the points of type depend on the points of type
.
The hierarchical version of the (stationary)
hard core process with types, with
hard core distances
and
parameters
, is a point process
in which each point of type
contributes a factor
to the
probability density of the point pattern.
If any pair of points
of types
and
lies closer than
units apart, the configuration of points is impossible (probability
density zero).
The nonstationary hierarchical hard core
process is similar except that
the contribution of each individual point
is a function
of location and type, rather than a constant beta.
The function ppm()
,
which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the hierarchical
hard core process pairwise interaction is
yielded by the function HierHard()
. See the examples below.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the HierHard interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix radii
.
The argument archy
can be used to specify a hierarchical
ordering of the types. It can be either a vector of integers
or a character vector matching the possible types.
The default is the sequence
meaning that type
depends on types
.
The matrix iradii
must be square, with entries
which are either positive numbers, or zero or NA
.
A value of zero or NA
indicates that no hard core interaction term
should be included for this combination of types.
Note that only the hard core distances are
specified in HierHard
. The canonical
parameters
are estimated by
ppm()
, not fixed in HierHard()
.
An object of class "interact"
describing the interpoint interaction
structure of the hierarchical hard core process with
hard core distances .
Adrian Baddeley [email protected]
, Rolf Turner [email protected]
and Ege Rubak [email protected].
Grabarnik, P. and Sarkka, A. (2009) Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Hogmander, H. and Sarkka, A. (1999) Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058.
MultiHard
for the corresponding
symmetrical interaction.
h <- matrix(c(4, NA, 10, 15), 2, 2) HierHard(h) # prints a sensible description of itself ppm(ants ~1, HierHard(h)) # fit the stationary hierarchical hard core process to ants data
h <- matrix(c(4, NA, 10, 15), 2, 2) HierHard(h) # prints a sensible description of itself ppm(ants ~1, HierHard(h)) # fit the stationary hierarchical hard core process to ants data
An object describing the family of all hierarchical pairwise interaction Gibbs point processes.
Advanced Use Only!
This structure would not normally be touched by the user. It describes the hierarchical pairwise interaction family of point process models.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Other families:
pairwise.family
,
pairsat.family
,
ord.family
,
inforder.family
.
Hierarchical Strauss interaction:
HierStrauss
.
Creates an instance of the hierarchical Strauss point process model which can then be fitted to point pattern data.
HierStrauss(radii, types=NULL, archy=NULL)
HierStrauss(radii, types=NULL, archy=NULL)
radii |
Matrix of interaction radii |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
archy |
Optional: the hierarchical order. See Details. |
This is a hierarchical point process model
for a multitype point pattern
(Hogmander and
Sarkka, 1999;
Grabarnik and Sarkka, 2009).
It is appropriate for analysing multitype point pattern data
in which the types are ordered so that
the points of type depend on the points of type
.
The hierarchical version of the (stationary)
Strauss process with types, with interaction radii
and
parameters
and
is a point process
in which each point of type
contributes a factor
to the
probability density of the point pattern, and a pair of points
of types
and
closer than
units apart contributes a factor
to the density
provided
.
The nonstationary hierarchical Strauss process is similar except that
the contribution of each individual point
is a function
of location and type, rather than a constant beta.
The function ppm()
,
which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the hierarchical
Strauss process pairwise interaction is
yielded by the function HierStrauss()
. See the examples below.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the HierStrauss interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix radii
.
The argument archy
can be used to specify a hierarchical
ordering of the types. It can be either a vector of integers
or a character vector matching the possible types.
The default is the sequence
meaning that type
depends on types
.
The matrix radii
must be symmetric, with entries
which are either positive numbers or NA
.
A value of NA
indicates that no interaction term should be included
for this combination of types.
Note that only the interaction radii are
specified in HierStrauss
. The canonical
parameters and
are estimated by
ppm()
, not fixed in HierStrauss()
.
An object of class "interact"
describing the interpoint interaction
structure of the hierarchical Strauss process with
interaction radii .
Adrian Baddeley [email protected]
, Rolf Turner [email protected]
and Ege Rubak [email protected].
Grabarnik, P. and Sarkka, A. (2009) Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Hogmander, H. and Sarkka, A. (1999) Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058.
MultiStrauss
for the corresponding
symmetrical interaction.
r <- matrix(10 * c(3,4,4,3), nrow=2,ncol=2) HierStrauss(r) # prints a sensible description of itself ppm(ants ~1, HierStrauss(r, , c("Messor", "Cataglyphis"))) # fit the stationary hierarchical Strauss process to ants data
r <- matrix(10 * c(3,4,4,3), nrow=2,ncol=2) HierStrauss(r) # prints a sensible description of itself ppm(ants ~1, HierStrauss(r, , c("Messor", "Cataglyphis"))) # fit the stationary hierarchical Strauss process to ants data
Creates an instance of the hierarchical Strauss-hard core point process model which can then be fitted to point pattern data.
HierStraussHard(iradii, hradii=NULL, types=NULL, archy=NULL)
HierStraussHard(iradii, hradii=NULL, types=NULL, archy=NULL)
iradii |
Matrix of interaction radii |
hradii |
Optional matrix of hard core distances |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
archy |
Optional: the hierarchical order. See Details. |
This is a hierarchical point process model
for a multitype point pattern
(Hogmander and
Sarkka, 1999;
Grabarnik and Sarkka, 2009).
It is appropriate for analysing multitype point pattern data
in which the types are ordered so that
the points of type depend on the points of type
.
The hierarchical version of the (stationary)
Strauss hard core process with types, with interaction radii
, hard core distances
and
parameters
and
is a point process
in which each point of type
contributes a factor
to the
probability density of the point pattern, and a pair of points
of types
and
closer than
units apart contributes a factor
to the density
provided
. If any pair of points
of types
and
lies closer than
units apart, the configuration of points is impossible (probability
density zero).
The nonstationary hierarchical Strauss hard core
process is similar except that
the contribution of each individual point
is a function
of location and type, rather than a constant beta.
The function ppm()
,
which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the hierarchical
Strauss hard core process pairwise interaction is
yielded by the function HierStraussHard()
. See the examples below.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the HierStraussHard interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix radii
.
The argument archy
can be used to specify a hierarchical
ordering of the types. It can be either a vector of integers
or a character vector matching the possible types.
The default is the sequence
meaning that type
depends on types
.
The matrices iradii
and hradii
must be square, with entries
which are either positive numbers or zero or NA
.
A value of zero or NA
indicates that no interaction term
should be included for this combination of types.
Note that only the interaction radii and hard core distances are
specified in HierStraussHard
. The canonical
parameters and
are estimated by
ppm()
, not fixed in HierStraussHard()
.
An object of class "interact"
describing the interpoint interaction
structure of the hierarchical Strauss-hard core process with
interaction radii and hard core distances
.
Adrian Baddeley [email protected]
, Rolf Turner [email protected]
and Ege Rubak [email protected].
Grabarnik, P. and Sarkka, A. (2009) Modelling the spatial structure of forest stands by multivariate point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Hogmander, H. and Sarkka, A. (1999) Multitype spatial point patterns with hierarchical interactions. Biometrics 55, 1051–1058.
MultiStraussHard
for the corresponding
symmetrical interaction.
r <- matrix(c(30, NA, 40, 30), nrow=2,ncol=2) h <- matrix(c(4, NA, 10, 15), 2, 2) HierStraussHard(r, h) # prints a sensible description of itself ppm(ants ~1, HierStraussHard(r, h)) # fit the stationary hierarchical Strauss-hard core process to ants data
r <- matrix(c(30, NA, 40, 30), nrow=2,ncol=2) h <- matrix(c(4, NA, 10, 15), 2, 2) HierStraussHard(r, h) # prints a sensible description of itself ppm(ants ~1, HierStraussHard(r, h)) # fit the stationary hierarchical Strauss-hard core process to ants data
Creates an instance of a hybrid point process model which can then be fitted to point pattern data.
Hybrid(...)
Hybrid(...)
... |
Two or more interactions (objects of class |
A hybrid (Baddeley, Turner, Mateu and Bevan, 2013) is a point process model created by combining two or more point process models, or an interpoint interaction created by combining two or more interpoint interactions.
The hybrid of two point processes, with probability densities
and
respectively,
is the point process with probability density
where is a normalising constant.
Equivalently, the hybrid of two point processes with conditional intensities
and
is the point process with conditional intensity
The hybrid of point processes is defined in a similar way.
The function ppm
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of a hybrid interaction is
yielded by the function Hybrid()
.
The arguments ...
will be interpreted as interpoint interactions
(objects of class "interact"
) and the result will be the hybrid
of these interactions. Each argument must either be an
interpoint interaction (object of class "interact"
),
or a point process model (object of class "ppm"
) from which the
interpoint interaction will be extracted.
The arguments ...
may also be given in the form
name=value
. This is purely cosmetic: it can be used to attach
simple mnemonic names to the component interactions, and makes the
printed output from print.ppm
neater.
An object of class "interact"
describing an interpoint interaction structure.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013)
Hybrids of Gibbs point process models and their implementation.
Journal of Statistical Software 55:11, 1–43.
DOI: 10.18637/jss.v055.i11
Hybrid(Strauss(0.1), Geyer(0.2, 3)) Hybrid(Ha=Hardcore(0.05), St=Strauss(0.1), Ge=Geyer(0.2, 3)) fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2))) fit ctr <- rmhcontrol(nrep=5e4, expand=1) plot(simulate(fit, control=ctr)) # hybrid components can be models (including hybrid models) Hybrid(fit, S=Softcore(0.5)) # plot.fii only works if every component is a pairwise interaction fit2 <- ppm(swedishpines, ~1, Hybrid(DG=DiggleGratton(2,10), S=Strauss(5))) plot(fitin(fit2)) plot(fitin(fit2), separate=TRUE, mar.panel=rep(4,4))
Hybrid(Strauss(0.1), Geyer(0.2, 3)) Hybrid(Ha=Hardcore(0.05), St=Strauss(0.1), Ge=Geyer(0.2, 3)) fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2))) fit ctr <- rmhcontrol(nrep=5e4, expand=1) plot(simulate(fit, control=ctr)) # hybrid components can be models (including hybrid models) Hybrid(fit, S=Softcore(0.5)) # plot.fii only works if every component is a pairwise interaction fit2 <- ppm(swedishpines, ~1, Hybrid(DG=DiggleGratton(2,10), S=Strauss(5))) plot(fitin(fit2)) plot(fitin(fit2), separate=TRUE, mar.panel=rep(4,4))
An object describing the family of all hybrid interactions.
Advanced Use Only!
This structure would not normally be touched by the user. It describes the family of all hybrid point process models.
If you need to create a specific hybrid interaction model for use in
modelling, use the function Hybrid
.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Use Hybrid
to make hybrid interactions.
Other families:
pairwise.family
,
pairsat.family
,
ord.family
,
inforder.family
.
Information criteria for selecting the intensity function model of a Poisson, cluster or Cox point process.
ic(object) ## S3 method for class 'ppm' ic(object) ## S3 method for class 'kppm' ic(object)
ic(object) ## S3 method for class 'ppm' ic(object) ## S3 method for class 'kppm' ic(object)
object |
Fitted point process model
(object of class |
This function returns information criteria for selecting the intensity function model of a Poisson, Cox or cluster point process fitted by first order composite likelihood (i.e. using the Poisson likelihood function).
Degrees of freedom for the information criteria are given by
the trace of
where
is the sensitivity matrix
and
is the variance matrix for the log composite
likelihood score function. In case of a Poisson process,
is
the number of parameters in the model for the intensity function.
The composite Bayesian information criterion (cbic) is
where
is the maximal log first-order composite likelihood
(Poisson loglikelihood for the intensity function) and
is the observed number of points.
It reduces to the BIC criterion in case of a Poisson process.
The composite information criterion (cic) is
and reduces to the AIC in case of a Poisson process.
NOTE: the information criteria are for selecting the intensity function model (a set of covariates) within a given model class. They cannot be used to choose among different types of cluster or Cox point process models (e.g. can not be used to choose between Thomas and LGCP models).
A list with entries loglike
, cbic
, cic
and
df
.
Here loglike
is the fitted log first-order composite likelihood,
cbic
is composite Bayesian information criterion,
cic
is is the composite likelihood criterion and
df
is the adjusted degrees of freedom for the
fitted intensity function model.
Achmad Choiruddin, Jean-Francois Coeurjolly and Rasmus Waagepetersen.
Choiruddin, A., Coeurjolly, J.F. and Waagepetersen, R. (2020) Information criteria for inhomogeneous spatial point processes. Australian and New Zealand Journal of Statistics. To appear.
if(interactive()) { # model with one covariate fit1 <- kppm(bei~elev,data=bei.extra) ic1 <- ic(fit1) # model with two covariates fit2 <- kppm(bei~elev+grad,data=bei.extra) ic2 <- ic(fit2) # smallest cbic for fit1 but smallest cic for fit2 }
if(interactive()) { # model with one covariate fit1 <- kppm(bei~elev,data=bei.extra) ic1 <- ic(fit1) # model with two covariates fit2 <- kppm(bei~elev+grad,data=bei.extra) ic2 <- ic(fit2) # smallest cbic for fit1 but smallest cic for fit2 }
Update the fitted intensity of a fitted cluster point process model.
improve.kppm(object, type=c("quasi", "wclik1", "clik1"), rmax = NULL, eps.rmax = 0.01, dimyx = 50, maxIter = 100, tolerance = 1e-06, fast = TRUE, vcov = FALSE, fast.vcov = FALSE, verbose = FALSE, save.internals = FALSE)
improve.kppm(object, type=c("quasi", "wclik1", "clik1"), rmax = NULL, eps.rmax = 0.01, dimyx = 50, maxIter = 100, tolerance = 1e-06, fast = TRUE, vcov = FALSE, fast.vcov = FALSE, verbose = FALSE, save.internals = FALSE)
object |
Fitted cluster point process model (object of class |
type |
A character string indicating the method of estimation.
Current options are |
rmax |
Optional. The dependence range. Not usually specified by the user. |
eps.rmax |
Numeric. A small positive number which is used to determine |
dimyx |
Pixel array dimensions. See Details. |
maxIter |
Integer. Maximum number of iterations of iterative weighted least squares (Fisher scoring). |
tolerance |
Numeric. Tolerance value specifying when to stop iterative weighted least squares (Fisher scoring). |
fast |
Logical value indicating whether tapering should be used to make the computations faster (requires the package Matrix). |
vcov |
Logical value indicating whether to calculate the asymptotic variance covariance/matrix. |
fast.vcov |
Logical value indicating whether tapering should be used for the variance/covariance matrix to make the computations faster (requires the package Matrix). Caution: This is expected to underestimate the true asymptotic variances/covariances. |
verbose |
A logical indicating whether the details of computations should be printed. |
save.internals |
A logical indicating whether internal quantities should be saved in the returned object (mostly for development purposes). |
This function reestimates the intensity parameters in a fitted "kppm"
object. If type="clik1"
estimates are based on the first order
composite (Poisson) likelihood, which ignores dependence between the
points. Note that type="clik1"
is mainly included for testing
purposes and is not recommended for the typical user;
instead the more efficient kppm
with improve.type="none"
should be used.
When type="quasi"
or type="wclik1"
the dependence
structure between the points is incorporated in the estimation
procedure by using the estimated pair correlation function in the
estimating equation.
In all cases the estimating equation is based on dividing the
observation window into small subregions and count the number of points
in each subregion. To do this the observation window is first
converted into a digital mask by as.mask
where the
resolution is controlled by the argument dimyx
. The
computational time grows with the cube of the number of subregions, so fine
grids may take very long to compute (or even run out of memory).
A fitted cluster point process model of class "kppm"
.
Abdollah Jalilian [email protected] and Rasmus Plenge Waagepetersen [email protected]. Adapted for spatstat by Adrian Baddeley [email protected] and Ege Rubak [email protected].
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes, Biometrics, 63, 252-258.
Guan, Y. and Shen, Y. (2010) A weighted estimating equation approach to inference for inhomogeneous spatial point processes, Biometrika, 97, 867-880.
Guan, Y., Jalilian, A. and Waagepetersen, R. (2015) Quasi-likelihood for spatial point processes. Journal of the Royal Statistical Society, Series B 77, 677–697.
# fit a Thomas process using minimum contrast estimation method # to model interaction between points of the pattern fit0 <- kppm(bei ~ elev + grad, data = bei.extra) # fit the log-linear intensity model with quasi-likelihood method fit1 <- improve.kppm(fit0, type="quasi") # compare coef(fit0) coef(fit1)
# fit a Thomas process using minimum contrast estimation method # to model interaction between points of the pattern fit0 <- kppm(bei ~ elev + grad, data = bei.extra) # fit the log-linear intensity model with quasi-likelihood method fit1 <- improve.kppm(fit0, type="quasi") # compare coef(fit0) coef(fit1)
Computes the influence measure for a fitted spatial point process model.
## S3 method for class 'ppm' influence(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
## S3 method for class 'ppm' influence(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
model |
Fitted point process model (object of class |
... |
Ignored. |
drop |
Logical. Whether to include ( |
iScore , iHessian
|
Components of the score vector and Hessian matrix for the irregular parameters, if required. See Details. |
iArgs |
List of extra arguments for the functions |
Given a fitted spatial point process model model
,
this function computes the influence measure
described in Baddeley, Chang and Song (2013)
and Baddeley, Rubak and Turner (2019).
The function influence
is generic,
and influence.ppm
is the method for objects of class
"ppm"
representing point process models.
The influence of a point process model is a value attached to each data point
(i.e. each point of the point pattern to which the model
was fitted).
The influence value at a data point
represents the change in the maximised log (pseudo)likelihood
that occurs when the point
is deleted.
A relatively large value of
indicates a
data point with a large influence on the fitted model.
If the point process model trend has irregular parameters that were
fitted (using ippm
)
then the influence calculation requires the first and second
derivatives of the log trend with respect to the irregular parameters.
The argument iScore
should be a list,
with one entry for each irregular parameter, of R functions that compute the
partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular
parameter. The argument iHessian
should be a list,
with entries where
is the number of irregular
parameters, of R functions that compute the second order
partial derivatives of the
log trend with respect to each pair of irregular parameters.
The result of influence.ppm
is
an object of class "influence.ppm"
. It can be printed and plotted.
It can be converted to a marked
point pattern by as.ppp
(see as.ppp.influence.ppm
).
There are also methods for [
,
as.owin
, domain
,
shift
, integral
and Smooth
.
An object of class "influence.ppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
Baddeley, A., Rubak, E. and Turner, R. (2019) Leverage and influence diagnostics for Gibbs spatial point processes. Spatial Statistics 29, 15–48.
leverage.ppm
,
dfbetas.ppm
,
ppmInfluence
,
plot.influence.ppm
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(influence(fit))
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) plot(influence(fit))
An object describing the family of all Gibbs point processes with infinite interaction order.
Advanced Use Only!
This structure would not normally be touched by the user. It describes the interaction structure of Gibbs point processes which have infinite order of interaction, such as the area-interaction process AreaInter.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
AreaInter
to create the area interaction process
structure.
Other families:
pairwise.family
,
pairsat.family
,
ord.family
.
Computes the integral (total value) of a measure over its domain.
## S3 method for class 'msr' integral(f, domain=NULL, weight=NULL, ...)
## S3 method for class 'msr' integral(f, domain=NULL, weight=NULL, ...)
f |
A signed measure or vector-valued measure (object of class
|
domain |
Optional window specifying the domain of integration. Alternatively a tessellation. |
weight |
Optional. A pixel image (object of class |
... |
Ignored. |
The integral (total value) of the measure over its domain is calculated.
If domain
is a window (class "owin"
) then the integration
will be restricted to this window. If domain
is a tessellation
(class "tess"
) then the integral of f
in each
tile of domain
will be computed.
For a multitype measure m
, use split.msr
to separate the contributions for each type of point,
as shown in the Examples.
If weight
is given, it should be a pixel image or a function of
coordinates and
returning numerical values.
Then each increment of the measure
will be multiplied by the corresponding value of
weight
.
Effectively, weight
becomes the integrand, and
the result is the integral of weight
with respect to the
measure f
.
A numeric value, vector, or matrix.
integral(f)
returns a numeric value (for a signed measure)
or a vector of values (for a vector-valued measure).
If domain
is a tessellation then integral(f, domain)
returns a numeric vector with one entry for each tile
(if f
is a signed measure)
or a numeric matrix with one row for each tile
(if f
is a vector-valued measure).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- ppm(cells ~ x) rr <- residuals(fit) integral(rr) # vector-valued measure rs <- residuals(fit, type="score") integral(rs) # multitype fitA <- ppm(amacrine ~ x) rrA <- residuals(fitA) sapply(split(rrA), integral) # multitype and vector-valued rsA <- residuals(fitA, type="score") sapply(split(rsA), integral) ## integral over a subregion integral(rr, domain=square(0.5)) ## integrals over the tiles of a tessellation integral(rr, domain=quadrats(cells, 2)) ## weighted integral integral(rr, weight=function(x,y){y})
fit <- ppm(cells ~ x) rr <- residuals(fit) integral(rr) # vector-valued measure rs <- residuals(fit, type="score") integral(rs) # multitype fitA <- ppm(amacrine ~ x) rrA <- residuals(fitA) sapply(split(rrA), integral) # multitype and vector-valued rsA <- residuals(fitA, type="score") sapply(split(rsA), integral) ## integral over a subregion integral(rr, domain=square(0.5)) ## integrals over the tiles of a tessellation integral(rr, domain=quadrats(cells, 2)) ## weighted integral integral(rr, weight=function(x,y){y})
Extracts the intensity of a determinantal point process model.
## S3 method for class 'detpointprocfamily' intensity(X, ...) ## S3 method for class 'dppm' intensity(X, ...)
## S3 method for class 'detpointprocfamily' intensity(X, ...) ## S3 method for class 'dppm' intensity(X, ...)
X |
A determinantal point process model (object of class
|
... |
Ignored. |
A numeric value (if the model is stationary), a pixel image
(if the model is non-stationary) or NA
if the intensity is
unknown for the model.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Computes the intensity of a fitted point process model.
## S3 method for class 'ppm' intensity(X, ..., approx=c("Poisson", "DPP"))
## S3 method for class 'ppm' intensity(X, ..., approx=c("Poisson", "DPP"))
X |
A fitted point process model (object of class |
... |
Arguments passed to |
approx |
Character string (partially matched) specifying the type of approximation to the intensity for a non-Poisson model. |
This is a method for the generic function intensity
for fitted point process models (class "ppm"
).
The intensity of a point process model is the expected number of random points per unit area.
If X
is a Poisson point process model, the intensity of the
process is computed exactly.
The result is a numerical value if X
is a stationary Poisson point process, and a pixel image if X
is non-stationary. (In the latter case, the resolution of the pixel
image is controlled by the arguments ...
which are passed
to predict.ppm
.)
If X
is a Gibbs point process model that is not a Poisson model,
the intensity is computed approximately:
if approx="Poisson"
(the default),
the intensity is computed using the Poisson-saddlepoint approximation
(Baddeley and Nair, 2012a, 2012b, 2017; Anderssen et al, 2014).
This approximation is currently available for pairwise-interaction
models (Baddeley and Nair, 2012a, 2012b)
and for the area-interaction model and Geyer saturation model
(Baddeley and Nair, 2017).
If the model is non-stationary. the pseudostationary solution
(Baddeley and Nair, 2012b; Anderssen et al, 2014) is used.
The result is a pixel image,
whose resolution is controlled by the arguments ...
which are passed to predict.ppm
.
if approx="DPP"
, the intensity is calculated using
the approximation of (Coeurjolly and Lavancier, 2018) based on a
determinantal point process. This approximation is more accurate
than the Poisson saddlepoint approximation, for inhibitory
interactions. However the DPP approximation is only available
for stationary pairwise interaction models.
A numeric value (if the model is stationary) or a pixel image.
Adrian Baddeley [email protected], Gopalan Nair, and Frederic Lavancier.
Anderssen, R.S., Baddeley, A., DeHoog, F.R. and Nair, G.M. (2014) Solution of an integral equation arising in spatial point process theory. Journal of Integral Equations and Applications 26 (4) 437–453.
Baddeley, A. and Nair, G. (2012a) Fast approximation of the intensity of Gibbs point processes. Electronic Journal of Statistics 6 1155–1169.
Baddeley, A. and Nair, G. (2012b)
Approximating the moments of a spatial point process.
Stat 1, 1, 18–30.
DOI: 10.1002/sta4.5
Baddeley, A. and Nair, G. (2017) Poisson-saddlepoint approximation for Gibbs point processes with infinite-order interaction: in memory of Peter Hall. Journal of Applied Probability 54, 4, 1008–1026.
Coeurjolly, J.-F. and Lavancier, F. (2018) Approximation intensity for pairwise interaction Gibbs point processes using determinantal point processes. Electronic Journal of Statistics 12 3181–3203.
fitP <- ppm(swedishpines ~ 1) intensity(fitP) fitS <- ppm(swedishpines ~ 1, Strauss(9)) intensity(fitS) intensity(fitS, approx="D") fitSx <- ppm(swedishpines ~ x, Strauss(9)) lamSx <- intensity(fitSx) fitG <- ppm(swedishpines ~ 1, Geyer(9, 1)) lamG <- intensity(fitG) fitA <- ppm(swedishpines ~ 1, AreaInter(7)) lamA <- intensity(fitA)
fitP <- ppm(swedishpines ~ 1) intensity(fitP) fitS <- ppm(swedishpines ~ 1, Strauss(9)) intensity(fitS) intensity(fitS, approx="D") fitSx <- ppm(swedishpines ~ x, Strauss(9)) lamSx <- intensity(fitSx) fitG <- ppm(swedishpines ~ 1, Geyer(9, 1)) lamG <- intensity(fitG) fitA <- ppm(swedishpines ~ 1, AreaInter(7)) lamA <- intensity(fitA)
Computes the intensity of a fitted spatial logistic regression model, treated as a point process model.
## S3 method for class 'slrm' intensity(X, ...)
## S3 method for class 'slrm' intensity(X, ...)
X |
A fitted spatial logistic regression model (object of class |
... |
Arguments passed to |
This is a method for the generic function intensity
for spatial logistic regression models (class "slrm"
).
The fitted spatial logistic regression model X
is interpreted
as a point process model. The intensity of a point process model is
defined as the expected number of random points per unit area. The
fitted probabilities of presence according to X
are converted
to intensity values.
The result is a numerical value if X
is stationary, and a pixel image if X
is non-stationary. In the latter case, the resolution of the pixel
image is controlled by the arguments ...
which are passed
to predict.slrm
.
A numeric value (if the model is stationary) or a pixel image.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K.,
Schuhmacher, D., Shah, R. and Turner, R. (2010)
Spatial logistic regression and change-of-support
for spatial Poisson point processes.
Electronic Journal of Statistics
4, 1151–1201.
DOI: 10.1214/10-EJS581
fitS <- slrm(swedishpines ~ 1) intensity(fitS) fitX <- slrm(swedishpines ~ x) intensity(fitX)
fitS <- slrm(swedishpines ~ 1) intensity(fitS) fitX <- slrm(swedishpines ~ x) intensity(fitX)
Given a point process model, report the order of interpoint interaction.
interactionorder(object) ## S3 method for class 'ppm' interactionorder(object) ## S3 method for class 'interact' interactionorder(object) ## S3 method for class 'isf' interactionorder(object) ## S3 method for class 'fii' interactionorder(object)
interactionorder(object) ## S3 method for class 'ppm' interactionorder(object) ## S3 method for class 'interact' interactionorder(object) ## S3 method for class 'isf' interactionorder(object) ## S3 method for class 'fii' interactionorder(object)
object |
A point process model (class |
This function determines the order of interpoint interaction in a Gibbs point process model (or a related object).
The interaction order is defined as the largest number k
such
that the probability density of the model contains terms involving k
points at a time.
For example, in a pairwise interaction
process such as the Strauss process, the probability density contains
interaction terms between each pair of points, but does not contain
any terms that involve three points at a time, so the interaction order is 2.
Poisson point processes have interaction order 1.
Pairwise-interaction processes have interaction order 2.
Point processes with the triplet interaction Triplets
have interaction order 3. The Geyer saturation model
Geyer
and the area-interaction model
AreaInter
have infinite order of interaction.
A positive integer, or Inf
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
interactionorder(ppm(cells ~ 1)) interactionorder(Strauss(0.1)) interactionorder(Triplets(0.1)) interactionorder(Geyer(0.1, 2)) interactionorder(Hybrid(Strauss(0.1), Triplets(0.2)))
interactionorder(ppm(cells ~ 1)) interactionorder(Strauss(0.1)) interactionorder(Triplets(0.1)) interactionorder(Geyer(0.1, 2)) interactionorder(Hybrid(Strauss(0.1), Triplets(0.2)))
Experimental extension to ppm
which finds optimal values of the irregular trend parameters in a
point process model.
ippm(Q, ..., iScore=NULL, start=list(), covfunargs=start, nlm.args=list(stepmax=1/2), silent=FALSE, warn.unused=TRUE)
ippm(Q, ..., iScore=NULL, start=list(), covfunargs=start, nlm.args=list(stepmax=1/2), silent=FALSE, warn.unused=TRUE)
Q , ...
|
Arguments passed to |
iScore |
Optional. A named list of R functions that compute the partial derivatives of the logarithm of the trend, with respect to each irregular parameter. See Details. |
start |
Named list containing initial values of the irregular parameters over which to optimise. |
covfunargs |
Argument passed to |
nlm.args |
Optional list of arguments passed to |
silent |
Logical. Whether to print warnings if the optimization algorithm fails to converge. |
warn.unused |
Logical. Whether to print a warning if some of the parameters
in |
This function is an experimental extension to the
point process model fitting command ppm
.
The extension allows the trend of the model to include irregular parameters,
which will be maximised by a Newton-type iterative
method, using nlm
.
For the sake of explanation,
consider a Poisson point process with intensity function
at location
. Assume that
where are
parameters to be estimated,
is a spatial covariate
function, and
is some known function.
Then the parameters
are called regular because they
appear in a loglinear form; the parameter
is called irregular.
To fit this model using ippm
, we specify the
intensity using the trend
formula
in the same way as usual for ppm
.
The trend formula is a representation of the log intensity.
In the above example the log intensity is
So the model above would be encoded with the trend formula
~Z + offset(log(f))
. Note that the irregular part of the model
is an offset term, which means that it is included in the log trend
as it is, without being multiplied by another regular parameter.
The optimisation runs faster if we specify the derivative
of with
respect to
. We call this the
irregular score. To specify this, the user must write an R function
that computes the irregular score for any value of
at any location
(x,y)
.
Thus, to code such a problem,
The argument trend
should define the
log intensity, with the irregular part as an offset;
The argument start
should be a list
containing initial values of each of the irregular parameters;
The argument iScore
, if provided,
must be a list (with one entry
for each entry of start
) of functions
with arguments x,y,...
, that evaluate the partial derivatives
of with
respect to each irregular parameter.
The coded example below illustrates the model with two irregular
parameters and irregular term
Arguments ...
passed to ppm
may
also include interaction
. In this case the model is not
a Poisson point process but a more general Gibbs point process;
the trend formula trend
determines the first-order trend
of the model (the first order component of the conditional intensity),
not the intensity.
A fitted point process model (object of class "ppm"
)
which also belongs to the special class "ippm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
nd <- 32 gamma0 <- 3 delta0 <- 5 POW <- 3 # Terms in intensity Z <- function(x,y) { -2*y } f <- function(x,y,gamma,delta) { 1 + exp(gamma - delta * x^POW) } # True intensity lamb <- function(x,y,gamma,delta) { 200 * exp(Z(x,y)) * f(x,y,gamma,delta) } # Simulate realisation lmax <- max(lamb(0,0,gamma0,delta0), lamb(1,1,gamma0,delta0)) set.seed(42) X <- rpoispp(lamb, lmax=lmax, win=owin(), gamma=gamma0, delta=delta0) # Partial derivatives of log f DlogfDgamma <- function(x,y, gamma, delta) { topbit <- exp(gamma - delta * x^POW) topbit/(1 + topbit) } DlogfDdelta <- function(x,y, gamma, delta) { topbit <- exp(gamma - delta * x^POW) - (x^POW) * topbit/(1 + topbit) } # irregular score Dlogf <- list(gamma=DlogfDgamma, delta=DlogfDdelta) # fit model ippm(X ~Z + offset(log(f)), covariates=list(Z=Z, f=f), iScore=Dlogf, start=list(gamma=1, delta=1), nlm.args=list(stepmax=1), nd=nd)
nd <- 32 gamma0 <- 3 delta0 <- 5 POW <- 3 # Terms in intensity Z <- function(x,y) { -2*y } f <- function(x,y,gamma,delta) { 1 + exp(gamma - delta * x^POW) } # True intensity lamb <- function(x,y,gamma,delta) { 200 * exp(Z(x,y)) * f(x,y,gamma,delta) } # Simulate realisation lmax <- max(lamb(0,0,gamma0,delta0), lamb(1,1,gamma0,delta0)) set.seed(42) X <- rpoispp(lamb, lmax=lmax, win=owin(), gamma=gamma0, delta=delta0) # Partial derivatives of log f DlogfDgamma <- function(x,y, gamma, delta) { topbit <- exp(gamma - delta * x^POW) topbit/(1 + topbit) } DlogfDdelta <- function(x,y, gamma, delta) { topbit <- exp(gamma - delta * x^POW) - (x^POW) * topbit/(1 + topbit) } # irregular score Dlogf <- list(gamma=DlogfDgamma, delta=DlogfDdelta) # fit model ippm(X ~Z + offset(log(f)), covariates=list(Z=Z, f=f), iScore=Dlogf, start=list(gamma=1, delta=1), nlm.args=list(stepmax=1), nd=nd)
Check that an object inherits the class dppm
is.dppm(x)
is.dppm(x)
x |
Any object. |
A single logical value.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Tests where a point process model or point process interaction is a hybrid of several interactions.
is.hybrid(x) ## S3 method for class 'ppm' is.hybrid(x) ## S3 method for class 'interact' is.hybrid(x)
is.hybrid(x) ## S3 method for class 'ppm' is.hybrid(x) ## S3 method for class 'interact' is.hybrid(x)
x |
A point process model (object of class |
A hybrid (Baddeley, Turner, Mateu and Bevan, 2012) is a point process model created by combining two or more point process models, or an interpoint interaction created by combining two or more interpoint interactions.
The function is.hybrid
is generic, with methods for
point process models (objects of class "ppm"
)
and point process interactions
(objects of class "interact"
).
These functions return TRUE
if the object x
is a hybrid,
and FALSE
if it is not a hybrid.
Hybrids of two or more interpoint interactions
are created by the function Hybrid
.
Such a hybrid interaction can then be fitted to point pattern data
using ppm
.
TRUE
if the object is a hybrid, and FALSE
otherwise.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013)
Hybrids of Gibbs point process models and their implementation.
Journal of Statistical Software 55:11, 1–43.
DOI: 10.18637/jss.v055.i11
S <- Strauss(0.1) is.hybrid(S) H <- Hybrid(Strauss(0.1), Geyer(0.2, 3)) is.hybrid(H) fit <- ppm(redwood, ~1, H) is.hybrid(fit)
S <- Strauss(0.1) is.hybrid(S) H <- Hybrid(Strauss(0.1), Geyer(0.2, 3)) is.hybrid(H) fit <- ppm(redwood, ~1, H) is.hybrid(fit)
Tests whether a fitted point process model involves “marks” attached to the points.
## S3 method for class 'ppm' is.marked(X, ...)
## S3 method for class 'ppm' is.marked(X, ...)
X |
Fitted point process model (object of class |
... |
Ignored. |
“Marks” are observations attached to each point of a point pattern.
For example the longleaf
dataset contains
the locations of trees, each tree being marked by its diameter;
the amacrine
dataset gives the locations of cells
of two types (on/off) and the type of cell may be regarded as a mark attached
to the location of the cell.
The argument X
is a fitted point process model
(an object of class "ppm"
) typically obtained
by fitting a model to point pattern data using ppm
.
This function returns TRUE
if the original data
(to which the model X
was fitted) were a marked point pattern.
Note that this is not the same as testing whether the model involves terms that depend on the marks (i.e. whether the fitted model ignores the marks in the data). See the Examples for a trick to do this.
If this function returns TRUE
, the implications are
(for example) that
any simulation of this model will require simulation of random marks
as well as random point locations.
Logical value, equal to TRUE
if
X
is a model that was fitted to a marked point pattern dataset.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
X <- lansing # Multitype point pattern --- trees marked by species fit1 <- ppm(X, ~ marks, Poisson()) is.marked(fit1) fit2 <- ppm(X, ~ 1, Poisson()) is.marked(fit2) ## test whether the model formula involves marks "marks" %in% spatstat.utils::variablesinformula(formula(fit2)) # Unmarked point pattern fit3 <- ppm(cells, ~ 1, Poisson()) is.marked(fit3) # FALSE
X <- lansing # Multitype point pattern --- trees marked by species fit1 <- ppm(X, ~ marks, Poisson()) is.marked(fit1) fit2 <- ppm(X, ~ 1, Poisson()) is.marked(fit2) ## test whether the model formula involves marks "marks" %in% spatstat.utils::variablesinformula(formula(fit2)) # Unmarked point pattern fit3 <- ppm(cells, ~ 1, Poisson()) is.marked(fit3) # FALSE
Tests whether a fitted point process model involves “marks” attached to the points that classify the points into several types.
## S3 method for class 'ppm' is.multitype(X, ...)
## S3 method for class 'ppm' is.multitype(X, ...)
X |
Fitted point process model (object of class |
... |
Ignored. |
“Marks” are observations attached to each point of a point pattern.
For example the longleaf
dataset
contains the locations of trees, each tree being marked by its diameter;
the amacrine
dataset gives the locations of cells
of two types (on/off) and the type of cell may be regarded as a mark attached
to the location of the cell.
The argument X
is a fitted point process model
(an object of class "ppm"
) typically obtained
by fitting a model to point pattern data using ppm
.
This function returns TRUE
if the original data
(to which the model X
was fitted) were a multitype point pattern.
Note that this is not the same as testing whether the model involves terms that depend on the marks (i.e. whether the fitted model ignores the marks in the data). Currently we have not implemented a test for this.
If this function returns TRUE
, the implications are
(for example) that
any simulation of this model will require simulation of random marks
as well as random point locations.
Logical value, equal to TRUE
if
X
is a model that was fitted to a multitype point pattern dataset.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
is.multitype
,
is.multitype.ppp
X <- lansing # Multitype point pattern --- trees marked by species fit1 <- ppm(X, ~ marks, Poisson()) is.multitype(fit1) # TRUE fit2 <- ppm(X, ~ 1, Poisson()) is.multitype(fit2) # TRUE # Unmarked point pattern fit3 <- ppm(cells, ~ 1, Poisson()) is.multitype(fit3) # FALSE
X <- lansing # Multitype point pattern --- trees marked by species fit1 <- ppm(X, ~ marks, Poisson()) is.multitype(fit1) # TRUE fit2 <- ppm(X, ~ 1, Poisson()) is.multitype(fit2) # TRUE # Unmarked point pattern fit3 <- ppm(cells, ~ 1, Poisson()) is.multitype(fit3) # FALSE
Given a point process model (either a model that has been fitted to data, or a model specified by its parameters), determine whether the model is a Poisson cluster process.
is.poissonclusterprocess(model) ## S3 method for class 'kppm' is.poissonclusterprocess(model) ## S3 method for class 'zclustermodel' is.poissonclusterprocess(model) ## Default S3 method: is.poissonclusterprocess(model)
is.poissonclusterprocess(model) ## S3 method for class 'kppm' is.poissonclusterprocess(model) ## S3 method for class 'zclustermodel' is.poissonclusterprocess(model) ## Default S3 method: is.poissonclusterprocess(model)
model |
Any kind of object representing a spatial point process model, either a model fitted to data, or a specification of a point process model. |
The argument model
represents a fitted spatial point process
model (such as an object of class "ppm"
, "kppm"
or
similar) or a specification of a point process model
(such as an object of class "zclustermodel"
).
This function returns TRUE
if the model
is a Poisson cluster
process, and FALSE
otherwise.
The function is.poissonclusterprocess
is generic, with methods
for classes kppm
and zclustermodel
, and a default
method.
A logical value.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fut <- kppm(redwood ~ 1, "Thomas") is.poissonclusterprocess(fut) fot <- slrm(cells ~ x, dimyx=16) is.poissonclusterprocess(fot)
fut <- kppm(redwood ~ 1, "Thomas") is.poissonclusterprocess(fut) fot <- slrm(cells ~ x, dimyx=16) is.poissonclusterprocess(fot)
Checks whether its argument is a fitted point process model
(object of class "ppm"
, "kppm"
,
"lppm"
or "slrm"
).
is.ppm(x) is.kppm(x) is.lppm(x) is.slrm(x)
is.ppm(x) is.kppm(x) is.lppm(x) is.slrm(x)
x |
Any object. |
These functions test whether the object x
is a fitted point process model object of the specified class.
The result of is.ppm(x)
is TRUE
if x
has "ppm"
amongst its classes, and
otherwise FALSE
. Similarly for the other functions.
A single logical value.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Given a point process model (either a model that has been fitted to data, or a model specified by its parameters), determine whether the model is a stationary point process, and whether it is a Poisson point process.
## S3 method for class 'ppm' is.stationary(x) ## S3 method for class 'kppm' is.stationary(x) ## S3 method for class 'slrm' is.stationary(x) ## S3 method for class 'dppm' is.stationary(x) ## S3 method for class 'detpointprocfamily' is.stationary(x) ## S3 method for class 'ppm' is.poisson(x) ## S3 method for class 'kppm' is.poisson(x) ## S3 method for class 'slrm' is.poisson(x) ## S3 method for class 'interact' is.poisson(x)
## S3 method for class 'ppm' is.stationary(x) ## S3 method for class 'kppm' is.stationary(x) ## S3 method for class 'slrm' is.stationary(x) ## S3 method for class 'dppm' is.stationary(x) ## S3 method for class 'detpointprocfamily' is.stationary(x) ## S3 method for class 'ppm' is.poisson(x) ## S3 method for class 'kppm' is.poisson(x) ## S3 method for class 'slrm' is.poisson(x) ## S3 method for class 'interact' is.poisson(x)
x |
A fitted spatial point process model
(object of class |
The argument x
represents a fitted spatial point process model
or a similar object.
is.stationary(x)
returns TRUE
if x
represents
a stationary point process, and FALSE
if not.
is.poisson(x)
returns TRUE
if x
represents
a Poisson point process, and FALSE
if not.
The functions is.stationary
and is.poisson
are generic,
with methods for the classes "ppm"
(Gibbs point process models),
"kppm"
(cluster or Cox point process models),
"slrm"
(spatial logistic regression models) and
"rmhmodel"
(model specifications for the
Metropolis-Hastings algorithm).
Additionally is.stationary
has a method for
classes "detpointprocfamily"
and "dppm"
(both determinantal point processes) and
is.poisson
has a method for
class "interact"
(interaction structures for Gibbs models).
is.poisson.kppm
will return FALSE
, unless
the model x
is degenerate:
either x
has zero intensity so that its realisations are empty
with probability 1, or it is a log-Gaussian Cox process
where the log intensity has zero variance.
is.poisson.slrm
will always return TRUE
,
by convention.
A logical value.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
is.marked
to determine whether a model is a marked
point process.
summary.ppm
for detailed information about a
fitted model.
Model-fitting functions
ppm
,
dppm
,
kppm
,
slrm
.
fit <- ppm(cells ~ x) is.stationary(fit) is.poisson(fit) fut <- kppm(redwood ~ 1, "MatClust") is.stationary(fut) is.poisson(fut) fot <- slrm(cells ~ x) is.stationary(fot) is.poisson(fot)
fit <- ppm(cells ~ x) is.stationary(fit) is.poisson(fit) fut <- kppm(redwood ~ 1, "MatClust") is.stationary(fut) is.poisson(fut) fot <- slrm(cells ~ x) is.stationary(fot) is.poisson(fot)
Objects of class "isf"
are used internally by the
spatstat package to represent the structure
of the interpoint interactions in a family of point process models.
Advanced Use Only!
An object of class "isf"
(Interaction Structure Family)
is used internally by the spatstat package to
represent the common mathematical and algorithmic structure of
the interpoint interactions in a family of point process models.
The existing objects of class "isf"
are:
pairwise.family |
pairwise interaction |
triplet.family |
triplet interaction |
pairsat.family |
saturated pairwise interaction |
hierpair.family |
hierarchical pairwise interaction |
inforder.family |
infinite order interaction |
hybrid.family |
hybrids of several interactions |
ord.family |
Ord interactions |
The information contained in these objects enables the spatstat package to select the appropriate algorithm for fitting, predicting and simulating each point process model.
For example, in order to fit a model that involves pairwise
interactions, the model-fitting function ppm
would use
information contained in pairwise.family
to select the
appropriate algorithms.
An object of class "isf"
is essentially a list of functions
for various tasks.
The internal format is undocumented and may be changed
without notice.
An object of class "isf"
, essentially a list of functions
for various tasks.
The internal format is undocumented and may be changed without notice.
Adrian Baddeley [email protected].
Given a point process model fitted to a point pattern dataset,
this function computes the compensator
of the function based on the fitted model
(as well as the usual nonparametric estimates
of
based on the data alone).
Comparison between the nonparametric and model-compensated
functions serves as a diagnostic for the model.
Kcom(object, r = NULL, breaks = NULL, ..., correction = c("border", "isotropic", "translate"), conditional = !is.poisson(object), restrict = FALSE, model = NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), compute.var = TRUE, truecoef = NULL, hi.res = NULL)
Kcom(object, r = NULL, breaks = NULL, ..., correction = c("border", "isotropic", "translate"), conditional = !is.poisson(object), restrict = FALSE, model = NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), compute.var = TRUE, truecoef = NULL, hi.res = NULL)
object |
Object to be analysed.
Either a fitted point process model (object of class |
r |
Optional.
Vector of values of the argument |
breaks |
This argument is for advanced use only. |
... |
Ignored. |
correction |
Optional vector of character strings specifying the edge
correction(s) to be used. See |
conditional |
Optional. Logical value indicating whether to compute the estimates for the conditional case. See Details. |
restrict |
Logical value indicating
whether to compute the restriction estimator ( |
model |
Optional. A fitted point process model (object of
class |
trend , interaction , rbord
|
Optional. Arguments passed to |
compute.var |
Logical value indicating whether to compute the
Poincare variance bound for the residual |
truecoef |
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with |
hi.res |
Optional. List of parameters passed to |
This command provides a diagnostic for the goodness-of-fit of
a point process model fitted to a point pattern dataset.
It computes an estimate of the function of the
dataset, together with a model compensator of the
function, which should be approximately equal if the model is a good
fit to the data.
The first argument, object
, is usually a fitted point process model
(object of class "ppm"
), obtained from the
model-fitting function ppm
.
For convenience, object
can also be a point pattern
(object of class "ppp"
). In that case, a point process
model will be fitted to it, by calling ppm
using the arguments
trend
(for the first order trend),
interaction
(for the interpoint interaction)
and rbord
(for the erosion distance in the border correction
for the pseudolikelihood). See ppm
for details
of these arguments.
The algorithm first extracts the original point pattern dataset
(to which the model was fitted) and computes the
standard nonparametric estimates of the function.
It then also computes the model compensator of the
function. The different function estimates are returned
as columns in a data frame (of class
"fv"
).
The argument correction
determines the edge correction(s)
to be applied. See Kest
for explanation of the principle
of edge corrections. The following table gives the options
for the correction
argument, and the corresponding
column names in the result:
correction |
description of correction | nonparametric | compensator |
"isotropic" |
Ripley isotropic correction | iso |
icom |
"translate" |
Ohser-Stoyan translation correction | trans |
tcom |
"border" |
border correction | border |
bcom
|
The nonparametric estimates can all be expressed in the form
where is the
-th data point,
is the distance between
and
, and
is
a term that serves to correct edge effects and to re-normalise the
sum. The corresponding model compensator is
where the integral is over all locations in
the observation window,
denotes the conditional intensity
of the model at the location
, and
denotes the
data point pattern
augmented by adding the extra point
.
If the fitted model is a Poisson point process, then the formulae above are exactly what is computed. If the fitted model is not Poisson, the formulae above are modified slightly to handle edge effects.
The modification is determined by the arguments
conditional
and restrict
.
The value of conditional
defaults to FALSE
for Poisson models
and TRUE
for non-Poisson models.
If conditional=FALSE
then the formulae above are not modified.
If conditional=TRUE
, then the algorithm calculates
the restriction estimator if restrict=TRUE
,
and calculates the reweighting estimator if restrict=FALSE
.
See Appendix D of Baddeley, Rubak and Moller (2011).
Thus, by default, the reweighting estimator is computed
for non-Poisson models.
The nonparametric estimates of are approximately unbiased
estimates of the
-function, assuming the point process is
stationary. The model compensators are unbiased estimates
of the mean values of the corresponding nonparametric estimates,
assuming the model is true. Thus, if the model is a good fit, the mean value
of the difference between the nonparametric estimates and model compensators
is approximately zero.
A function value table (object of class "fv"
),
essentially a data frame of function values.
There is a plot method for this class. See fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Related functions:
Kres
,
Kest
.
Alternative functions:
Gcom
,
psstG
, psstA
, psst
.
Point process models: ppm
.
fit0 <- ppm(cells, ~1) # uniform Poisson if(interactive()) { plot(Kcom(fit0)) # compare the isotropic-correction estimates plot(Kcom(fit0), cbind(iso, icom) ~ r) # uniform Poisson is clearly not correct } fit1 <- ppm(cells, ~1, Strauss(0.08)) K1 <- Kcom(fit1) K1 if(interactive()) { plot(K1) plot(K1, cbind(iso, icom) ~ r) plot(K1, cbind(trans, tcom) ~ r) # how to plot the difference between nonparametric estimates and compensators plot(K1, iso - icom ~ r) # fit looks approximately OK; try adjusting interaction distance } fit2 <- ppm(cells, ~1, Strauss(0.12)) K2 <- Kcom(fit2) if(interactive()) { plot(K2) plot(K2, cbind(iso, icom) ~ r) plot(K2, iso - icom ~ r) }
fit0 <- ppm(cells, ~1) # uniform Poisson if(interactive()) { plot(Kcom(fit0)) # compare the isotropic-correction estimates plot(Kcom(fit0), cbind(iso, icom) ~ r) # uniform Poisson is clearly not correct } fit1 <- ppm(cells, ~1, Strauss(0.08)) K1 <- Kcom(fit1) K1 if(interactive()) { plot(K1) plot(K1, cbind(iso, icom) ~ r) plot(K1, cbind(trans, tcom) ~ r) # how to plot the difference between nonparametric estimates and compensators plot(K1, iso - icom ~ r) # fit looks approximately OK; try adjusting interaction distance } fit2 <- ppm(cells, ~1, Strauss(0.12)) K2 <- Kcom(fit2) if(interactive()) { plot(K2) plot(K2, cbind(iso, icom) ~ r) plot(K2, iso - icom ~ r) }
Returns the theoretical function or the pair correlation function
of a point process model.
Kmodel(model, ...) pcfmodel(model, ...)
Kmodel(model, ...) pcfmodel(model, ...)
model |
A fitted point process model of some kind. |
... |
Ignored. |
For certain types of point process models, it is possible to
write down a mathematical expression for the function
or the pair correlation function of the model.
The functions Kmodel
and pcfmodel
give the
theoretical -function and the theoretical pair correlation
function for a point process model that has been fitted to data.
The functions Kmodel
and pcfmodel
are generic,
with methods for the classes "kppm"
(cluster processes
and Cox processes) and "ppm"
(Gibbs processes).
The return value is a function
in the R language,
which takes one argument r
.
Evaluation of this function, on a numeric vector r
,
yields values of the desired function or pair correlation
function at these distance values.
A function
in the R language,
which takes one argument r
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Kest
or pcf
to estimate the function or pair correlation function
nonparametrically from data.
Kmodel.kppm
for the method for cluster processes
and Cox processes.
Kmodel.ppm
for the method for Gibbs processes.
Returns the theoretical -function
or theoretical pair correlation function of a determinantal point
process model as a function of one argument
.
## S3 method for class 'dppm' Kmodel(model, ...) ## S3 method for class 'dppm' pcfmodel(model, ...) ## S3 method for class 'detpointprocfamily' Kmodel(model, ...) ## S3 method for class 'detpointprocfamily' pcfmodel(model, ...)
## S3 method for class 'dppm' Kmodel(model, ...) ## S3 method for class 'dppm' pcfmodel(model, ...) ## S3 method for class 'detpointprocfamily' Kmodel(model, ...) ## S3 method for class 'detpointprocfamily' pcfmodel(model, ...)
model |
Model of class |
... |
Ignored (not quite true – there is some undocumented internal use) |
A function in the R language, with one numeric argument r
,
that can be used to evaluate the theoretical -function or
pair correlation function of the model at distances
r
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) KMatern <- Kmodel(model) pcfMatern <- pcfmodel(model) plot(KMatern, xlim = c(0,0.05)) plot(pcfMatern, xlim = c(0,0.05))
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) KMatern <- Kmodel(model) pcfMatern <- pcfmodel(model) plot(KMatern, xlim = c(0,0.05)) plot(pcfMatern, xlim = c(0,0.05))
Returns the theoretical function or the pair correlation function
of a cluster point process model or Cox point process model.
## S3 method for class 'kppm' Kmodel(model, ...) ## S3 method for class 'kppm' pcfmodel(model, ...)
## S3 method for class 'kppm' Kmodel(model, ...) ## S3 method for class 'kppm' pcfmodel(model, ...)
model |
A fitted cluster point process model (object of
class |
... |
Ignored. |
For certain types of point process models, it is possible to
write down a mathematical expression for the function
or the pair correlation function of the model. In particular this
is possible for a fitted cluster point process model
(object of class
"kppm"
obtained from kppm
).
The functions Kmodel
and pcfmodel
are generic.
The functions documented here are the methods for the class "kppm"
.
The return value is a function
in the R language,
which takes one argument r
.
Evaluation of this function, on a numeric vector r
,
yields values of the desired function or pair correlation
function at these distance values.
A function
in the R language,
which takes one argument r
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Kest
or pcf
to estimate the function or pair correlation function
nonparametrically from data.
kppm
to fit cluster models.
Kmodel
for the generic functions.
Kmodel.ppm
for the method for Gibbs processes.
fit <- kppm(redwood, ~x, "MatClust") K <- Kmodel(fit) K(c(0.1, 0.2)) curve(K(x), from=0, to=0.25)
fit <- kppm(redwood, ~x, "MatClust") K <- Kmodel(fit) K(c(0.1, 0.2)) curve(K(x), from=0, to=0.25)
Returns the theoretical function or the pair correlation function
of a fitted Gibbs point process model.
## S3 method for class 'ppm' Kmodel(model, ...) ## S3 method for class 'ppm' pcfmodel(model, ...)
## S3 method for class 'ppm' Kmodel(model, ...) ## S3 method for class 'ppm' pcfmodel(model, ...)
model |
A fitted Poisson or Gibbs point process model (object of
class |
... |
Ignored. |
This function computes an approximation to the function
or the pair correlation function of a Gibbs point process.
The functions Kmodel
and pcfmodel
are generic.
The functions documented here are the methods for the class
"ppm"
.
The approximation is only available for stationary pairwise-interaction models. It uses the second order Poisson-saddlepoint approximation (Baddeley and Nair, 2012b) which is a combination of the Poisson-Boltzmann-Emden and Percus-Yevick approximations.
The return value is a function
in the R language,
which takes one argument r
.
Evaluation of this function, on a numeric vector r
,
yields values of the desired function or pair correlation
function at these distance values.
A function
in the R language,
which takes one argument r
.
Adrian Baddeley [email protected] and Gopalan Nair.
Baddeley, A. and Nair, G. (2012a) Fast approximation of the intensity of Gibbs point processes. Electronic Journal of Statistics 6 1155–1169.
Baddeley, A. and Nair, G. (2012b)
Approximating the moments of a spatial point process.
Stat 1, 1, 18–30.
DOI: 10.1002/sta4.5
Kest
or pcf
to estimate the function or pair correlation function
nonparametrically from data.
ppm
to fit Gibbs models.
Kmodel
for the generic functions.
Kmodel.kppm
for the method for cluster/Cox processes.
fit <- ppm(swedishpines, ~1, Strauss(8)) p <- pcfmodel(fit) K <- Kmodel(fit) p(6) K(8) curve(K(x), from=0, to=15)
fit <- ppm(swedishpines, ~1, Strauss(8)) p <- pcfmodel(fit) K <- Kmodel(fit) p(6) K(8) curve(K(x), from=0, to=15)
Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern.
kppm(X, ...) ## S3 method for class 'formula' kppm(X, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), ..., data=NULL) ## S3 method for class 'ppp' kppm(X, trend = ~1, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), data = NULL, ..., covariates=data, subset, method = c("mincon", "clik2", "palm", "adapcl"), penalised = FALSE, improve.type = c("none", "clik1", "wclik1", "quasi"), improve.args = list(), weightfun=NULL, control=list(), stabilize=TRUE, algorithm, trajectory=FALSE, statistic="K", statargs=list(), rmax = NULL, epsilon=0.01, covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL, ppm.improve.type=c("none", "ho", "enet"), ppm.improve.args=list()) ## S3 method for class 'quad' kppm(X, trend = ~1, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), data = NULL, ..., covariates=data, subset, method = c("mincon", "clik2", "palm", "adapcl"), penalised = FALSE, improve.type = c("none", "clik1", "wclik1", "quasi"), improve.args = list(), weightfun=NULL, control=list(), stabilize=TRUE, algorithm, trajectory=FALSE, statistic="K", statargs=list(), rmax = NULL, epsilon=0.01, covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL, ppm.improve.type=c("none", "ho", "enet"), ppm.improve.args=list())
kppm(X, ...) ## S3 method for class 'formula' kppm(X, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), ..., data=NULL) ## S3 method for class 'ppp' kppm(X, trend = ~1, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), data = NULL, ..., covariates=data, subset, method = c("mincon", "clik2", "palm", "adapcl"), penalised = FALSE, improve.type = c("none", "clik1", "wclik1", "quasi"), improve.args = list(), weightfun=NULL, control=list(), stabilize=TRUE, algorithm, trajectory=FALSE, statistic="K", statargs=list(), rmax = NULL, epsilon=0.01, covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL, ppm.improve.type=c("none", "ho", "enet"), ppm.improve.args=list()) ## S3 method for class 'quad' kppm(X, trend = ~1, clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"), data = NULL, ..., covariates=data, subset, method = c("mincon", "clik2", "palm", "adapcl"), penalised = FALSE, improve.type = c("none", "clik1", "wclik1", "quasi"), improve.args = list(), weightfun=NULL, control=list(), stabilize=TRUE, algorithm, trajectory=FALSE, statistic="K", statargs=list(), rmax = NULL, epsilon=0.01, covfunargs=NULL, use.gam=FALSE, nd=NULL, eps=NULL, ppm.improve.type=c("none", "ho", "enet"), ppm.improve.args=list())
X |
A point pattern dataset (object of class |
trend |
An R formula, with no left hand side, specifying the form of the log intensity. |
clusters |
Character string determining the cluster model.
Partially matched.
Options are |
data , covariates
|
The values of spatial covariates (other than the Cartesian coordinates) required by the model. A named list of pixel images, functions, windows, tessellations or numeric constants. |
... |
Additional arguments. See Details. |
subset |
Optional.
A subset of the spatial domain,
to which the model-fitting should be restricted.
A window (object of class |
method |
The fitting method. Either
|
penalised |
Logical value specifying whether the objective function (the composite likelihood or contrast) should be modified by adding a penalty against extreme values of cluster scale. |
improve.type |
Method for updating the initial estimate of the trend.
Initially the trend is estimated as if the process
is an inhomogeneous Poisson process.
The default, |
improve.args |
Additional arguments passed to |
weightfun |
Optional weighting function |
control |
List of control parameters passed to the optimization function
|
stabilize |
Logical value specifying whether to numerically stabilize
the optimization algorithm, by specifying suitable default values of
|
algorithm |
Character string determining the mathematical algorithm
to be used to solve the fitting problem.
If |
trajectory |
Logical value specifying whether to save the history of all function evaluations performed by the optimization algorithm. |
statistic |
Name of the summary statistic to be used
for minimum contrast estimation: either |
statargs |
Optional list of arguments to be used when calculating
the |
rmax |
Maximum value of interpoint distance to use in the composite likelihood. |
epsilon |
Tuning parameter for the adaptive composite likelihood method. |
covfunargs , use.gam , nd , eps
|
Arguments passed to |
ppm.improve.type , ppm.improve.args
|
Arguments controlling the initial fit of the trend.
Passed to |
This function fits a clustered point process model to the
point pattern dataset X
.
The model may be either a Neyman-Scott cluster process
or another Cox process.
The type of model is determined by the argument clusters
.
Currently the options
are clusters="Thomas"
for the Thomas process,
clusters="MatClust"
for the Matern cluster process,
clusters="Cauchy"
for the Neyman-Scott cluster process
with Cauchy kernel,
clusters="VarGamma"
for the Neyman-Scott cluster process
with Variance Gamma kernel (requires an additional argument nu
to be passed through the dots; see rVarGamma
for details),
and clusters="LGCP"
for the log-Gaussian Cox process (may
require additional arguments passed through ...
; see
rLGCP
for details on argument names).
The first four models are Neyman-Scott cluster processes.
The algorithm first estimates the intensity function
of the point process using ppm
.
The argument X
may be a point pattern
(object of class "ppp"
) or a quadrature scheme
(object of class "quad"
). The intensity is specified by
the trend
argument.
If the trend formula is ~1
(the default)
then the model is homogeneous. The algorithm begins by
estimating the intensity as the number of points divided by
the area of the window.
Otherwise, the model is inhomogeneous.
The algorithm begins by fitting a Poisson process with log intensity
of the form specified by the formula trend
.
(See ppm
for further explanation).
The argument X
may also be a formula
in the
R language. The right hand side of the formula gives the
trend
as described above. The left hand side of the formula
gives the point pattern dataset to which the model should be fitted.
If improve.type="none"
this is the final estimate of the
intensity. Otherwise, the intensity estimate is updated, as explained in
improve.kppm
. Additional arguments to
improve.kppm
are passed as a named list in
improve.args
.
The cluster parameters of the model are then fitted either by minimum contrast estimation, or by a composite likelihood method (maximum composite likelihood, maximum Palm likelihood, or by solving the adaptive composite likelihood estimating equation).
If method = "mincon"
(the default) clustering parameters of
the model will be fitted
by minimum contrast estimation, that is, by matching the theoretical
-function of the model to the empirical
-function
of the data, as explained in
mincontrast
.
For a homogeneous model ( trend = ~1
)
the empirical -function of the data is computed
using
Kest
,
and the parameters of the cluster model are estimated by
the method of minimum contrast.
For an inhomogeneous model,
the inhomogeneous function is estimated
by
Kinhom
using the fitted intensity.
Then the parameters of the cluster model
are estimated by the method of minimum contrast using the
inhomogeneous function. This two-step estimation
procedure is due to Waagepetersen (2007).
If statistic="pcf"
then instead of using the
-function, the algorithm will use
the pair correlation function
pcf
for homogeneous
models and the inhomogeneous pair correlation function
pcfinhom
for inhomogeneous models.
In this case, the smoothing parameters of the pair correlation
can be controlled using the argument statargs
,
as shown in the Examples.
Additional arguments ...
will be passed to
clusterfit
to control the minimum contrast fitting
algorithm.
The optimisation is performed by the generic
optimisation algorithm optim
.
If method = "clik2"
the clustering parameters of the
model will be fitted by maximising the second-order composite likelihood
(Guan, 2006). The log composite likelihood is
where the sums are taken over all pairs of data points
separated by a distance
less than
rmax
,
and the double integral is taken over all pairs of locations
in the spatial window of the data.
Here
is the
pair correlation function of the model with
cluster parameters
.
The function in the composite likelihood
is a weighting function and may be chosen arbitrarily.
It is specified by the argument
weightfun
.
If this is missing or NULL
then the default is
a threshold weight function,
, where
is
rmax/2
.
If it is specified, the argument weightfun
should
be a function
in the R language with one argument.
Alternatively weightfun
may be
one of the strings "threshold"
or "taper"
representing the functions
and
respectively.
The optimisation is performed by the generic
optimisation algorithm optim
.
If method = "palm"
the clustering parameters of the
model will be fitted by maximising the Palm loglikelihood
(Tanaka et al, 2008)
with the same notation as above. Here
is the Palm intensity of
the model at location
given there is a point at
.
The optimisation is performed by the generic
optimisation algorithm optim
.
If method = "cladap"
the clustering parameters of the
model will be fitted by solving the adaptive second order
composite likelihood estimating equation (Lavancier et al, 2021).
The estimating function is
where the sum is taken over all distinct pairs of points.
Here is the
pair correlation function with
parameters
.
The partial derivative with respect to
is
, and
denotes
the fitted intensity function of the model.
The tuning parameter is
independent of the data. It can be specified by the
argument
epsilon
and has default value .
The function in the estimating function
is a weighting function of bounded support
.
It is specified by the argument
weightfun
.
If this is missing or NULL
then the default is
The estimating equation is solved using the
nonlinear equation solver
nleqslv
from the package nleqslv. The package nleqslv
must be installed in order to use this option.
If penalised=TRUE
, the fitting procedure is modified by
adding a penalty against extreme values of the cluster scale,
as proposed by Baddeley et al (2022).
If trajectory=TRUE
, the resulting object contains the history
of all points in the cluster parameter space which were evaluated by
the optimization algorithm. The trajectory can be extracted by
traj(fit)
or traj(obsurf(fit))
where fit
is the
fitted model object.
An object of class "kppm"
representing the fitted model.
There are methods for printing, plotting, predicting, simulating
and updating objects of this class.
For Neyman-Scott models, the fitting procedure searches for the best-fitting values of the parameters that control the intensity of parents and the physical scale of the clusters. (Any parameters that control the shape of the clusters must be specified separately and are assumed to be fixed.)
The fitted object fit
contains the fitted cluster parameters as
the element fit$par
in the format described below.
Initial estimates for these cluster
parameters can be specified using the argument startpar
in the
same format.
The cluster parameters will be stored in a named numeric vector
par
of length 2. The first value is always kappa
,
the intensity of parents (cluster centres).
The format is as follows:
for clusters="Thomas"
,
a vector c(kappa, sigma2)
where
sigma2
is the square of the cluster standard deviation;
for clusters="MatClust"
,
a vector c(kappa, R)
where
R
is the radius of the cluster;
for clusters="Cauchy"
,
a vector c(kappa, eta2)
where
eta2 = code{4 * scale^2}
where scale
is the scale parameter for the model
as used in rCauchy
;
for clusters="VarGamma"
,
a vector c(kappa, eta)
where
eta
is equivalent to the
scale parameter omega
used in rVarGamma
.
For clusters="VarGamma"
it will be necessary to specify
the shape parameter nu
as described in the help for
rVarGamma
. This is specified separately as an argument
nu
in the call to kppm
.
The following details allow greater control over the fitting procedure.
For the first three fitting methods
(method="mincon", "clik2"
and "palm"
),
the optimisation is performed by the generic
optimisation algorithm optim
.
The behaviour of this algorithm can be controlled
by the following arguments to kppm
:
startpar
determines the initial estimates of the cluster parameters.
algorithm
determines the particular optimization
method. This argument is passed to optim
as the
argument method
. Options are listed in the help for
optim
. The default is the Nelder-Mead
simplex method.
control
is a named list of control parameters,
documented in the help for optim
.
Useful control arguments include
trace
, maxit
and abstol
.
lower
and upper
specify bounds for the
cluster parameters, when algorithm="L-BFGS-B"
or
algorithm="Brent"
, as described in the help for optim
.
For method="adapcl"
, the estimating equation is solved
using the nonlinear equation solver nleqslv
from the package nleqslv.
The package nleqslv must be installed in order to use this
option.
The behaviour of this algorithm can be controlled
by the following arguments to kppm
:
startpar
determines the initial estimates of the cluster parameters.
algorithm
determines the method for solving
the equation. This argument is passed to nleqslv
as the
argument method
. Options are listed in the help for
nleqslv
.
globStrat
determines the global strategy
to be applied. This argument is is passed to nleqslv
as the argument global
. Options are listed in the help for
nleqslv
.
control
is a named list of control parameters,
documented in the help for nleqslv
.
To fit a log-Gaussian Cox model,
specify clusters="LGCP"
and use additional arguments
to specify the covariance structure. These additional arguments can
be given individually in the call to kppm
, or they can be
collected together in a list called covmodel
.
For example a Matern model with parameter could be specified
either by
kppm(X, clusters="LGCP", model="matern", nu=0.5)
or by
kppm(X, clusters="LGCP", covmodel=list(model="matern", nu=0.5))
.
The argument model
specifies the type of covariance
model: the default is model="exp"
for an exponential covariance.
Additional arguments specify the shape parameters of the covariance
model. For example if model="matern"
then the additional argument
nu
is required.
The available models are as follows:
model="exponential"
:the exponential covariance function
where is the (fitted) variance parameter,
and
is the (fitted) scale parameter.
No shape parameters are required.
model="gauss"
:the Gaussian covariance function
where is the (fitted) variance parameter,
and
is the (fitted) scale parameter.
No shape parameters are required.
model="stable"
:the stable covariance function
where is the (fitted) variance parameter,
is the (fitted) scale parameter,
and
is the shape parameter
alpha
.
The parameter alpha
must be given, either as a stand-alone
argument, or as an entry in the list covmodel
.
model="gencauchy"
:the generalised Cauchy covariance function
where is the (fitted) variance parameter,
is the (fitted) scale parameter,
and
and
are the shape parameters
alpha
and beta
.
The parameters alpha
and beta
must be given, either as stand-alone arguments, or as entries
in the list covmodel
.
model="matern"
:the Whittle-Matern covariance function
where is the (fitted) variance parameter,
is the (fitted) scale parameter,
and
is the shape parameter
nu
.
The parameter nu
must be given, either as a stand-alone
argument, or as an entry in the list covmodel
.
Note that it is not possible to use anisotropic covariance models
because the kppm
technique assumes the pair correlation function
is isotropic.
See ppm.ppp
for a list of common error messages
and warnings originating from the first stage of model-fitting.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected], with contributions from Abdollah Jalilian [email protected] and Rasmus Plenge Waagepetersen [email protected]. Adaptive composite likelihood method contributed by Chiara Fend and modified by Adrian Baddeley. Penalised optimization developed by Adrian Baddeley, Tilman Davies [email protected] and Martin Hazelton [email protected].
Baddeley, A., Davies, T.M., Hazelton, M.L., Rakshit, S. and Turner,
R. (2022) Fundamental problems in fitting spatial cluster
process models. Spatial Statistics 52, 100709.
DOI: 10.1016/j.spasta.2022.100709
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502–1512.
Guan, Y., Jalilian, A. and Waagepetersen, R. (2015) Quasi-likelihood for spatial point processes. Journal of the Royal Statistical Society, Series B 77, 677-697.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2012) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119–137.
Lavancier, F., Poinas, A., and Waagepetersen, R. (2021) Adaptive estimating function inference for nonstationary determinantal point processes. Scandinavian Journal of Statistics, 48 (1), 87–107.
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43–57.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
Methods for kppm
objects:
plot.kppm
,
fitted.kppm
,
predict.kppm
,
simulate.kppm
,
update.kppm
,
vcov.kppm
,
methods.kppm
,
as.ppm.kppm
,
as.fv.kppm
,
Kmodel.kppm
,
pcfmodel.kppm
.
See also improve.kppm
for improving the fit of a
kppm
object.
Minimum contrast fitting algorithm:
higher level interface clusterfit
;
low-level algorithm mincontrast
.
Alternative fitting algorithms:
thomas.estK
,
matclust.estK
,
lgcp.estK
,
cauchy.estK
,
vargamma.estK
,
thomas.estpcf
,
matclust.estpcf
,
lgcp.estpcf
,
cauchy.estpcf
,
vargamma.estpcf
.
Summary statistics:
Kest
,
Kinhom
,
pcf
,
pcfinhom
.
For fitting Poisson or Gibbs point process models, see ppm
.
online <- interactive() if(!online) op <- spatstat.options(npixel=32, ndummy.min=16) # method for point patterns kppm(redwood, ~1, "Thomas") # method for formulas kppm(redwood ~ 1, "Thomas") # different models for clustering if(online) kppm(redwood ~ x, "MatClust") kppm(redwood ~ x, "MatClust", statistic="pcf", statargs=list(stoyan=0.2)) kppm(redwood ~ x, cluster="Cauchy", statistic="K") kppm(redwood, cluster="VarGamma", nu = 0.5, statistic="pcf") # log-Gaussian Cox process (LGCP) models kppm(redwood ~ 1, "LGCP", statistic="pcf") kppm(redwood ~ x, "LGCP", statistic="pcf", model="matern", nu=0.3, control=list(maxit=10)) # Different fitting techniques fitc <- kppm(redwood ~ 1, "Thomas", method="c") fitp <- kppm(redwood ~ 1, "Thomas", method="p") # penalised fit fitmp <- kppm(redwood ~ 1, "Thomas", penalised=TRUE) # quasi-likelihood improvement fitq <- kppm(redwood ~ x, "Thomas", improve.type = "quasi") if(!online) spatstat.options(op)
online <- interactive() if(!online) op <- spatstat.options(npixel=32, ndummy.min=16) # method for point patterns kppm(redwood, ~1, "Thomas") # method for formulas kppm(redwood ~ 1, "Thomas") # different models for clustering if(online) kppm(redwood ~ x, "MatClust") kppm(redwood ~ x, "MatClust", statistic="pcf", statargs=list(stoyan=0.2)) kppm(redwood ~ x, cluster="Cauchy", statistic="K") kppm(redwood, cluster="VarGamma", nu = 0.5, statistic="pcf") # log-Gaussian Cox process (LGCP) models kppm(redwood ~ 1, "LGCP", statistic="pcf") kppm(redwood ~ x, "LGCP", statistic="pcf", model="matern", nu=0.3, control=list(maxit=10)) # Different fitting techniques fitc <- kppm(redwood ~ 1, "Thomas", method="c") fitp <- kppm(redwood ~ 1, "Thomas", method="p") # penalised fit fitmp <- kppm(redwood ~ 1, "Thomas", penalised=TRUE) # quasi-likelihood improvement fitq <- kppm(redwood ~ x, "Thomas", improve.type = "quasi") if(!online) spatstat.options(op)
Given a point process model fitted to a point pattern dataset,
this function computes the residual function,
which serves as a diagnostic for goodness-of-fit of the model.
Kres(object, ...)
Kres(object, ...)
object |
Object to be analysed.
Either a fitted point process model (object of class |
... |
Arguments passed to |
This command provides a diagnostic for the goodness-of-fit of
a point process model fitted to a point pattern dataset.
It computes a residual version of the function of the
dataset, which should be approximately zero if the model is a good
fit to the data.
In normal use, object
is a fitted point process model
or a point pattern. Then Kres
first calls Kcom
to compute both the nonparametric estimate of the function
and its model compensator. Then
Kres
computes the
difference between them, which is the residual -function.
Alternatively, object
may be a function value table
(object of class "fv"
) that was returned by
a previous call to Kcom
. Then Kres
computes the
residual from this object.
A function value table (object of class "fv"
),
essentially a data frame of function values.
There is a plot method for this class. See fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Related functions:
Kcom
,
Kest
.
Alternative functions:
Gres
,
psstG
, psstA
, psst
.
Point process models: ppm
.
fit0 <- ppm(cells ~1) # uniform Poisson K0 <- Kres(fit0) K0 plot(K0) # isotropic-correction estimate plot(K0, ires ~ r) # uniform Poisson is clearly not correct fit1 <- ppm(cells ~1, Strauss(0.08)) K1 <- Kres(fit1) if(interactive()) { plot(K1, ires ~ r) # fit looks approximately OK; try adjusting interaction distance plot(Kres(cells, interaction=Strauss(0.12))) } # How to make envelopes E <- envelope(fit1, Kres, model=fit1, nsim=19) plot(E) # For computational efficiency Kc <- Kcom(fit1) K1 <- Kres(Kc)
fit0 <- ppm(cells ~1) # uniform Poisson K0 <- Kres(fit0) K0 plot(K0) # isotropic-correction estimate plot(K0, ires ~ r) # uniform Poisson is clearly not correct fit1 <- ppm(cells ~1, Strauss(0.08)) K1 <- Kres(fit1) if(interactive()) { plot(K1, ires ~ r) # fit looks approximately OK; try adjusting interaction distance plot(Kres(cells, interaction=Strauss(0.12))) } # How to make envelopes E <- envelope(fit1, Kres, model=fit1, nsim=19) plot(E) # For computational efficiency Kc <- Kcom(fit1) K1 <- Kres(Kc)
Computes Lambert's W-function.
LambertW(x)
LambertW(x)
x |
Vector of nonnegative numbers. |
Lambert's W-function is the inverse function of
.
That is,
is the function such that
This command LambertW
computes for each entry
in the argument
x
.
If the library gsl has been installed, then the function
lambert_W0
in that library is invoked. Otherwise,
values of the W-function are computed by root-finding, using the
function uniroot
.
Computation using gsl is about 100 times faster.
If any entries of x
are infinite or NA
, the corresponding
results are NA
.
Numeric vector.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Corless, R, Gonnet, G, Hare, D, Jeffrey, D and Knuth, D (1996), On the Lambert W function. Computational Mathematics, 5, 325–359.
Roy, R and Olver, F (2010), Lambert W function. In Olver, F, Lozier, D and Boisvert, R (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press.
LambertW(exp(1))
LambertW(exp(1))
Creates the Lennard-Jones pairwise interaction structure which can then be fitted to point pattern data.
LennardJones(sigma0=NA)
LennardJones(sigma0=NA)
sigma0 |
Optional. Initial estimate of the parameter |
In a pairwise interaction point process with the
Lennard-Jones pair potential (Lennard-Jones, 1924)
each pair of points in the point pattern,
a distance apart,
contributes a factor
to the probability density,
where and
are
positive parameters to be estimated.
See Examples for a plot of this expression.
This potential causes very strong inhibition between points at short
range, and attraction between points at medium range.
The parameter is called the
characteristic diameter and controls the scale of interaction.
The parameter
is called the well depth
and determines the strength of attraction.
The potential switches from inhibition to attraction at
.
The maximum value of the pair potential is
occuring at distance
.
Interaction is usually considered to be negligible for distances
.
This potential is used to model interactions between uncharged molecules in statistical physics.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Lennard-Jones pairwise interaction is
yielded by the function LennardJones()
.
See the examples below.
An object of class "interact"
describing the Lennard-Jones interpoint interaction
structure.
To avoid numerical instability,
the interpoint distances d
are rescaled
when fitting the model.
Distances are rescaled by dividing by sigma0
.
In the formula for above,
the interpoint distance
will be replaced by
d/sigma0
.
The rescaling happens automatically by default.
If the argument sigma0
is missing or NA
(the default),
then sigma0
is taken to be the minimum
nearest-neighbour distance in the data point pattern (in the
call to ppm
).
If the argument sigma0
is given, it should be a positive
number, and it should be a rough estimate of the
parameter .
The “canonical regular parameters” estimated by ppm
are
and
.
Fitting the Lennard-Jones model is extremely unstable, because
of the strong dependence between the functions
and
. The fitting algorithm often fails to
converge. Try increasing the number of
iterations of the GLM fitting algorithm, by setting
gcontrol=list(maxit=1e3)
in the call to ppm
.
Errors are likely to occur if this model is fitted to a point pattern dataset
which does not exhibit both short-range inhibition and
medium-range attraction between points. The values of the parameters
and
may be
NA
(because the fitted canonical parameters have opposite sign, which
usually occurs when the pattern is completely random).
An absence of warnings does not mean that the fitted model is sensible.
A negative value of may be obtained (usually when
the pattern is strongly clustered); this does not correspond
to a valid point process model, but the software does not issue a warning.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Lennard-Jones, J.E. (1924) On the determination of molecular fields. Proc Royal Soc London A 106, 463–477.
ppm
,
pairwise.family
,
ppm.object
badfit <- ppm(cells ~1, LennardJones(), rbord=0.1) badfit fit <- ppm(unmark(longleaf) ~1, LennardJones(), rbord=1) fit plot(fitin(fit)) # Note the Longleaf Pines coordinates are rounded to the nearest decimetre # (multiple of 0.1 metres) so the apparent inhibition may be an artefact
badfit <- ppm(cells ~1, LennardJones(), rbord=0.1) badfit fit <- ppm(unmark(longleaf) ~1, LennardJones(), rbord=1) fit plot(fitin(fit)) # Note the Longleaf Pines coordinates are rounded to the nearest decimetre # (multiple of 0.1 metres) so the apparent inhibition may be an artefact
Computes the leverage measure for a fitted spatial point process model.
leverage(model, ...) ## S3 method for class 'ppm' leverage(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
leverage(model, ...) ## S3 method for class 'ppm' leverage(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
model |
Fitted point process model (object of class |
... |
Ignored, except for the arguments |
drop |
Logical. Whether to include ( |
iScore , iHessian
|
Components of the score vector and Hessian matrix for the irregular parameters, if required. See Details. |
iArgs |
List of extra arguments for the functions |
The function leverage
is generic, and
leverage.ppm
is the method for objects of class "ppm"
.
Given a fitted spatial point process model model
,
the function leverage.ppm
computes the leverage of the model,
described in Baddeley, Chang and Song (2013)
and Baddeley, Rubak and Turner (2019).
The leverage of a spatial point process model
is a function of spatial location, and is typically
displayed as a colour pixel image.
The leverage value at a spatial location
represents the
change in the fitted trend of the fitted point process model that would have
occurred if a data point were to have occurred at the location
.
A relatively large value of
indicates a
part of the space where the data have a potentially
strong effect on the fitted model (specifically, a strong effect
on the intensity or conditional intensity of the fitted model)
due to the values of the covariates.
If the point process model trend has irregular parameters that were
fitted (using ippm
)
then the leverage calculation requires the first and second
derivatives of the log trend with respect to the irregular parameters.
The argument iScore
should be a list,
with one entry for each irregular parameter, of R functions that compute the
partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular
parameter. The argument iHessian
should be a list,
with entries where
is the number of irregular
parameters, of R functions that compute the second order
partial derivatives of the log trend with respect to each
pair of irregular parameters.
The result of leverage.ppm
is an object of
class "leverage.ppm"
. It can be printed or plotted.
It can be converted to a pixel image
by as.im
(see as.im.leverage.ppm
).
There are also methods for contour
, persp
,
[
, as.function
,
as.owin
, domain
,
Smooth
,
integral
, and mean
.
An object of class "leverage.ppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
Baddeley, A., Rubak, E. and Turner, R. (2019) Leverage and influence diagnostics for Gibbs spatial point processes. Spatial Statistics 29, 15–48.
influence.ppm
,
dfbetas.ppm
,
ppmInfluence
,
plot.leverage.ppm
as.function.leverage.ppm
if(offline <- !interactive()) op <- spatstat.options(npixel=32, ndummy.min=16) X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) le <- leverage(fit) if(!offline) plot(le) mean(le) if(offline) spatstat.options(op)
if(offline <- !interactive()) op <- spatstat.options(npixel=32, ndummy.min=16) X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) le <- leverage(fit) if(!offline) plot(le) mean(le) if(offline) spatstat.options(op)
For a fitted spatial logistic regression model, these functions compute diagnostics of leverage and influence.
## S3 method for class 'slrm' leverage(model, ...) ## S3 method for class 'slrm' influence(model, ...) ## S3 method for class 'slrm' dfbetas(model, ...) ## S3 method for class 'slrm' dffit(object, ...)
## S3 method for class 'slrm' leverage(model, ...) ## S3 method for class 'slrm' influence(model, ...) ## S3 method for class 'slrm' dfbetas(model, ...) ## S3 method for class 'slrm' dffit(object, ...)
model , object
|
A fitted spatial logistic regression model
(object of class |
... |
Arguments passed to methods. |
These functions are methods for the generics
leverage
, influence
,
dfbetas
and dffit
for the class "slrm"
.
These functions adapt the standard diagnostics for logistic regression
(see influence.measures
) to a
fitted spatial logistic regression model (object of class
"slrm"
). This adaptation was described by Baddeley, Chang
and Song (2013).
leverage.slrm
computes the leverage value (diagonal of the hat
matrix) for the covariate data in each pixel.
The result is a pixel image.
influence.slrm
computes the likelihood influence
for the data (covariates and presence/absence of points) in each pixel.
The result is a pixel image.
dfbetas.slrm
computes the parameter influence
for the data (covariates and presence/absence of points) in each pixel.
The result is a list of pixel images, one image for each of the
model coefficients in coef(model)
. The list can be plotted
immediately.
dffit.slrm
computes the total influence
for the data (covariates and presence/absence of points) in each pixel.
The result is a pixel image.
A pixel image, or a list of pixel images.
Adrian Baddeley [email protected].
Baddeley, A., Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
leverage.ppm
,
influence.ppm
,
dfbetas.ppm
,
dffit.ppm
H <- unmark(humberside) fit <- slrm(H ~ x+y, dimyx=32) plot(leverage(fit)) plot(influence(fit)) plot(dfbetas(fit)) plot(dffit(fit))
H <- unmark(humberside) fit <- slrm(H ~ x+y, dimyx=32) plot(leverage(fit)) plot(influence(fit)) plot(dfbetas(fit)) plot(dffit(fit))
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast.
lgcp.estK(X, startpar=c(var=1,scale=1), covmodel=list(model="exponential"), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
lgcp.estK(X, startpar=c(var=1,scale=1), covmodel=list(model="exponential"), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the log-Gaussian Cox process model. |
covmodel |
Specification of the covariance model for the log-Gaussian field. See Details. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by the Method of Minimum Contrast, using the K function of the point pattern.
The shape of the covariance of the LGCP must be specified: the default is the exponential covariance function, but other covariance models can be selected.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The function of the point pattern will be computed
using
Kest
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits a log-Gaussian Cox point process (LGCP)
model to X
, by finding the parameters of the LGCP model
which give the closest match between the
theoretical function of the LGCP model
and the observed
function.
For a more detailed explanation of the Method of Minimum Contrast,
see
mincontrast
.
The model fitted is a stationary, isotropic log-Gaussian Cox process
(Moller and Waagepetersen, 2003, pp. 72-76).
To define this process we start with
a stationary Gaussian random field in the two-dimensional plane,
with constant mean
and covariance function
.
Given
, we generate a Poisson point process
with intensity
function
at
location
. Then
is a log-Gaussian Cox process.
The -function of the LGCP is
The intensity of the LGCP is
The covariance function is parametrised in the form
where and
are parameters
controlling the strength and the scale of autocorrelation,
respectively, and
is a known covariance function
determining the shape of the covariance.
The strength and scale parameters
and
will be estimated by the algorithm as the values
var
and scale
respectively.
The template covariance function must be specified
as explained below.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
The template covariance function is specified
using the argument
covmodel
. This should be of the form
list(model="modelname", ...)
where
modelname
is a string identifying the template model
as explained below, and ...
are optional arguments of the
form tag=value
giving the values of parameters controlling the
shape of the template model.
The default is the exponential covariance
so that the scaled covariance is
For a list of available models see kppm
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
This function is considerably slower than lgcp.estpcf
because of the computation time required for the integral
in the -function.
Computation can be accelerated, at the cost of less accurate results,
by setting spatstat.options(fastK.lgcp=TRUE)
.
Rasmus Plenge Waagepetersen [email protected]. Adapted for spatstat by Adrian Baddeley [email protected]. Further modifications by Rasmus Waagepetersen and Shen Guochun, and by Ege Rubak [email protected].
Moller, J, Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
and
lgcp.estpcf
for alternative methods of fitting LGCP.
matclust.estK
,
thomas.estK
for other models.
mincontrast
for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
Kest
for the function.
if(interactive()) { u <- lgcp.estK(redwood) print(u) plot(u) } else { # faster - better starting point u <- lgcp.estK(redwood, c(var=1.05, scale=0.1)) } if(FALSE) { ## takes several minutes! lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3)) }
if(interactive()) { u <- lgcp.estK(redwood) print(u) plot(u) } else { # faster - better starting point u <- lgcp.estK(redwood, c(var=1.05, scale=0.1)) } if(FALSE) { ## takes several minutes! lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3)) }
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.
lgcp.estpcf(X, startpar=c(var=1,scale=1), covmodel=list(model="exponential"), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
lgcp.estpcf(X, startpar=c(var=1,scale=1), covmodel=list(model="exponential"), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the log-Gaussian Cox process model. |
covmodel |
Specification of the covariance model for the log-Gaussian field. See Details. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by the Method of Minimum Contrast, using the estimated pair correlation function of the point pattern.
The shape of the covariance of the LGCP must be specified: the default is the exponential covariance function, but other covariance models can be selected.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf
or one of its relatives.
The algorithm fits a log-Gaussian Cox point process (LGCP)
model to X
, by finding the parameters of the LGCP model
which give the closest match between the
theoretical pair correlation function of the LGCP model
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The model fitted is a stationary, isotropic log-Gaussian Cox process
(Moller and Waagepetersen, 2003, pp. 72-76).
To define this process we start with
a stationary Gaussian random field in the two-dimensional plane,
with constant mean
and covariance function
.
Given
, we generate a Poisson point process
with intensity
function
at
location
. Then
is a log-Gaussian Cox process.
The theoretical pair correlation function of the LGCP is
The intensity of the LGCP is
The covariance function takes the form
where and
are parameters
controlling the strength and the scale of autocorrelation,
respectively, and
is a known covariance function
determining the shape of the covariance.
The strength and scale parameters
and
will be estimated by the algorithm.
The template covariance function
must be specified
as explained below.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
The template covariance function is specified
using the argument
covmodel
. This should be of the form
list(model="modelname", ...)
where
modelname
is a string identifying the template model
as explained below, and ...
are optional arguments of the
form tag=value
giving the values of parameters controlling the
shape of the template model.
The default is the exponential covariance
so that the scaled covariance is
For a list of available models see kppm
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Adrian Baddeley [email protected] with modifications by Shen Guochun and Rasmus Plenge Waagepetersen [email protected] and Ege Rubak [email protected].
Moller, J., Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
and lgcp.estK
for alternative methods of fitting LGCP.
matclust.estpcf
,
thomas.estpcf
for other models.
mincontrast
for the generic minimum contrast
fitting algorithm, including important parameters that affect
the accuracy of the fit.
pcf
for the pair correlation function.
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1)) u plot(u) lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1)) u plot(u) lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
Extracts the log Palm likelihood, deviance, and AIC of a fitted determinantal point process model.
## S3 method for class 'dppm' logLik(object, ...) ## S3 method for class 'dppm' AIC(object, ..., k=2) ## S3 method for class 'dppm' extractAIC(fit, scale=0, k=2, ...) ## S3 method for class 'dppm' nobs(object, ...)
## S3 method for class 'dppm' logLik(object, ...) ## S3 method for class 'dppm' AIC(object, ..., k=2) ## S3 method for class 'dppm' extractAIC(fit, scale=0, k=2, ...) ## S3 method for class 'dppm' nobs(object, ...)
object , fit
|
Fitted point process model.
An object of class |
... |
Ignored. |
scale |
Ignored. |
k |
Numeric value specifying the weight of the equivalent degrees of freedom in the AIC. See Details. |
These functions are methods for the generic commands
logLik
,
extractAIC
and
nobs
for the class "dppm"
.
An object of class "dppm"
represents a fitted
Cox or cluster point process model.
It is obtained from the model-fitting function dppm
.
These methods apply only when the model was fitted
by maximising the Palm likelihood (Tanaka et al, 2008)
by calling dppm
with the argument method="palm"
.
The method logLik.dppm
computes the
maximised value of the log Palm likelihood for the fitted model object
.
The methods AIC.dppm
and extractAIC.dppm
compute the
Akaike Information Criterion AIC for the fitted model
based on the Palm likelihood (Tanaka et al, 2008)
where is the maximised Palm likelihood of the fitted model,
and
is the effective degrees of freedom
of the model.
The method nobs.dppm
returns the number of points
in the original data point pattern to which the model was fitted.
The R function step
uses these methods, but it does
not work for determinantal models yet due to a missing implementation
of update.dppm
.
logLik
returns a numerical value, belonging to the class
"logLik"
, with an attribute "df"
giving the degrees of
freedom.
AIC
returns a numerical value.
extractAIC
returns a numeric vector of length 2
containing the degrees of freedom and the AIC value.
nobs
returns an integer value.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43–57.
fit <- dppm(swedishpines ~ x, dppGauss(), method="palm") nobs(fit) logLik(fit) extractAIC(fit) AIC(fit)
fit <- dppm(swedishpines ~ x, dppGauss(), method="palm") nobs(fit) logLik(fit) extractAIC(fit) AIC(fit)
Extracts the log composite likelihood, deviance, and AIC of a fitted Cox or cluster point process model.
## S3 method for class 'kppm' logLik(object, ...) ## S3 method for class 'kppm' AIC(object, ..., k=2) ## S3 method for class 'kppm' extractAIC(fit, scale=0, k=2, ...) ## S3 method for class 'kppm' nobs(object, ...)
## S3 method for class 'kppm' logLik(object, ...) ## S3 method for class 'kppm' AIC(object, ..., k=2) ## S3 method for class 'kppm' extractAIC(fit, scale=0, k=2, ...) ## S3 method for class 'kppm' nobs(object, ...)
object , fit
|
Fitted point process model.
An object of class |
... |
Ignored. |
scale |
Ignored. |
k |
Numeric value specifying the weight of the equivalent degrees of freedom in the AIC. See Details. |
These functions are methods for the generic commands
logLik
,
extractAIC
and
nobs
for the class "kppm"
.
An object of class "kppm"
represents a fitted
Cox or cluster point process model.
It is obtained from the model-fitting function kppm
.
These methods apply only when the model was fitted
by maximising a composite likelihood:
either the Palm likelihood (Tanaka et al, 2008)
or the second order composite likelihood (Guan, 2006),
by calling kppm
with the argument method="palm"
or method="clik2"
respectively.
The method logLik.kppm
computes the
maximised value of the log composite likelihood
for the fitted model object
.
The methods AIC.kppm
and extractAIC.kppm
compute the
Akaike Information Criterion AIC for the fitted model
based on the composite likelihood
where is the maximised composite likelihood of the fitted model,
and
is the effective degrees of freedom
of the model.
The method nobs.kppm
returns the number of points
in the original data point pattern to which the model was fitted.
The R function step
uses these methods.
logLik
returns a numerical value, belonging to the class
"logLik"
, with an attribute "df"
giving the degrees of
freedom.
AIC
returns a numerical value.
extractAIC
returns a numeric vector of length 2
containing the degrees of freedom and the AIC value.
nobs
returns an integer value.
The values of log-likelihood and AIC returned by these functions
are based on the composite likelihood of the cluster process or Cox
process model. They are available only when the model was fitted
using method="palm"
or method="clik2"
.
For model comparison and model selection,
it is valid to compare the logLik
values,
or to compare the AIC
values, but only when
all the models are of class "kppm"
and were fitted using the same method
.
For method="palm"
some theoretical justification was provided by
Tanaka et al (2008).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal of the American Statistical Association 101, 1502–1512.
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott point processes. Biometrical Journal 50, 43–57.
fit <- kppm(redwood ~ x, "Thomas", method="palm") nobs(fit) logLik(fit) extractAIC(fit) AIC(fit) step(fit)
fit <- kppm(redwood ~ x, "Thomas", method="palm") nobs(fit) logLik(fit) extractAIC(fit) AIC(fit) step(fit)
For a point process model that has been fitted to multiple point patterns, these functions extract the log likelihood and AIC, or analogous quantities based on the pseudolikelihood.
## S3 method for class 'mppm' logLik(object, ..., warn=TRUE) ## S3 method for class 'mppm' AIC(object, ..., k=2, takeuchi=TRUE) ## S3 method for class 'mppm' extractAIC(fit, scale = 0, k = 2, ..., takeuchi = TRUE) ## S3 method for class 'mppm' nobs(object, ...) ## S3 method for class 'mppm' getCall(x, ...) ## S3 method for class 'mppm' terms(x, ...)
## S3 method for class 'mppm' logLik(object, ..., warn=TRUE) ## S3 method for class 'mppm' AIC(object, ..., k=2, takeuchi=TRUE) ## S3 method for class 'mppm' extractAIC(fit, scale = 0, k = 2, ..., takeuchi = TRUE) ## S3 method for class 'mppm' nobs(object, ...) ## S3 method for class 'mppm' getCall(x, ...) ## S3 method for class 'mppm' terms(x, ...)
object , fit , x
|
Fitted point process model (fitted to multiple point
patterns). An object of class |
... |
Ignored. |
warn |
If |
scale |
Ignored. |
k |
Numeric value specifying the weight of the equivalent degrees of freedom in the AIC. See Details. |
takeuchi |
Logical value specifying whether to use the Takeuchi penalty
( |
These functions are methods for the generic commands
logLik
,
AIC
,
extractAIC
,
terms
and
getCall
for the class "mppm"
.
An object of class "mppm"
represents a fitted
Poisson or Gibbs point process model fitted to several point patterns.
It is obtained from the model-fitting function mppm
.
The method logLik.mppm
extracts the
maximised value of the log likelihood for the fitted model
(as approximated by quadrature using the Berman-Turner approximation).
If object
is not a Poisson process, the maximised log
pseudolikelihood is returned, with a warning.
The Akaike Information Criterion AIC for a fitted model is defined as
where is the maximised likelihood of the fitted model,
and
is a penalty for model complexity,
usually equal to the effective degrees of freedom of the model.
The method
extractAIC.mppm
returns the analogous quantity
in which
is replaced by
,
the quadrature approximation
to the likelihood (if
fit
is a Poisson model)
or the pseudolikelihood (if fit
is a Gibbs model).
The term is calculated
as follows. If
takeuchi=FALSE
then is
the number of fitted parameters. If
takeuchi=TRUE
then
where
and
are the estimated variance and hessian,
respectively, of the composite score.
These two choices are equivalent for a Poisson process.
The method nobs.mppm
returns the total number of points
in the original data point patterns to which the model was fitted.
The method getCall.mppm
extracts the original call to
mppm
which caused the model to be fitted.
The method terms.mppm
extracts the covariate terms in the
model formula as a terms
object. Note that these terms do not
include the interaction component of the model.
The R function step
uses these methods.
See the help files for the corresponding generic functions.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders)) logLik(fit) AIC(fit) nobs(fit) getCall(fit)
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders)) logLik(fit) AIC(fit) nobs(fit) getCall(fit)
Extracts the log likelihood, deviance, and AIC of a fitted Poisson point process model, or analogous quantities based on the pseudolikelihood or logistic likelihood for a fitted Gibbs point process model.
## S3 method for class 'ppm' logLik(object, ..., new.coef=NULL, warn=TRUE, absolute=FALSE) ## S3 method for class 'ppm' deviance(object, ...) ## S3 method for class 'ppm' AIC(object, ..., k=2, takeuchi=TRUE) ## S3 method for class 'ppm' extractAIC(fit, scale=0, k=2, ..., takeuchi=TRUE) ## S3 method for class 'ppm' nobs(object, ...)
## S3 method for class 'ppm' logLik(object, ..., new.coef=NULL, warn=TRUE, absolute=FALSE) ## S3 method for class 'ppm' deviance(object, ...) ## S3 method for class 'ppm' AIC(object, ..., k=2, takeuchi=TRUE) ## S3 method for class 'ppm' extractAIC(fit, scale=0, k=2, ..., takeuchi=TRUE) ## S3 method for class 'ppm' nobs(object, ...)
object , fit
|
Fitted point process model.
An object of class |
... |
Ignored. |
warn |
If |
absolute |
Logical value indicating whether to include constant terms in the loglikelihood. |
scale |
Ignored. |
k |
Numeric value specifying the weight of the equivalent degrees of freedom in the AIC. See Details. |
new.coef |
New values for the canonical parameters of the model.
A numeric vector of the same length as |
takeuchi |
Logical value specifying whether to use the Takeuchi penalty
( |
These functions are methods for the generic commands
logLik
,
deviance
,
extractAIC
and
nobs
for the class "ppm"
.
An object of class "ppm"
represents a fitted
Poisson or Gibbs point process model.
It is obtained from the model-fitting function ppm
.
The method logLik.ppm
computes the
maximised value of the log likelihood for the fitted model object
(as approximated by quadrature using the Berman-Turner approximation)
is extracted. If object
is not a Poisson process, the maximised log
pseudolikelihood is returned, with a warning (if warn=TRUE
).
The Akaike Information Criterion AIC for a fitted model is defined as
where is the maximised likelihood of the fitted model,
and
is a penalty for model complexity,
usually equal to the effective degrees of freedom of the model.
The method
extractAIC.ppm
returns the analogous quantity
in which
is replaced by
,
the quadrature approximation
to the likelihood (if
fit
is a Poisson model)
or the pseudolikelihood or logistic likelihood
(if fit
is a Gibbs model).
The term is calculated
as follows. If
takeuchi=FALSE
then is
the number of fitted parameters. If
takeuchi=TRUE
then
where
and
are the estimated variance and hessian,
respectively, of the composite score.
These two choices are equivalent for a Poisson process.
The method nobs.ppm
returns the number of points
in the original data point pattern to which the model was fitted.
The R function step
uses these methods.
logLik
returns a numerical value, belonging to the class
"logLik"
, with an attribute "df"
giving the degrees of
freedom.
AIC
returns a numerical value.
extractAIC
returns a numeric vector of length 2
containing the degrees of freedom and the AIC value.
nobs
returns an integer value.
The values of logLik
and AIC
returned by these functions
are based on the pseudolikelihood of the Gibbs point process
model. If the model is a Poisson process, then the pseudolikelihood
is the same as the likelihood, but for other Gibbs models,
the pseudolikelihood is different from the likelihood (and the
likelihood of a Gibbs model is hard to compute).
For model comparison and model selection,
it is valid to compare the logLik
values,
or to compare the AIC
values, but only when
all the models are of class "ppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Varin, C. and Vidoni, P. (2005) A note on composite likelihood inference and model selection. Biometrika 92, 519–528.
ppm
,
as.owin
,
anova.ppm
,
coef.ppm
,
fitted.ppm
,
formula.ppm
,
model.frame.ppm
,
model.matrix.ppm
,
plot.ppm
,
predict.ppm
,
residuals.ppm
,
simulate.ppm
,
summary.ppm
,
terms.ppm
,
update.ppm
,
vcov.ppm
.
fit <- ppm(cells, ~x) nobs(fit) logLik(fit) deviance(fit) extractAIC(fit) AIC(fit) step(fit)
fit <- ppm(cells, ~x) nobs(fit) logLik(fit) deviance(fit) extractAIC(fit) AIC(fit) step(fit)
Computes the (maximised) loglikelihood of a fitted Spatial Logistic Regression model.
## S3 method for class 'slrm' logLik(object, ..., adjust = TRUE)
## S3 method for class 'slrm' logLik(object, ..., adjust = TRUE)
object |
a fitted spatial logistic regression model.
An object of class |
... |
Ignored. |
adjust |
Logical value indicating whether to adjust the loglikelihood of the model to make it comparable with a point process likelihood. See Details. |
This is a method for logLik
for fitted spatial logistic
regression models (objects of class "slrm"
, usually obtained
from the function slrm
). It computes the log-likelihood
of a fitted spatial logistic regression model.
If adjust=FALSE
, the loglikelihood is computed
using the standard formula for the loglikelihood of a
logistic regression model for a finite set of (pixel) observations.
If adjust=TRUE
then the loglikelihood is adjusted so that it
is approximately comparable with the likelihood of a point process
in continuous space, by subtracting the value
where
is the number of points in the original point pattern
dataset, and
is the area of one pixel.
A numerical value.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
X <- rpoispp(42) fit <- slrm(X ~ x+y) logLik(fit) logLik(fit, adjust=FALSE)
X <- rpoispp(42) fit <- slrm(X ~ x+y) logLik(fit) logLik(fit, adjust=FALSE)
Plot spatial point process residuals against a covariate
lurking(object, ...) ## S3 method for class 'ppm' lurking(object, covariate, type="eem", cumulative=TRUE, ..., plot.it = TRUE, plot.sd = is.poisson(object), clipwindow=default.clipwindow(object), rv = NULL, envelope=FALSE, nsim=39, nrank=1, typename, covname, oldstyle=FALSE, check=TRUE, verbose=TRUE, nx=128, splineargs=list(spar=0.5), internal=NULL) ## S3 method for class 'ppp' lurking(object, covariate, type="eem", cumulative=TRUE, ..., plot.it = TRUE, plot.sd = is.poisson(object), clipwindow=default.clipwindow(object), rv = NULL, envelope=FALSE, nsim=39, nrank=1, typename, covname, oldstyle=FALSE, check=TRUE, verbose=TRUE, nx=128, splineargs=list(spar=0.5), internal=NULL)
lurking(object, ...) ## S3 method for class 'ppm' lurking(object, covariate, type="eem", cumulative=TRUE, ..., plot.it = TRUE, plot.sd = is.poisson(object), clipwindow=default.clipwindow(object), rv = NULL, envelope=FALSE, nsim=39, nrank=1, typename, covname, oldstyle=FALSE, check=TRUE, verbose=TRUE, nx=128, splineargs=list(spar=0.5), internal=NULL) ## S3 method for class 'ppp' lurking(object, covariate, type="eem", cumulative=TRUE, ..., plot.it = TRUE, plot.sd = is.poisson(object), clipwindow=default.clipwindow(object), rv = NULL, envelope=FALSE, nsim=39, nrank=1, typename, covname, oldstyle=FALSE, check=TRUE, verbose=TRUE, nx=128, splineargs=list(spar=0.5), internal=NULL)
object |
The fitted point process model (an object of class |
covariate |
The covariate against which residuals should be plotted.
Either a numeric vector, a pixel image, or an |
type |
String indicating the type of residuals or weights to be computed.
Choices include |
cumulative |
Logical flag indicating whether to plot a
cumulative sum of marks ( |
... |
Arguments passed to |
plot.it |
Logical value indicating whether
plots should be shown. If |
plot.sd |
Logical value indicating whether
error bounds should be added to plot.
The default is |
clipwindow |
If not |
rv |
Usually absent.
If this argument is present, the point process residuals will not be
calculated from the fitted model |
envelope |
Logical value indicating whether to compute simulation envelopes
for the plot. Alternatively |
nsim |
Number of simulated point patterns to be generated
to produce the simulation envelope, if |
nrank |
Integer. Rank of the envelope value amongst the |
typename |
Usually absent. If this argument is present, it should be a string, and will be used (in the axis labels of plots) to describe the type of residuals. |
covname |
A string name for the covariate, to be used in axis labels of plots. |
oldstyle |
Logical flag indicating whether error bounds should be plotted
using the approximation given in the original paper
( |
check |
Logical flag indicating whether the integrity of the data structure
in |
verbose |
Logical value indicating whether to print progress reports during Monte Carlo simulation. |
nx |
Integer. Number of covariate values to be used in the plot. |
splineargs |
A list of arguments passed to |
internal |
Internal use only. |
This function generates a ‘lurking variable’ plot for a
fitted point process model.
Residuals from the model represented by object
are plotted against the covariate specified by covariate
.
This plot can be used to reveal departures from the fitted model,
in particular, to reveal that the point pattern depends on the covariate.
The function lurking
is generic, with methods for
ppm
and ppp
documented here, and possibly other methods.
The argument object
would usually be a fitted point process model
(object of class "ppm"
) produced by the
model-fitting algorithm ppm
). If
object
is a point pattern (object of class "ppp"
) then
the model is taken to be the uniform Poisson process (Complete
Spatial Randomness) fitted to this point pattern.
First the residuals from the fitted model (Baddeley et al, 2004)
are computed at each quadrature point,
or alternatively the ‘exponential energy marks’ (Stoyan and Grabarnik,
1991) are computed at each data point.
The argument type
selects the type of
residual or weight. See diagnose.ppm
for options
and explanation.
A lurking variable plot for point processes (Baddeley et al, 2004)
displays either the cumulative sum of residuals/weights
(if cumulative = TRUE
) or a kernel-weighted average of the
residuals/weights (if cumulative = FALSE
) plotted against
the covariate. The empirical plot (solid lines) is shown
together with its expected value assuming the model is true
(dashed lines) and optionally also the pointwise
two-standard-deviation limits (grey shading).
To be more precise, let denote the value of the covariate
at a spatial location
.
If cumulative=TRUE
then we plot against
,
where
is the sum of the residuals
over all quadrature points where the covariate takes
a value less than or equal to
, or the sum of the
exponential energy weights over all data points where the covariate
takes a value less than or equal to
.
If cumulative=FALSE
then we plot against
,
where
is the derivative of
,
computed approximately by spline smoothing.
For the point process residuals ,
while for the exponential energy weights
area of the subset of the window
satisfying
.
If the empirical and theoretical curves deviate substantially from one another, the interpretation is that the fitted model does not correctly account for dependence on the covariate. The correct form (of the spatial trend part of the model) may be suggested by the shape of the plot.
If plot.sd = TRUE
, then superimposed on the lurking variable
plot are the pointwise
two-standard-deviation error limits for calculated for the
inhomogeneous Poisson process. The default is
plot.sd = TRUE
for Poisson models and plot.sd = FALSE
for non-Poisson
models.
By default, the two-standard-deviation limits are calculated
from the exact formula for the asymptotic variance
of the residuals under the asymptotic normal approximation,
equation (37) of Baddeley et al (2006).
However, for compatibility with the original paper
of Baddeley et al (2005), if oldstyle=TRUE
,
the two-standard-deviation limits are calculated
using the innovation variance, an over-estimate of the true
variance of the residuals.
The argument covariate
is either a numeric vector, a pixel
image, or an R language expression.
If it is a numeric vector, it is assumed to contain
the values of the covariate for each of the quadrature points
in the fitted model. The quadrature points can be extracted by
quad.ppm(object)
.
If covariate
is a pixel image, it is assumed to contain the
values of the covariate at each location in the window. The values of
this image at the quadrature points will be extracted.
Alternatively, if covariate
is an expression
, it will be evaluated in the same environment
as the model formula used in fitting the model object
. It must
yield a vector of the same length as the number of quadrature points.
The expression may contain the terms x
and y
representing the
cartesian coordinates, and may also contain other variables that were
available when the model was fitted. Certain variable names are
reserved words; see ppm
.
Note that lurking variable plots for the and
coordinates
are also generated by
diagnose.ppm
, amongst other
types of diagnostic plots. This function is more general in that it
enables the user to plot the residuals against any chosen covariate
that may have been present.
For advanced use, even the values of the residuals/weights
can be altered. If the argument rv
is present,
the residuals will not be calculated from the fitted model
object
but will instead be taken directly from the object rv
.
If type = "eem"
then rv
should be similar to the
return value of eem
, namely, a numeric vector
with length equal to the number of data points in the original point
pattern. Otherwise, rv
should be
similar to the return value of residuals.ppm
,
that is, rv
should be an object of class
"msr"
(see msr
) representing a signed measure.
The (invisible) return value is an object
belonging to the class "lurk"
, for which there
are methods for plot
and print
.
This object is a list containing two dataframes
empirical
and theoretical
.
The first dataframe empirical
contains columns
covariate
and value
giving the coordinates of the
lurking variable plot. The second dataframe theoretical
contains columns covariate
, mean
and sd
giving the coordinates of the plot of the theoretical mean
and standard deviation.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2006) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
residuals.ppm
,
diagnose.ppm
,
residuals.ppm
,
qqplot.ppm
,
eem
,
ppm
(a <- lurking(nztrees, expression(x), type="raw")) fit <- ppm(nztrees ~x, Poisson(), nd=128) (b <- lurking(fit, expression(x), type="raw")) lurking(fit, expression(x), type="raw", cumulative=FALSE)
(a <- lurking(nztrees, expression(x), type="raw")) fit <- ppm(nztrees ~x, Poisson(), nd=128) (b <- lurking(fit, expression(x), type="raw")) lurking(fit, expression(x), type="raw", cumulative=FALSE)
Generate a lurking variable plot of spatial point process residuals against a covariate, for a model fitted to several point patterns.
## S3 method for class 'mppm' lurking(object, covariate, type="eem", ..., separate = FALSE, plot.it = TRUE, covname, oldstyle = FALSE, nx = 512, main="")
## S3 method for class 'mppm' lurking(object, covariate, type="eem", ..., separate = FALSE, plot.it = TRUE, covname, oldstyle = FALSE, nx = 512, main="")
object |
The fitted model. An object of class |
covariate |
The covariate to be used on the horizontal axis.
Either an |
type |
String indicating the type of residuals or weights to be computed.
Choices include |
... |
Additional arguments passed to |
separate |
Logical value indicating whether to compute a separate lurking
variable plot for each of the original point patterns.
If |
plot.it |
Logical value indicating whether
plots should be shown. If |
covname |
A string name for the covariate, to be used in axis labels of plots. |
oldstyle |
Logical flag indicating whether error bounds should be plotted
using the approximation given in the original paper
( |
nx |
Integer. Number of covariate values to be used in the plot. |
main |
Character string giving a main title for the plot. |
This function generates a ‘lurking variable’ plot for a
point process model fitted to several point patterns.
Residuals from the model represented by object
are plotted against the covariate specified by covariate
.
This plot can be used to reveal departures from the fitted model.
The function lurking
is generic. This is the method
for the class mppm
.
The argument object
must be a fitted point process model
object of class "mppm"
) produced by the
model-fitting algorithm mppm
.
If separate=FALSE
(the default),
the return value is an object
belonging to the class "lurk"
, for which there
are methods for plot
and print
.
See lurking
for details of the format.
If separate=TRUE
, the result is a list of such objects,
and also belongs to the class anylist
so that it can be
printed and plotted.
Adrian Baddeley [email protected], with thanks to Nicholas Read.
fit <- mppm(Points ~ Image + Group, demohyper) lurking(fit, expression(Image), type="P") lurking(fit, expression(Image), type="P", separate=TRUE)
fit <- mppm(Points ~ Image + Group, demohyper) lurking(fit, expression(Image), type="P") lurking(fit, expression(Image), type="P", separate=TRUE)
Fits the Matern Cluster point process to a point pattern dataset by the Method of Minimum Contrast.
matclust.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
matclust.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the Matern Cluster model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the Matern Cluster process. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
This algorithm fits the Matern Cluster point process model
to a point pattern dataset
by the Method of Minimum Contrast, using the function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The function of the point pattern will be computed
using
Kest
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits the Matern Cluster point process to X
,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical function of the Matern Cluster process
and the observed
function.
For a more detailed explanation of the Method of Minimum Contrast,
see
mincontrast
.
The Matern Cluster point process is described in Moller and Waagepetersen
(2003, p. 62). It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
are independent and uniformly distributed inside a circle of radius
centred on the parent point, where
is equal to
the parameter
scale
. The named vector of stating values can use
either R
or scale
as the name of the second component,
but the latter is recommended for consistency with other cluster models.
The theoretical -function of the Matern Cluster process is
where the radius R is the parameter scale
and
for , and
for
.
The theoretical intensity
of the Matern Cluster process
is
.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The Matern Cluster process can be simulated, using
rMatClust
.
Homogeneous or inhomogeneous Matern Cluster models can also be
fitted using the function kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Rasmus Plenge Waagepetersen [email protected]. Adapted for spatstat by Adrian Baddeley [email protected].
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
lgcp.estK
,
thomas.estK
,
mincontrast
,
Kest
,
rMatClust
to simulate the fitted model.
u <- matclust.estK(redwood, c(kappa=10, scale=0.1)) u plot(u)
u <- matclust.estK(redwood, c(kappa=10, scale=0.1)) u plot(u)
Fits the Matern Cluster point process to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.
matclust.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
matclust.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
X |
Data to which the Matern Cluster model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the Matern Cluster process. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
This algorithm fits the Matern Cluster point process model to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf
or one of its relatives.
The algorithm fits the Matern Cluster point process to X
,
by finding the parameters of the Matern Cluster model
which give the closest match between the
theoretical pair correlation function of the Matern Cluster process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The Matern Cluster point process is described in Moller and Waagepetersen
(2003, p. 62). It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
are independent and uniformly distributed inside a circle of radius
centred on the parent point, where
is equal to
the parameter
scale
. The named vector of stating values can use
either R
or scale
as the name of the second component,
but the latter is recommended for consistency with other cluster models.
The theoretical pair correlation function of the Matern Cluster process is
where the radius R is the parameter scale
and
for , and
for
.
The theoretical intensity
of the Matern Cluster process
is
.
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The Matern Cluster process can be simulated, using
rMatClust
.
Homogeneous or inhomogeneous Matern Cluster models can also be
fitted using the function kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Adrian Baddeley [email protected]
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
matclust.estK
,
thomas.estpcf
,
thomas.estK
,
lgcp.estK
,
mincontrast
,
pcf
,
rMatClust
to simulate the fitted model.
u <- matclust.estpcf(redwood, c(kappa=10, R=0.1)) u plot(u, legendpos="topright")
u <- matclust.estpcf(redwood, c(kappa=10, R=0.1)) u plot(u, legendpos="topright")
Given a measure A
(object of class "msr"
)
these functions find the discrete and continuous parts
of A
.
measureDiscrete(x) measureContinuous(x)
measureDiscrete(x) measureContinuous(x)
x |
A measure (object of class |
The functions measureDiscrete
and measureContinuous
return the discrete and continuous components, respectively,
of a measure.
If x
is a measure, then measureDiscrete(x)
is a measure
consisting only of the discrete (atomic) component of x
,
and measureContinuous(x)
is a measure consisting only of the
continuous (diffuse) component of x
.
Another measure (object of class "msr"
)
on the same spatial domain.
Adrian Baddeley [email protected].
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
msr
, with.msr
, split.msr
,
measurePositive
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp measureDiscrete(rp) measureContinuous(rp)
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp measureDiscrete(rp) measureContinuous(rp)
Given a measure A
(object of class "msr"
)
these functions find the positive part, negative part and variation
of A
.
measurePositive(x) measureNegative(x) measureVariation(x) totalVariation(x)
measurePositive(x) measureNegative(x) measureVariation(x) totalVariation(x)
x |
A measure (object of class |
The functions measurePositive
and measureNegative
return the positive and negative parts of the measure,
and measureVariation
returns the variation (sum of positive and
negative parts). The function totalVariation
returns the total
variation norm.
If is a signed measure,
it can be represented as
where and
are nonnegative measures called the positive and negative
parts of
.
In a nutshell, the positive part of
consists of all positive contributions or increments,
and the negative part consists of all negative contributions
multiplied by
-1
.
The variation is defined by
and is also a nonnegative measure.
The total variation norm is the integral of the variation.
The result of measurePositive
, measureNegative
and measureVariation
is another measure (object of class "msr"
)
on the same spatial domain.
The result of totalVariation
is a non-negative number.
Adrian Baddeley [email protected].
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
msr
, with.msr
, split.msr
,
measureDiscrete
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") measurePositive(rp) measureNegative(rp) measureVariation(rp) # total variation norm totalVariation(rp)
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") measurePositive(rp) measureNegative(rp) measureVariation(rp) # total variation norm totalVariation(rp)
Given a measure m
(object of class "msr"
)
and a spatially-varying weight function, construct the
weighted version of m
.
measureWeighted(m, w)
measureWeighted(m, w)
m |
A measure (object of class |
w |
A pixel image (object of class |
For any region of space B
,
the weighted measure wm
has the value
In any small region of space, the increment of the
weighted measure wm
is equal to the increment of the
original measure m
multiplied by the weight w
at that
location.
Another measure (object of class "msr"
)
on the same spatial domain.
Adrian Baddeley [email protected].
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
msr
, with.msr
, split.msr
,
measurePositive
fit <- ppm(cells ~ x) res <- residuals(fit) measureWeighted(res, function(x,y){x})
fit <- ppm(cells ~ x) res <- residuals(fit) measureWeighted(res, function(x,y){x})
These are methods for the class "dppm"
.
## S3 method for class 'dppm' coef(object, ...) ## S3 method for class 'dppm' formula(x, ...) ## S3 method for class 'dppm' print(x, ...) ## S3 method for class 'dppm' terms(x, ...) ## S3 method for class 'dppm' labels(object, ...)
## S3 method for class 'dppm' coef(object, ...) ## S3 method for class 'dppm' formula(x, ...) ## S3 method for class 'dppm' print(x, ...) ## S3 method for class 'dppm' terms(x, ...) ## S3 method for class 'dppm' labels(object, ...)
x , object
|
An object of class |
... |
Arguments passed to other methods. |
These functions are methods for the generic commands
coef
,
formula
,
print
,
terms
and
labels
for the class "dppm"
.
An object of class "dppm"
represents a fitted
determinantal point process model. It is obtained from dppm
.
The method coef.dppm
returns the vector of
regression coefficients of the fitted model.
It does not return the interaction parameters.
See the help files for the corresponding generic functions.
Adrian Baddeley [email protected]
dppm
, plot.dppm
,
predict.dppm
, simulate.dppm
,
as.ppm.dppm
.
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") coef(fit) formula(fit) tf <- terms(fit) labels(fit)
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") coef(fit) formula(fit) tf <- terms(fit) labels(fit)
These are methods specifically for the class "fii"
of
fitted interpoint interactions.
## S3 method for class 'fii' print(x, ...) ## S3 method for class 'fii' coef(object, ...) ## S3 replacement method for class 'fii' coef(object, ...) <- value ## S3 method for class 'fii' plot(x, ...) ## S3 method for class 'fii' summary(object,...) ## S3 method for class 'summary.fii' print(x, ...) ## S3 method for class 'summary.fii' coef(object, ...)
## S3 method for class 'fii' print(x, ...) ## S3 method for class 'fii' coef(object, ...) ## S3 replacement method for class 'fii' coef(object, ...) <- value ## S3 method for class 'fii' plot(x, ...) ## S3 method for class 'fii' summary(object,...) ## S3 method for class 'summary.fii' print(x, ...) ## S3 method for class 'summary.fii' coef(object, ...)
x , object
|
An object of class |
... |
Arguments passed to other methods. |
value |
Numeric vector containing new values for the fitted interaction coefficients. |
These are methods for the class "fii"
.
An object of class "fii"
represents a fitted interpoint
interaction. It is usually obtained by
using the command fitin
to extract the fitted
interaction part of a fitted point process model.
See fitin
for further explanation of this class.
The commands listed here are methods for the generic functions
print
,
summary
,
plot
,
coef
and coef<-
for objects of the class "fii"
.
Following the usual convention, summary.fii
returns an object of class
summary.fii
, for which there is a print method.
The effect is that, when the user types summary(x)
,
the summary is printed, but when the user types y <- summary(x)
,
the summary information is saved.
The method coef.fii
extracts the canonical coefficients of
the fitted interaction, and returns them as a numeric vector.
The method coef.summary.fii
transforms these values into
quantities that are more easily interpretable, in a format that
depends on the particular model.
There are also methods for the generic commands reach
and
as.interact
, described elsewhere.
The print
and plot
methods return NULL
.
The summary
method returns an object of class
summary.fii
.
coef.fii
returns a numeric vector.
coef.summary.fii
returns data whose structure
depends on the model.
Adrian Baddeley [email protected]
fitin
,
reach.fii
,
as.interact.fii
mod <- ppm(cells ~1, Strauss(0.1)) f <- fitin(mod) f summary(f) plot(f) coef(f) coef(summary(f))
mod <- ppm(cells ~1, Strauss(0.1)) f <- fitin(mod) f summary(f) plot(f) coef(f) coef(summary(f))
Methods for the class "influence.ppm"
.
## S3 method for class 'influence.ppm' as.ppp(X, ...) ## S3 method for class 'influence.ppm' as.owin(W, ..., fatal=TRUE) ## S3 method for class 'influence.ppm' domain(X, ...) ## S3 method for class 'influence.ppm' Smooth(X, ...) ## S3 method for class 'influence.ppm' Window(X, ...) ## S3 method for class 'influence.ppm' integral(f, domain, ...)
## S3 method for class 'influence.ppm' as.ppp(X, ...) ## S3 method for class 'influence.ppm' as.owin(W, ..., fatal=TRUE) ## S3 method for class 'influence.ppm' domain(X, ...) ## S3 method for class 'influence.ppm' Smooth(X, ...) ## S3 method for class 'influence.ppm' Window(X, ...) ## S3 method for class 'influence.ppm' integral(f, domain, ...)
X , W , f
|
An object of class |
domain |
Optional. Domain of integration: a window (class |
... |
Additional arguments. See Details. |
fatal |
Logical value indicating what to do if the data cannot be
converted to a window. If |
These functions are methods for the class "influence.ppm"
.
An object of this class
represents the influence measure of a fitted point process model
(see influence.ppm
).
For as.ppp
, domain
, integral
and Window
,
additional arguments (...
) are ignored.
For as.owin
and Smooth
,
additional arguments are passed to the method for class "ppp"
.
A window (object of class "owin"
) for
as.owin
, domain
and Window
.
A point pattern (object of class "ppp"
) for
as.ppp
. A numeric value or numeric vector for integral
.
A pixel image, or list of pixel images, for Smooth
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
influence.ppm
, plot.influence.ppm
,
[.influence.ppm
fit <- ppm(cells ~ x) a <- influence(fit) Window(a)
fit <- ppm(cells ~ x) a <- influence(fit) Window(a)
These are methods for the class "kppm"
.
## S3 method for class 'kppm' coef(object, ...) ## S3 method for class 'kppm' formula(x, ...) ## S3 method for class 'kppm' print(x, ...) ## S3 method for class 'kppm' terms(x, ...) ## S3 method for class 'kppm' labels(object, ...)
## S3 method for class 'kppm' coef(object, ...) ## S3 method for class 'kppm' formula(x, ...) ## S3 method for class 'kppm' print(x, ...) ## S3 method for class 'kppm' terms(x, ...) ## S3 method for class 'kppm' labels(object, ...)
x , object
|
An object of class |
... |
Arguments passed to other methods. |
These functions are methods for the generic commands
coef
,
formula
,
print
,
terms
and
labels
for the class "kppm"
.
An object of class "kppm"
represents a fitted
cluster point process model. It is obtained from kppm
.
The method coef.kppm
returns the vector of
regression coefficients of the fitted model.
It does not return the clustering parameters.
See the help files for the corresponding generic functions.
Adrian Baddeley [email protected]
kppm
, plot.kppm
,
predict.kppm
, simulate.kppm
,
update.kppm
, vcov.kppm
,
as.ppm.kppm
.
fit <- kppm(redwood ~ x, "MatClust") coef(fit) formula(fit) tf <- terms(fit) labels(fit)
fit <- kppm(redwood ~ x, "MatClust") coef(fit) formula(fit) tf <- terms(fit) labels(fit)
Methods for the class "leverage.ppm"
.
## S3 method for class 'leverage.ppm' as.im(X, ..., what=c("smooth", "nearest")) ## S3 method for class 'leverage.ppm' as.owin(W, ..., fatal=TRUE) ## S3 method for class 'leverage.ppm' domain(X, ...) ## S3 method for class 'leverage.ppm' integral(f, domain, ...) ## S3 method for class 'leverage.ppm' mean(x, ...) ## S3 method for class 'leverage.ppm' Smooth(X, ...) ## S3 method for class 'leverage.ppm' Window(X, ...)
## S3 method for class 'leverage.ppm' as.im(X, ..., what=c("smooth", "nearest")) ## S3 method for class 'leverage.ppm' as.owin(W, ..., fatal=TRUE) ## S3 method for class 'leverage.ppm' domain(X, ...) ## S3 method for class 'leverage.ppm' integral(f, domain, ...) ## S3 method for class 'leverage.ppm' mean(x, ...) ## S3 method for class 'leverage.ppm' Smooth(X, ...) ## S3 method for class 'leverage.ppm' Window(X, ...)
X , x , W , f
|
An object of class |
domain |
Optional. Domain of integration: a window (class |
... |
Additional arguments. See Details. |
fatal |
Logical value indicating what to do if the data cannot be
converted to a window. If |
what |
Character string (partially matched) specifying which image data
should be extracted. See |
These functions are methods for the class "leverage.ppm"
.
An object of this class
represents the leverage measure of a fitted point process model
(see leverage.ppm
).
For as.im
, domain
and Window
,
additional arguments (...
) are ignored.
For as.owin
, integral
, mean
and Smooth
,
additional arguments are passed to the method for class "im"
.
A window (object of class "owin"
) for
as.owin
, domain
and Window
.
A numeric value or numeric vector for integral
.
A pixel image, or list of pixel images, for as.im
and Smooth
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
leverage.ppm
, plot.leverage.ppm
,
[.leverage.ppm
, as.function.leverage.ppm
.
fit <- ppm(cells ~ x) a <- leverage(fit) integral(a)
fit <- ppm(cells ~ x) a <- leverage(fit) integral(a)
Methods for printing and plotting an objective function surface.
## S3 method for class 'objsurf' print(x, ...) ## S3 method for class 'objsurf' plot(x, ...) ## S3 method for class 'objsurf' image(x, ...) ## S3 method for class 'objsurf' contour(x, ...) ## S3 method for class 'objsurf' persp(x, ...) ## S3 method for class 'objsurf' summary(object, ...) ## S3 method for class 'summary.objsurf' print(x, ...)
## S3 method for class 'objsurf' print(x, ...) ## S3 method for class 'objsurf' plot(x, ...) ## S3 method for class 'objsurf' image(x, ...) ## S3 method for class 'objsurf' contour(x, ...) ## S3 method for class 'objsurf' persp(x, ...) ## S3 method for class 'objsurf' summary(object, ...) ## S3 method for class 'summary.objsurf' print(x, ...)
x , object
|
Object of class |
... |
Additional arguments passed to plot methods. |
These are methods for the generic functions
print
, plot
,
image
, contour
, persp
and summary
for the class "objsurf"
.
For print.objsurf
, print.summary.objsurf
,
plot.objsurf
and image.objsurf
the value is NULL
.
For contour.objsurf
and persp.objsurf
the value is described in the help for
contour.default
and persp.default
respectively.
For summary.objsurf
the result is a list, of class
summary.objsurf
, containing summary information. This list is
printed in sensible format by print.summary.objsurf
.
Adrian Baddeley [email protected] and Ege Rubak [email protected].
fit <- kppm(redwood ~ 1, "Thomas") os <- objsurf(fit) os summary(os) plot(os) contour(os, add=TRUE) persp(os)
fit <- kppm(redwood ~ 1, "Thomas") os <- objsurf(fit) os summary(os) plot(os) contour(os, add=TRUE) persp(os)
These are methods for the class "slrm"
.
## S3 method for class 'slrm' formula(x, ...) ## S3 method for class 'slrm' print(x, ...) ## S3 method for class 'slrm' summary(object, ...) ## S3 method for class 'slrm' terms(x, ...) ## S3 method for class 'slrm' labels(object, ...) ## S3 method for class 'slrm' deviance(object, ...) ## S3 method for class 'slrm' update(object, fmla, ..., evaluate = TRUE, env = parent.frame())
## S3 method for class 'slrm' formula(x, ...) ## S3 method for class 'slrm' print(x, ...) ## S3 method for class 'slrm' summary(object, ...) ## S3 method for class 'slrm' terms(x, ...) ## S3 method for class 'slrm' labels(object, ...) ## S3 method for class 'slrm' deviance(object, ...) ## S3 method for class 'slrm' update(object, fmla, ..., evaluate = TRUE, env = parent.frame())
x , object
|
An object of class |
... |
Arguments passed to other methods. |
fmla |
Optional. A formula, to replace the formula of the model. |
evaluate |
Logical value. If |
env |
Optional environment in which the model should be updated. |
These functions are methods for the generic commands
formula
,
update
,
print
,
summary
,
terms
,
labels
and
deviance
for the class "slrm"
.
An object of class "slrm"
represents a fitted
spatial logistic regression model. It is obtained from slrm
.
See the help files for the corresponding generic functions.
Adrian Baddeley [email protected]
slrm
, plot.slrm
,
predict.slrm
, simulate.slrm
,
vcov.slrm
,
coef.slrm
.
fit <- slrm(redwood ~ x) coef(fit) formula(fit) tf <- terms(fit) labels(fit) deviance(fit)
fit <- slrm(redwood ~ x) coef(fit) formula(fit) tf <- terms(fit) labels(fit) deviance(fit)
Methods for objects of class "traj"
.
## S3 method for class 'traj' print(x, ...) ## S3 method for class 'traj' plot(x, ..., show.ends=TRUE, add=FALSE, xlab=NULL, ylab=NULL) ## S3 method for class 'traj' lines(x, ..., directed=FALSE)
## S3 method for class 'traj' print(x, ...) ## S3 method for class 'traj' plot(x, ..., show.ends=TRUE, add=FALSE, xlab=NULL, ylab=NULL) ## S3 method for class 'traj' lines(x, ..., directed=FALSE)
x |
Object of class |
... |
Additional arguments passed to other methods. |
directed |
Logical value specifying whether to draw arrows instead of undirected lines. |
show.ends |
Logical value specifying whether to indicate the start and finish of the trajectory. The start is a blue circle; the finish is a red cross. |
add |
Logical value specifying whether to draw the trajectory on the
existing plot ( |
xlab , ylab
|
Optional labels for the horizontal and vertical axes. |
An object of class "traj"
represents
the history of evaluations of the objective function
performed when a cluster process model was fitted.
It is a data frame containing the input parameter values for
the objective function, and the corresponding value of the objective
function, that were considered by the optimisation algorithm.
These functions are methods for the generic
print
, plot
and lines
.
Null.
Adrian Baddeley [email protected].
fit <- kppm(redwood, pspace=list(save=TRUE)) h <- traj(fit) h plot(h) lines(h)
fit <- kppm(redwood, pspace=list(save=TRUE)) h <- traj(fit) h plot(h) lines(h)
Methods for the experimental class of cluster models.
## S3 method for class 'zclustermodel' pcfmodel(model, ...) ## S3 method for class 'zclustermodel' Kmodel(model, ...) ## S3 method for class 'zclustermodel' intensity(X, ...) ## S3 method for class 'zclustermodel' predict(object, ..., locations, type = "intensity", ngrid = NULL) ## S3 method for class 'zclustermodel' print(x, ...) ## S3 method for class 'zclustermodel' clusterradius(model,...,thresh=NULL, precision=FALSE) ## S3 method for class 'zclustermodel' reach(x, ..., epsilon)
## S3 method for class 'zclustermodel' pcfmodel(model, ...) ## S3 method for class 'zclustermodel' Kmodel(model, ...) ## S3 method for class 'zclustermodel' intensity(X, ...) ## S3 method for class 'zclustermodel' predict(object, ..., locations, type = "intensity", ngrid = NULL) ## S3 method for class 'zclustermodel' print(x, ...) ## S3 method for class 'zclustermodel' clusterradius(model,...,thresh=NULL, precision=FALSE) ## S3 method for class 'zclustermodel' reach(x, ..., epsilon)
model , object , x , X
|
Object of class |
... |
Arguments passed to other methods. |
locations |
Locations where prediction should be performed. A window or a point pattern. |
type |
Currently must equal |
ngrid |
Pixel grid dimensions for prediction, if |
thresh , epsilon
|
Tolerance thresholds |
precision |
Logical value stipulating whether the precision should also be returned. |
Experimental.
Same as for other methods.
Adrian Baddeley [email protected]
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1) m2 <- zclustermodel("VarGamma", kappa=10, mu=10, scale=0.1, nu=0.7) m m2 g <- pcfmodel(m) g(0.2) g2 <- pcfmodel(m2) g2(1) Z <- predict(m, locations=square(2)) Z2 <- predict(m2, locations=square(1)) varcount(m, square(1)) varcount(m2, square(1))
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1) m2 <- zclustermodel("VarGamma", kappa=10, mu=10, scale=0.1, nu=0.7) m m2 g <- pcfmodel(m) g(0.2) g2 <- pcfmodel(m2) g2(1) Z <- predict(m, locations=square(2)) Z2 <- predict(m2, locations=square(1)) varcount(m, square(1)) varcount(m2, square(1))
Methods for the experimental class of Gibbs models
## S3 method for class 'zgibbsmodel' as.interact(object) ## S3 method for class 'zgibbsmodel' as.isf(object) ## S3 method for class 'zgibbsmodel' interactionorder(object) ## S3 method for class 'zgibbsmodel' is.poisson(x) ## S3 method for class 'zgibbsmodel' is.stationary(x) ## S3 method for class 'zgibbsmodel' print(x, ...) ## S3 method for class 'zgibbsmodel' intensity(X, ..., approx=c("Poisson", "DPP"))
## S3 method for class 'zgibbsmodel' as.interact(object) ## S3 method for class 'zgibbsmodel' as.isf(object) ## S3 method for class 'zgibbsmodel' interactionorder(object) ## S3 method for class 'zgibbsmodel' is.poisson(x) ## S3 method for class 'zgibbsmodel' is.stationary(x) ## S3 method for class 'zgibbsmodel' print(x, ...) ## S3 method for class 'zgibbsmodel' intensity(X, ..., approx=c("Poisson", "DPP"))
object , x , X
|
Object of class |
... |
Additional arguments. |
approx |
Character string (partially matched) specifying the type of approximation. |
Experimental.
Same as for other methods.
Adrian Baddeley [email protected]
m <- zgibbsmodel(10, Strauss(0.1), -0.5) m is.poisson(m) is.stationary(m) interactionorder(m) as.interact(m) as.isf(m) intensity(m) intensity(m, approx="D")
m <- zgibbsmodel(10, Strauss(0.1), -0.5) m is.poisson(m) is.stationary(m) interactionorder(m) as.interact(m) as.isf(m) intensity(m) intensity(m, approx="D")
A general low-level algorithm for fitting theoretical point process models to point pattern data by the Method of Minimum Contrast.
mincontrast(observed, theoretical, startpar, ..., ctrl=list(q = 1/4, p = 2, rmin=NULL, rmax=NULL), fvlab=list(label=NULL, desc="minimum contrast fit"), explain=list(dataname=NULL, modelname=NULL, fname=NULL), action.bad.values=c("warn", "stop", "silent"), control=list(), stabilize=TRUE, pspace=NULL)
mincontrast(observed, theoretical, startpar, ..., ctrl=list(q = 1/4, p = 2, rmin=NULL, rmax=NULL), fvlab=list(label=NULL, desc="minimum contrast fit"), explain=list(dataname=NULL, modelname=NULL, fname=NULL), action.bad.values=c("warn", "stop", "silent"), control=list(), stabilize=TRUE, pspace=NULL)
observed |
Summary statistic, computed for the data.
An object of class |
theoretical |
An R language function that calculates the theoretical expected value of the summary statistic, given the model parameters. See Details. |
startpar |
Vector of initial values of the parameters of the
point process model (passed to |
... |
Additional arguments passed to the function |
ctrl |
Optional. List of arguments controlling the optimisation. See Details. |
fvlab |
Optional. List containing some labels for the return value. See Details. |
explain |
Optional. List containing strings that give a human-readable description of the model, the data and the summary statistic. |
action.bad.values |
String (partially matched) specifying what to do if
values of the summary statistic are |
control |
Optional. Argument passed to |
stabilize |
Logical value specifying whether to numerically stabilize
the optimization algorithm, by specifying suitable default values of
|
pspace |
For internal use by the package only. |
This function is a general algorithm for fitting point process models
by the Method of Minimum Contrast. If you want to fit the
Thomas process, see thomas.estK
.
If you want to fit a log-Gaussian Cox process, see
lgcp.estK
. If you want to fit the Matern cluster
process, see matclust.estK
.
The Method of Minimum Contrast (Pfanzagl, 1969; Diggle and Gratton, 1984)
is a general technique for fitting
a point process model to point pattern data. First a summary function
(typically the function) is computed from the data point
pattern. Second, the theoretical expected
value of this summary statistic under the point process model
is derived (if possible, as an algebraic expression involving the
parameters of the model) or estimated from simulations of the model.
Then the model is fitted by finding the optimal parameter values
for the model to give the closest match between the theoretical
and empirical curves.
The argument observed
should be an object of class "fv"
(see fv.object
) containing the values of a summary
statistic computed from the data point pattern. Usually this is the
function computed by
Kest
or one of its relatives.
The argument theoretical
should be a user-supplied function
that computes the theoretical expected value of the summary statistic.
It must have an argument named par
that will be the vector
of parameter values for the model (the length and format of this
vector are determined by the starting values in startpar
).
The function theoretical
should also expect a second argument
(the first argument other than par
)
containing values of the distance for which the theoretical
value of the summary statistic
should be computed.
The value returned by
theoretical
should be a vector of the
same length as the given vector of values.
The argument ctrl
determines the contrast criterion
(the objective function that will be minimised).
The algorithm minimises the criterion
where is the vector of parameters of the model,
is the observed value of the summary statistic
computed from the data,
is the
theoretical expected value of the summary statistic,
and
are two exponents. The default is
q = 1/4
,
p=2
so that the contrast criterion is the integrated squared
difference between the fourth roots of the two functions
(Waagepetersen, 2007).
The argument action.bad.values
specifies what to do if
some of the values of the summary statistic are NA
, NaN
or
infinite. If action.bad.values="stop"
, or if all of the values are bad,
then a fatal error occurs. Otherwise, the domain of the summary
function is shortened to avoid the bad values. The shortened domain is the
longest interval on which the function values are finite
(provided this interval is at least half the length of the original
domain). A warning is issued if action.bad.values="warn"
(the default)
and no warning is issued if action.bad.values="silent"
.
The other arguments just make things print nicely.
The argument fvlab
contains labels for the component
fit
of the return value.
The argument explain
contains human-readable strings
describing the data, the model and the summary statistic.
The "..."
argument of mincontrast
can be used to
pass extra arguments to the function theoretical
and/or to the optimisation function optim
.
In this case, the function theoretical
should also have a "..."
argument and should ignore it
(so that it ignores arguments intended for optim
).
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
opt |
The object returned from the optimizer |
crtl |
List of parameters determining the contrast objective. |
info |
List of explanatory strings. |
Rasmus Plenge Waagepetersen [email protected]. Adapted for spatstat by Adrian Baddeley [email protected].
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Pfanzagl, J. (1969). On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–276.
Waagepetersen, R. (2007). An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
lgcp.estK
,
matclust.estK
,
thomas.estK
,
Given a fitted model (of any kind), identify which of the covariates is involved in each term of the model.
model.depends(object) model.is.additive(object) model.covariates(object, fitted=TRUE, offset=TRUE) has.offset.term(object) has.offset(object)
model.depends(object) model.is.additive(object) model.covariates(object, fitted=TRUE, offset=TRUE) has.offset.term(object) has.offset(object)
object |
A fitted model of any kind. |
fitted , offset
|
Logical values determining which type of covariates to include. |
The object
can be a fitted model of any kind,
including models of the classes lm
, glm
and ppm
.
To be precise,
object
must belong to a class for which there are methods
for formula
, terms
and model.matrix
.
The command model.depends
determines the relationship between
the original covariates (the data supplied when object
was
fitted) and the canonical covariates (the columns of the design matrix).
It returns a logical matrix, with one row for each canonical
covariate, and one column for each of the original covariates,
with the i,j
entry equal to TRUE
if the
i
th canonical covariate depends on the j
th
original covariate.
If the model formula of object
includes offset terms
(see offset
), then the return value of model.depends
also has an attribute "offset"
. This is a logical value or
matrix with one row for each offset term and one column for each of
the original covariates, with the i,j
entry equal to TRUE
if the
i
th offset term depends on the j
th
original covariate.
The command model.covariates
returns a character vector
containing the names of all (original) covariates that were actually
used to fit the model. By default, this includes all covariates that
appear in the model formula, including offset terms as well as
canonical covariate terms. To omit the offset terms, set
offset=FALSE
. To omit the canonical covariate terms,
set fitted=FALSE
.
The command model.is.additive
determines whether the model
is additive, in the sense that there is no canonical covariate that
depends on two or more original covariates. It returns a logical value.
The command has.offset.term
is a faster way to determine whether the
model formula includes an offset
term.
The functions model.depends
and has.offset.term
only detect offset
terms which are present
in the model formula. They do not detect numerical offsets in the
model object, that were inserted using the offset
argument
in lm
, glm
etc. To detect the presence of offsets
of both kinds, use has.offset
.
A logical value or matrix.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
x <- 1:10 y <- 3*x + 2 z <- rep(c(-1,1), 5) fit <- lm(y ~ poly(x,2) + sin(z)) model.depends(fit) model.covariates(fit) model.is.additive(fit) fitoff1 <- lm(y ~ x + offset(z)) fitoff2 <- lm(y ~ x, offset=z) has.offset.term(fitoff1) has.offset(fitoff1) has.offset.term(fitoff2) has.offset(fitoff2)
x <- 1:10 y <- 3*x + 2 z <- rep(c(-1,1), 5) fit <- lm(y ~ poly(x,2) + sin(z)) model.depends(fit) model.covariates(fit) model.is.additive(fit) fitoff1 <- lm(y ~ x + offset(z)) fitoff2 <- lm(y ~ x, offset=z) has.offset.term(fitoff1) has.offset(fitoff1) has.offset.term(fitoff2) has.offset(fitoff2)
Given a fitted point process model, this function returns a data frame containing all the variables needed to fit the model using the Berman-Turner device.
## S3 method for class 'ppm' model.frame(formula, ...) ## S3 method for class 'kppm' model.frame(formula, ...) ## S3 method for class 'dppm' model.frame(formula, ...) ## S3 method for class 'slrm' model.frame(formula, ...)
## S3 method for class 'ppm' model.frame(formula, ...) ## S3 method for class 'kppm' model.frame(formula, ...) ## S3 method for class 'dppm' model.frame(formula, ...) ## S3 method for class 'slrm' model.frame(formula, ...)
formula |
A fitted point process model.
An object of class |
... |
Additional arguments passed to |
The function model.frame
is generic.
These functions are method for model.frame
for fitted point process models (objects of class "ppm"
,
"kppm"
, "slrm"
,
or "dppm"
).
The first argument should be a fitted point process model;
it has to be named formula
for consistency with the generic
function.
The result is a data frame containing all the variables used in
fitting the model. The data frame has one row for each quadrature point
used in fitting the model. The quadrature scheme can be extracted using
quad.ppm
.
A data.frame
containing all the variables used in the
fitted model, plus additional variables specified in ...
.
It has an additional attribute "terms"
containing information
about the model formula. For details see model.frame.glm
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
ppm
,
kppm
,
dppm
,
slrm
,
model.frame
,
model.matrix.ppm
fit <- ppm(cells ~ x) mf <- model.frame(fit) kfit <- kppm(redwood ~ x, "Thomas") kmf <- model.frame(kfit) sfit <- slrm(cells ~ x) smf <- model.frame(sfit)
fit <- ppm(cells ~ x) mf <- model.frame(fit) kfit <- kppm(redwood ~ x, "Thomas") kmf <- model.frame(kfit) sfit <- slrm(cells ~ x) smf <- model.frame(sfit)
For a point process model fitted to spatial point pattern data, this function computes pixel images of the covariates in the design matrix.
model.images(object, ...) ## S3 method for class 'ppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'kppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'dppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'slrm' model.images(object, ...)
model.images(object, ...) ## S3 method for class 'ppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'kppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'dppm' model.images(object, W = as.owin(object), ...) ## S3 method for class 'slrm' model.images(object, ...)
object |
The fitted point process model. An object of class |
W |
A window (object of class |
... |
Other arguments (such as |
This command is similar to model.matrix.ppm
except
that it computes pixel images of the covariates,
instead of computing the covariate values at certain points only.
The object
must be a fitted spatial point process model
object of class "ppm"
(produced by the model-fitting
function ppm
) or class "kppm"
(produced by the
fitting function kppm
)
or class "dppm"
(produced by the
fitting function dppm
)
or class "slrm"
(produced by slrm
).
The spatial covariates required by the model-fitting procedure
are computed at every pixel location in the window W
.
For slrm
objects, the
covariates are computed on the pixels that were used to fit the
model.
Note that the spatial covariates computed here are not necessarily the original covariates that were supplied when fitting the model. Rather, they are the canonical covariates, the covariates that appear in the loglinear representation of the (conditional) intensity and in the columns of the design matrix. For example, they might include dummy or indicator variables for different levels of a factor, depending on the contrasts that are in force.
The pixel resolution is determined by W
if W
is a mask (that is W$type = "mask"
).
Otherwise, the pixel resolution is determined by
spatstat.options
.
The format of the result depends on whether the original point pattern data were marked or unmarked.
If the original dataset was unmarked,
the result is a named list of pixel images (objects of class
"im"
) containing the values of the spatial covariates.
The names of the list elements are the names of the covariates
determined by model.matrix.lm
.
The result is also of class "solist"
so that it can
be plotted immediately.
If the original dataset was a multitype point pattern,
the result is a hyperframe
with one column for each possible type of points.
Each column is a named list of pixel images (objects of class
"im"
) containing the values of the spatial covariates.
The row names of the hyperframe are the names of the covariates
determined by model.matrix.lm
.
A list (of class "solist"
) or
array (of class "hyperframe"
) containing
pixel images (objects of class "im"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
model.matrix.ppm
,
model.matrix
,
ppm
,
ppm.object
,
dppm
,
kppm
,
slrm
,
im
,
im.object
,
plot.solist
,
spatstat.options
fit <- ppm(cells ~ x) model.images(fit) B <- owin(c(0.2, 0.4), c(0.3, 0.8)) model.images(fit, B) fit2 <- ppm(cells ~ cut(x,3)) model.images(fit2) fit3 <- slrm(japanesepines ~ x) model.images(fit3) fit4 <- ppm(amacrine ~ marks + x) model.images(fit4)
fit <- ppm(cells ~ x) model.images(fit) B <- owin(c(0.2, 0.4), c(0.3, 0.8)) model.images(fit, B) fit2 <- ppm(cells ~ cut(x,3)) model.images(fit2) fit3 <- slrm(japanesepines ~ x) model.images(fit3) fit4 <- ppm(amacrine ~ marks + x) model.images(fit4)
Given a point process model fitted to a list of point patterns, this function extracts the design matrix.
## S3 method for class 'mppm' model.matrix(object, ..., keepNA=TRUE, separate=FALSE)
## S3 method for class 'mppm' model.matrix(object, ..., keepNA=TRUE, separate=FALSE)
object |
A point process model fitted to several point patterns.
An object of class |
... |
Other arguments (such as |
keepNA |
Logical. Determines whether rows containing |
separate |
Logical value indicating whether to split the model matrix into sub-matrices corresponding to each of the original point patterns. |
This command is a method for the generic function
model.matrix
. It extracts the design matrix of a
point process model fitted to several point patterns.
The argument object
must be a fitted point process model
(object of class "mppm"
) produced by the
fitting algorithm mppm
). This represents a
point process model that has been fitted
to a list of several point pattern datasets. See mppm
for information.
The result is a matrix with one column for every constructed covariate in the model, and one row for every quadrature point.
If separate=TRUE
this matrix will be split into
sub-matrices corresponding to the original point patterns,
and the result will be a list containing these matrices.
A matrix (or list of matrices). Columns of the matrix are canonical covariates in the model.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- mppm(Points ~ Image + x, demohyper) head(model.matrix(fit)) # matrix with three columns: '(Intercept)', 'x' and 'Image'
fit <- mppm(Points ~ Image + x, demohyper) head(model.matrix(fit)) # matrix with three columns: '(Intercept)', 'x' and 'Image'
Given a point process model that has been fitted to spatial point pattern data, this function extracts the design matrix of the model.
## S3 method for class 'ppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'kppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'dppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'ippm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE, irregular=FALSE)
## S3 method for class 'ppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'kppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'dppm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE) ## S3 method for class 'ippm' model.matrix(object, data=model.frame(object, na.action=NULL), ..., Q=NULL, keepNA=TRUE, irregular=FALSE)
object |
The fitted point process model. An object of class |
data |
A model frame, containing the data required for the Berman-Turner device. |
Q |
A point pattern (class |
keepNA |
Logical. Determines whether rows containing NA values will be deleted or retained. |
... |
Other arguments (such as |
irregular |
Logical value indicating whether to include the irregular score components. |
These commands are methods for the generic function
model.matrix
.
They extract the design matrix of a spatial point process model
(class "ppm"
or "kppm"
or "dppm"
).
More precisely, this command extracts the design matrix of the generalised linear model associated with a spatial point process model.
The object
must be a fitted point process model
(object of class "ppm"
or "kppm"
or "dppm"
)
fitted to spatial point pattern data.
Such objects are produced by the model-fitting
functions ppm
, kppm
,
and dppm
.
The methods model.matrix.ppm
,
model.matrix.kppm
,
and model.matrix.dppm
extract the model matrix for the GLM.
The result is a matrix, with one row for every quadrature point in the fitting procedure, and one column for every constructed covariate in the design matrix.
If there are NA
values in the covariates,
the argument keepNA
determines whether to retain or delete
the corresponding rows of the model matrix. The default
keepNA=TRUE
is to retain them. Note that this differs from
the default behaviour of many other methods for model.matrix
,
which typically delete rows containing NA
.
The quadrature points themselves can be extracted using
quad.ppm
.
A matrix. Columns of the matrix are canonical covariates in the model.
Rows of the matrix correspond to quadrature points
in the fitting procedure (provided keepNA=TRUE
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
model.matrix
,
model.images
,
ppm
,
kppm
,
dppm
,
ippm
,
ppm.object
,
quad.ppm
,
residuals.ppm
fit <- ppm(cells ~ x) head(model.matrix(fit)) model.matrix(fit, Q=runifpoint(5)) kfit <- kppm(redwood ~ x, "Thomas") m <- model.matrix(kfit)
fit <- ppm(cells ~ x) head(model.matrix(fit)) model.matrix(fit, Q=runifpoint(5)) kfit <- kppm(redwood ~ x, "Thomas") m <- model.matrix(kfit)
This function extracts the design matrix of a spatial logistic regression model.
## S3 method for class 'slrm' model.matrix(object, ..., keepNA=TRUE)
## S3 method for class 'slrm' model.matrix(object, ..., keepNA=TRUE)
object |
A fitted spatial logistic regression model. An object of class
|
... |
Other arguments (such as |
keepNA |
Logical. Determines whether rows containing |
This command is a method for the generic function
model.matrix
. It extracts the design matrix of a
spatial logistic regression.
The object
must be a fitted
spatial logistic regression
(object of class "slrm"
).
Such objects are produced by the model-fitting
function slrm
.
Usually the result is a matrix with one column for every constructed covariate in the model, and one row for every pixel in the grid used to fit the model.
If object
was fitted using split pixels (by calling
slrm
using the argument splitby
) then the
matrix has one row for every pixel or half-pixel.
A matrix. Columns of the matrix are canonical covariates in the model.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
model.matrix
,
model.images
,
slrm
.
fit <- slrm(japanesepines ~x) head(model.matrix(fit)) # matrix with two columns: '(Intercept)' and 'x'
fit <- slrm(japanesepines ~x) head(model.matrix(fit)) # matrix with two columns: '(Intercept)' and 'x'
Fits a Gibbs point process model to several point patterns simultaneously.
mppm(formula, data, interaction=Poisson(), ..., iformula=NULL, random=NULL, weights=NULL, use.gam = FALSE, reltol.pql=1e-3, gcontrol=list())
mppm(formula, data, interaction=Poisson(), ..., iformula=NULL, random=NULL, weights=NULL, use.gam = FALSE, reltol.pql=1e-3, gcontrol=list())
formula |
A formula describing the systematic part of the model.
Variables in the formula are names of columns in |
data |
A hyperframe (object of class |
interaction |
Interpoint interaction(s) appearing in the model.
Either an object of class |
... |
Arguments passed to |
iformula |
Optional. A formula (with no left hand side)
describing the interaction to be applied to each case.
Each variable name in the formula should either be the name of a column
in the hyperframe |
random |
Optional. A formula (with no left hand side)
describing a random effect. Variable names in the formula
may be any of the column names of |
weights |
Optional. Numeric vector of case weights for each row of |
use.gam |
Logical flag indicating whether to fit the model
using |
reltol.pql |
Relative tolerance for successive steps in
the penalised quasi-likelihood algorithm,
used when the model includes random effects.
The algorithm terminates when the root mean square of the
relative change in coefficients is less than |
gcontrol |
List of arguments to control the fitting algorithm.
Arguments are passed to |
This function fits a common point process model to a dataset containing several different point patterns.
It extends the capabilities of the function ppm
to deal with data such as
replicated observations of spatial point patterns
two groups of spatial point patterns
a designed experiment in which the response from each unit is a point pattern.
The syntax of this function is similar to that of
standard R model-fitting functions like lm
and
glm
. The first argument formula
is an R formula
describing the systematic part of the model. The second argument
data
contains the responses and the explanatory variables.
Other arguments determine the stochastic structure of the model.
Schematically,
the data are regarded as the results of a designed experiment
involving experimental units. Each unit has a
‘response’, and optionally some ‘explanatory variables’
(covariates) describing the experimental conditions for that unit.
In this context,
the response from each unit is a point pattern.
The value of a particular covariate for each unit can be
either a single value (numerical, logical or factor),
or a spatial covariate.
A ‘spatial’ covariate is a quantity that depends on spatial location,
for example, the soil acidity or altitude at each location.
For the purposes of
mppm
, a spatial covariate must be stored
as a pixel image (object of class "im"
) which gives the values
of the covariate at a fine grid of locations.
The argument data
is a hyperframe (a generalisation of
a data frame, see hyperframe
). This is like a data frame
except that the entries can be objects of any class.
The hyperframe has one row for each experimental unit,
and one column for each variable (response or explanatory variable).
The formula
should be an R formula.
The left hand side of formula
determines the ‘response’
variable. This should be a single name, which
should correspond to a column in data
.
The right hand side of formula
determines the
spatial trend of the model. It specifies the linear predictor,
and effectively represents the logarithm
of the spatial trend.
Variables in the formula must be the names of columns of
data
, or one of the reserved names
Cartesian coordinates of location
Mark attached to point
which is a factor representing the
serial number ( to
) of the point pattern,
i.e. the row number in the data hyperframe.
The column of responses in data
must consist of point patterns (objects of class "ppp"
).
The individual point pattern responses
can be defined in different spatial windows.
If some of the point patterns are marked, then they must all be
marked, and must have the same type of marks.
The scope of models that can be fitted to each pattern is the same as the
scope of ppm
, that is, Gibbs point processes with
interaction terms that belong to a specified list, including
for example the Poisson process, Strauss process, Geyer's saturation
model, and piecewise constant pairwise interaction models.
Additionally, it is possible to include random effects
as explained in the section on Random Effects below.
The stochastic part of the model is determined by
the arguments interaction
and (optionally) iformula
.
In the simplest case, interaction
is
an object of class "interact"
,
determining the interpoint interaction structure of the point
process model, for all experimental units.
Alternatively, interaction
may be a hyperframe,
whose entries are objects of class "interact"
.
It should have the same number of rows as data
.
If interaction
consists of only one column,
then the entry in row i
is taken to be the
interpoint interaction for the i
th experimental unit
(corresponding to the i
th row of data
).
If interaction
has more than one column,
then the argument iformula
is also required.
Each row of interaction
determines
several interpoint interaction structures that might be applied
to the corresponding row of data
.
The choice of interaction is determined by iformula
;
this should be an R formula,
without a left hand side.
For example if interaction
has two columns called
A
and B
then iformula = ~B
indicates that the
interpoint interactions are taken from the second column.
Variables in iformula
typically refer to column names of interaction
.
They can also be names of columns in
data
, but only for columns of numeric, logical or factor
values. For example iformula = ~B * group
(where group
is a column of data
that contains a factor) causes the
model with interpoint interaction B
to be fitted
with different interaction parameters for each level of group
.
An object of class "mppm"
representing the
fitted model.
There are methods for
print
, summary
, coef
,
AIC
, anova
, fitted
, fixef
, logLik
,
plot
, predict
, ranef
, residuals
,
summary
, terms
and vcov
for this class.
The default methods for update
and formula
also work on this class.
It is also possible to include random effects in the
trend term. The argument random
is a formula,
with no left-hand side, that specifies the structure of the
random effects. The formula should be recognisable to
lme
(see the description of the argument random
for lme
).
The names in the formula random
may be any of the covariates
supplied by data
.
Additionally the formula may involve the name
id
, which is a factor representing the
serial number ( to
) of the point pattern in the
list
X
.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Baddeley, A., Bischof, L., Sintorn, I.-M., Haggarty, S., Bell, M. and Turner, R. Analysis of a designed experiment where the response is a spatial point pattern. In preparation.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Bell, M. and Grunwald, G. (2004) Mixed models for the analysis of replicated spatial point patterns. Biostatistics 5, 633–648.
ppm
,
print.mppm
,
summary.mppm
,
coef.mppm
,
# Waterstriders data H <- hyperframe(Y = waterstriders) mppm(Y ~ 1, data=H) mppm(Y ~ 1, data=H, Strauss(7)) mppm(Y ~ id, data=H) mppm(Y ~ x, data=H) # Synthetic data from known model n <- 10 H <- hyperframe(V=1:n, U=runif(n, min=-1, max=1), M=factor(letters[1 + (1:n) %% 3])) H$Z <- setcov(square(1)) H$U <- with(H, as.im(U, as.rectangle(Z))) H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z)))) fit <- mppm(Y ~Z + U + V, data=H)
# Waterstriders data H <- hyperframe(Y = waterstriders) mppm(Y ~ 1, data=H) mppm(Y ~ 1, data=H, Strauss(7)) mppm(Y ~ id, data=H) mppm(Y ~ x, data=H) # Synthetic data from known model n <- 10 H <- hyperframe(V=1:n, U=runif(n, min=-1, max=1), M=factor(letters[1 + (1:n) %% 3])) H$Z <- setcov(square(1)) H$U <- with(H, as.im(U, as.rectangle(Z))) H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z)))) fit <- mppm(Y ~Z + U + V, data=H)
Defines an object representing a signed measure or vector-valued measure on a spatial domain.
msr(qscheme, discrete, density, check=TRUE)
msr(qscheme, discrete, density, check=TRUE)
qscheme |
A quadrature scheme (object of class |
discrete |
Vector or matrix containing the values (masses) of the discrete component
of the measure, for each of the data points in |
density |
Vector or matrix containing values of the density of the
diffuse component of the measure, for each of the
quadrature points in |
check |
Logical. Whether to check validity of the arguments. |
This function creates an object that represents a signed or vector valued measure on the two-dimensional plane. It is not normally called directly by the user.
A signed measure is a classical mathematical object (Diestel and Uhl, 1977) which can be visualised as a collection of electric charges, positive and/or negative, spread over the plane. Electric charges may be concentrated at specific points (atoms), or spread diffusely over a region.
An object of class "msr"
represents a signed (i.e. real-valued)
or vector-valued measure in the spatstat package.
Spatial residuals for point process models
(Baddeley et al, 2005, 2008) take the form of a real-valued
or vector-valued measure. The function
residuals.ppm
returns an object of
class "msr"
representing the residual measure.
Various other diagnostic tools such as dfbetas.ppm
and
dffit.ppm
also return an object of class "msr"
.
The function msr
would not normally be called directly by the
user. It is the low-level creator function that
makes an object of class "msr"
from raw data.
The first argument qscheme
is a quadrature scheme (object of
class "quad"
). It is typically created by quadscheme
or
extracted from a fitted point process model using
quad.ppm
. A quadrature scheme contains both data points
and dummy points. The data points of qscheme
are used as the locations
of the atoms of the measure. All quadrature points
(i.e. both data points and dummy points)
of qscheme
are used as sampling points for the density
of the continuous component of the measure.
The argument discrete
gives the values of the
atomic component of the measure for each data point in qscheme
.
It should be either a numeric vector with one entry for each
data point, or a numeric matrix with one row
for each data point.
The argument density
gives the values of the density
of the diffuse component of the measure, at each
quadrature point in qscheme
.
It should be either a numeric vector with one entry for each
quadrature point, or a numeric matrix with one row
for each quadrature point.
If both discrete
and density
are vectors
(or one-column matrices) then the result is a signed (real-valued) measure.
Otherwise, the result is a vector-valued measure, with the dimension
of the vector space being determined by the number of columns
in the matrices discrete
and/or density
.
(If one of these is a -column matrix and the other
is a 1-column matrix, then the latter is replicated to
columns).
The class "msr"
has methods for print
,
plot
and [
.
There is also a function Smooth.msr
for smoothing a measure.
An object of class "msr"
.
Objects of class "msr"
, representing measures, are returned by the
functions residuals.ppm
, dfbetas.ppm
,
dffit.ppm
and possibly by other functions.
There are methods for printing and plotting a measure, along with
many other operations, which can be listed by
typing methods(class="msr")
.
The print
and summary
methods report basic information
about a measure, such as the total value of the measure, and the
spatial domain on which it is defined.
The plot
method displays the measure. It is documented separately in
plot.msr
.
A measure can be smoothed using Smooth.msr
, yielding a
pixel image which is sometimes easier to interpret than the plot
of the measure itself.
The subset operator [
can be used to restrict the measure to
a subregion of space, or to extract one of the scalar components of a
vector-valued measure. It is documented separately in
[.msr
.
The total value of a measure, or the value on a subregion,
can be obtained using integral.msr
.
The value of a measure m
on a subregion
B
can be obtained by integral(m, domain=B)
or
integral(m[B])
. The values of a measure m
on each tile
of a tessellation A
can be obtained by
integral(m, domain=A)
.
Some mathematical operations on measures are supported, such as multiplying a measure by a single number, or adding two measures.
Measures can be separated into components in different ways using
as.layered.msr
, unstack.msr
and split.msr
.
Internal components of the data structure of an "msr"
object
can be extracted using with.msr
.
Adrian Baddeley [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
Diestel, J. and Uhl, J.J. Jr (1977) Vector measures. Providence, RI, USA: American Mathematical Society.
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
plot.msr
,
Smooth.msr
,
[.msr
,
with.msr
,
split.msr
,
Ops.msr
,
measureVariation
,
measureWeighted
,
measureContinuous
.
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp rs <- residuals(fit, type="score") rs colnames(rs) # An equivalent way to construct the Pearson residual measure by hand Q <- quad.ppm(fit) lambda <- fitted(fit) slam <- sqrt(lambda) Z <- is.data(Q) m <- msr(Q, discrete=1/slam[Z], density = -slam) m
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp rs <- residuals(fit, type="score") rs colnames(rs) # An equivalent way to construct the Pearson residual measure by hand Q <- quad.ppm(fit) lambda <- fitted(fit) slam <- sqrt(lambda) Z <- is.data(Q) m <- msr(Q, discrete=1/slam[Z], density = -slam) m
Creates an instance of the multitype hard core point process model which can then be fitted to point pattern data.
MultiHard(hradii, types=NULL)
MultiHard(hradii, types=NULL)
hradii |
Matrix of hard core radii |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
This is a multitype version of the hard core process.
A pair of points
of types and
must not lie closer than
units apart.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the MultiStrauss interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix hradii
.
The matrix hradii
must be symmetric, with entries
which are either positive numbers or NA
.
A value of NA
indicates that no distance constraint should be applied
for this combination of types.
Note that only the hardcore radii
are specified in MultiHard
.
The canonical parameters
are estimated by
ppm()
, not fixed in
MultiHard()
.
An object of class "interact"
describing the interpoint interaction
structure of the multitype hard core process with
hard core radii .
In order that ppm
can fit the multitype hard core
model correctly to a point pattern X
, this pattern must
be marked, with markformat
equal to vector
and the
mark vector marks(X)
must be a factor. If the argument
types
is specified it is interpreted as a set of factor
levels and this set must equal levels(marks(X))
.
Before spatstat version 1.37-0
,
the syntax of this function was different:
MultiHard(types=NULL, hradii)
.
The new code attempts to handle the old syntax as well.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
ppm
,
pairwise.family
,
ppm.object
,
MultiStrauss
,
MultiStraussHard
,
Strauss
.
See ragsMultiHard
and rmh
for
simulation.
h <- matrix(c(1,2,2,1), nrow=2,ncol=2) # prints a sensible description of itself MultiHard(h) # Fit the stationary multitype hardcore process to `amacrine' # with hard core operating only between cells of the same type. h <- 0.02 * matrix(c(1, NA, NA, 1), nrow=2,ncol=2) ppm(amacrine ~1, MultiHard(h))
h <- matrix(c(1,2,2,1), nrow=2,ncol=2) # prints a sensible description of itself MultiHard(h) # Fit the stationary multitype hardcore process to `amacrine' # with hard core operating only between cells of the same type. h <- 0.02 * matrix(c(1, NA, NA, 1), nrow=2,ncol=2) ppm(amacrine ~1, MultiHard(h))
Creates an instance of the multitype Strauss point process model which can then be fitted to point pattern data.
MultiStrauss(radii, types=NULL)
MultiStrauss(radii, types=NULL)
radii |
Matrix of interaction radii |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
The (stationary) multitype
Strauss process with types, with interaction radii
and
parameters
and
is the pairwise interaction point process
in which each point of type
contributes a factor
to the
probability density of the point pattern, and a pair of points
of types
and
closer than
units apart contributes a factor
to the density.
The nonstationary multitype Strauss process is similar except that
the contribution of each individual point
is a function
of location and type, rather than a constant beta.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the multitype
Strauss process pairwise interaction is
yielded by the function MultiStrauss()
. See the examples below.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the MultiStrauss interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrix radii
.
The matrix radii
must be symmetric, with entries
which are either positive numbers or NA
.
A value of NA
indicates that no interaction term should be included
for this combination of types.
Note that only the interaction radii are
specified in MultiStrauss
. The canonical
parameters and
are estimated by
ppm()
, not fixed in MultiStrauss()
.
An object of class "interact"
describing the interpoint interaction
structure of the multitype Strauss process with
interaction radii .
In order that ppm
can fit the multitype Strauss
model correctly to a point pattern X
, this pattern must
be marked, with markformat
equal to vector
and the
mark vector marks(X)
must be a factor. If the argument
types
is specified it is interpreted as a set of factor
levels and this set must equal levels(marks(X))
.
Before spatstat version 1.37-0
,
the syntax of this function was different:
MultiStrauss(types=NULL, radii)
.
The new code attempts to handle the old syntax as well.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
ppm
,
pairwise.family
,
ppm.object
,
Strauss
,
MultiHard
r <- matrix(c(1,2,2,1), nrow=2,ncol=2) MultiStrauss(r) # prints a sensible description of itself r <- 0.03 * matrix(c(1,2,2,1), nrow=2,ncol=2) X <- amacrine ppm(X ~1, MultiStrauss(r)) # fit the stationary multitype Strauss process to `amacrine' ppm(X ~polynom(x,y,3), MultiStrauss(r, c("off","on"))) # fit a nonstationary multitype Strauss process with log-cubic trend
r <- matrix(c(1,2,2,1), nrow=2,ncol=2) MultiStrauss(r) # prints a sensible description of itself r <- 0.03 * matrix(c(1,2,2,1), nrow=2,ncol=2) X <- amacrine ppm(X ~1, MultiStrauss(r)) # fit the stationary multitype Strauss process to `amacrine' ppm(X ~polynom(x,y,3), MultiStrauss(r, c("off","on"))) # fit a nonstationary multitype Strauss process with log-cubic trend
Creates an instance of the multitype/hard core Strauss point process model which can then be fitted to point pattern data.
MultiStraussHard(iradii, hradii, types=NULL)
MultiStraussHard(iradii, hradii, types=NULL)
iradii |
Matrix of interaction radii |
hradii |
Matrix of hard core radii |
types |
Optional; vector of all possible types (i.e. the possible levels
of the |
This is a hybrid of the multitype Strauss process
(see MultiStrauss
) and the hard core process
(case of the Strauss process).
A pair of points
of types
and
must not lie closer than
units apart;
if the pair lies more than
and less than
units apart, it
contributes a factor
to the probability density.
The argument types
need not be specified in normal use.
It will be determined automatically from the point pattern data set
to which the MultiStraussHard interaction is applied,
when the user calls ppm
.
However, the user should be confident that
the ordering of types in the dataset corresponds to the ordering of
rows and columns in the matrices iradii
and hradii
.
The matrices iradii
and hradii
must be symmetric, with entries
which are either positive numbers or NA
.
A value of NA
indicates that no interaction term should be included
for this combination of types.
Note that only the interaction radii and hardcore radii
are specified in MultiStraussHard
.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
MultiStraussHard()
.
An object of class "interact"
describing the interpoint interaction
structure of the multitype/hard core Strauss process with
interaction radii
and hard core radii
.
In order that ppm
can fit the multitype/hard core
Strauss model correctly to a point pattern X
, this pattern
must be marked, with markformat
equal to vector
and the mark vector marks(X)
must be a factor. If the
argument types
is specified it is interpreted as a set of
factor levels and this set must equal levels(marks(X))
.
Before spatstat version 1.37-0
,
the syntax of this function was different:
MultiStraussHard(types=NULL, iradii, hradii)
.
The new code attempts to handle the old syntax as well.
Adrian Baddeley [email protected]
, Rolf Turner [email protected]
and Ege Rubak [email protected]
ppm
,
pairwise.family
,
ppm.object
,
MultiStrauss
,
MultiHard
,
Strauss
r <- matrix(3, nrow=2,ncol=2) h <- matrix(c(1,2,2,1), nrow=2,ncol=2) MultiStraussHard(r,h) # prints a sensible description of itself r <- 0.04 * matrix(c(1,2,2,1), nrow=2,ncol=2) h <- 0.02 * matrix(c(1,NA,NA,1), nrow=2,ncol=2) X <- amacrine fit <- ppm(X ~1, MultiStraussHard(r,h)) # fit stationary multitype hardcore Strauss process to `amacrine'
r <- matrix(3, nrow=2,ncol=2) h <- matrix(c(1,2,2,1), nrow=2,ncol=2) MultiStraussHard(r,h) # prints a sensible description of itself r <- 0.04 * matrix(c(1,2,2,1), nrow=2,ncol=2) h <- 0.02 * matrix(c(1,NA,NA,1), nrow=2,ncol=2) X <- amacrine fit <- ppm(X ~1, MultiStraussHard(r,h)) # fit stationary multitype hardcore Strauss process to `amacrine'
Returns a summary function which is constant with value equal to the number of points in the point pattern.
npfun(X, ..., r)
npfun(X, ..., r)
X |
Point pattern. |
... |
Ignored. |
r |
Vector of values of the distance argument |
This function is normally not called by the user. Instead it is passed
as an argument to the function psst
.
Object of class "fv"
representing a constant function.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
fit0 <- ppm(cells, ~1, nd=10) v <- psst(fit0, npfun)
fit0 <- ppm(cells, ~1, nd=10) v <- psst(fit0, npfun)
For a model that was fitted by optimisation, compute the values of the objective function in a neighbourhood of the optimal value.
objsurf(x, ...) ## S3 method for class 'dppm' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, enclose=FALSE, ratio = 1.5, verbose = TRUE) ## S3 method for class 'kppm' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, enclose=FALSE, ratio = 1.5, verbose = TRUE) ## S3 method for class 'minconfit' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, ratio = 1.5, verbose = TRUE)
objsurf(x, ...) ## S3 method for class 'dppm' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, enclose=FALSE, ratio = 1.5, verbose = TRUE) ## S3 method for class 'kppm' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, enclose=FALSE, ratio = 1.5, verbose = TRUE) ## S3 method for class 'minconfit' objsurf(x, ..., ngrid = 32, xlim=NULL, ylim=NULL, ratio = 1.5, verbose = TRUE)
x |
Some kind of model that was fitted
by finding the optimal value of an objective function.
An object of class |
... |
Extra arguments are usually ignored. |
ngrid |
Number of grid points to evaluate along each axis.
Either a single integer, or a pair of integers.
For example |
xlim , ylim
|
Optional. Numeric vectors of length 2, specifying the limits for the two parameters to be considered. |
enclose |
Logical value specifying whether the default values of
|
ratio |
Number greater than 1 determining the default ranges of parameter values. See Details. |
verbose |
Logical value indicating whether to print progress reports. |
The object x
should be some kind of model that was fitted
by maximising or minimising the value of an objective function.
The objective function will be evaluated on a grid of
values of the model parameters.
Currently the following types of objects are accepted:
an object of class "dppm"
representing a
determinantal point process.
See dppm
.
an object of class "kppm"
representing a
cluster point process or Cox point process.
See kppm
.
an object of class "minconfit"
representing a
minimum-contrast fit between a summary function and its
theoretical counterpart.
See mincontrast
.
The result is an object of class "objsurf"
which can be
printed and plotted: see methods.objsurf
.
The range of parameter values to be considered is determined by
xlim
and ylim
.
The default values of xlim
and ylim
are
chosen as follows.
if enclose=FALSE
(the default),
the default values of xlim
and ylim
are
the ranges from opt/ratio
to opt * ratio
where opt
is the optimal parameter value on the surface.
If enclose=TRUE
,
and if x
contains a trajectory
(history of function evaluations),
then xlim
and ylim
will be the ranges
of parameter values examined in the trajectory.
An object of class "objsurf"
which can be
printed and plotted.
Essentially a list containing entries x
, y
, z
giving the parameter values and objective function values.
There are methods for plot
, print
, summary
,
image
, contour
and persp
.
Adrian Baddeley [email protected] and Ege Rubak [email protected].
methods.objsurf
,
kppm
,
mincontrast
fit <- kppm(redwood ~ 1, "Thomas") os <- objsurf(fit) if(interactive()) { plot(os) contour(os, add=TRUE) persp(os) }
fit <- kppm(redwood ~ 1, "Thomas") os <- objsurf(fit) if(interactive()) { plot(os) contour(os, add=TRUE) persp(os) }
These group generic methods for the class "msr"
allow the arithmetic operators
+
, -
, *
and /
to be applied
directly to measures.
## S3 methods for group generics have prototypes: Ops(e1, e2)
## S3 methods for group generics have prototypes: Ops(e1, e2)
e1 , e2
|
objects of class |
Arithmetic operators on a measure A
are only defined
in some cases. The arithmetic operator is effectively applied
to the value of A(W)
for every spatial domain W
.
If the result is a measure, then this operation is valid.
If A
is a measure (object of class "msr"
)
then the operations -A
and +A
are defined.
If A
and B
are measures with the same dimension
(i.e. both are scalar-valued, or both are k
-dimensional vector-valued)
then A + B
and A - B
are defined.
If A
is a measure and z
is a numeric value,
then A * z
and A / z
are defined,
and z * A
is defined.
Another measure (object of class "msr"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp -rp 2 * rp rp /2 rp - rp rr <- residuals(fit, type="raw") rp - rr
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rp -rp 2 * rp rp /2 rp - rp rr <- residuals(fit, type="raw") rp - rr
Creates an instance of an Ord-type interaction point process model which can then be fitted to point pattern data.
Ord(pot, name)
Ord(pot, name)
pot |
An S language function giving the user-supplied interaction potential. |
name |
Character string. |
Ord's point process model (Ord, 1977) is a Gibbs point process
of infinite order. Each point in the point pattern
contributes a factor
where
is the area of the
tile associated with
in the Dirichlet tessellation of
.
Ord (1977) proposed fitting this model to forestry data
when has a simple “threshold” form. That model is
implemented in our function
OrdThresh
.
The present function Ord
implements the case of a
completely general Ord potential
specified as an S language function
pot
.
This is experimental.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.
An object describing the family of all Ord interaction point processes
Advanced Use Only!
This structure would not normally be touched by the user. It describes the family of point process models introduced by Ord (1977).
If you need to create a specific Ord-type model for use in
analysis, use the function OrdThresh
or
Ord
.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.
pairwise.family
,
pairsat.family
,
Ord
,
OrdThresh
Creates an instance of Ord's point process model which can then be fitted to point pattern data.
OrdThresh(r)
OrdThresh(r)
r |
Positive number giving the threshold value for Ord's model. |
Ord's point process model (Ord, 1977) is a Gibbs point process
of infinite order. Each point in the point pattern
contributes a factor
where
is the area of the
tile associated with
in the Dirichlet tessellation of
. The function
is
simply
if
and
if
,
where
is called the threshold value.
This function creates an instance of Ord's model with a given
value of . It can then be fitted to point process data
using
ppm
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.
Creates an instance of a pairwise interaction point process model with piecewise constant potential function. The model can then be fitted to point pattern data.
PairPiece(r)
PairPiece(r)
r |
vector of jump points for the potential function |
A pairwise interaction point process in a bounded region is a stochastic point process with probability density of the form
where represent the
points of the pattern. The first product on the right hand side is
over all points of the pattern; the second product is over all
unordered pairs of points of the pattern.
Thus each point of the pattern contributes a factor
to the probability density, and each pair of
points
contributes a factor
to the density.
The pairwise interaction term is called
piecewise constant
if it depends only on the distance between
and
,
say
, and
is a piecewise constant
function (a function which is constant except for jumps at a finite
number of places). The use of piecewise constant interaction terms
was first suggested by Takacs (1986).
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the piecewise constant pairwise
interaction is yielded by the function PairPiece()
.
See the examples below.
The entries of r
must be strictly increasing, positive numbers.
They are interpreted as the points of discontinuity of .
It is assumed that
for all
where
is the maximum value in
r
. Thus the
model has as many regular parameters (see ppm
)
as there are entries in r
. The -th regular parameter
is the logarithm of the value of the
interaction function
on the interval
.
If r
is a single number, this model is similar to the
Strauss process, see Strauss
. The difference is that
in PairPiece
the interaction function is continuous on the
right, while in Strauss
it is continuous on the left.
The analogue of this model for multitype point processes has not yet been implemented.
An object of class "interact"
describing the interpoint interaction
structure of a point process. The process is a pairwise interaction process,
whose interaction potential is piecewise constant, with jumps
at the distances given in the vector .
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Takacs, R. (1986) Estimator for the pair potential of a Gibbsian point process. Statistics 17, 429–433.
ppm
,
pairwise.family
,
ppm.object
,
Strauss
rmh.ppm
PairPiece(c(0.1,0.2)) # prints a sensible description of itself ppm(cells ~1, PairPiece(r = c(0.05, 0.1, 0.2))) # fit a stationary piecewise constant pairwise interaction process ppm(cells ~polynom(x,y,3), PairPiece(c(0.05, 0.1))) # nonstationary process with log-cubic polynomial trend
PairPiece(c(0.1,0.2)) # prints a sensible description of itself ppm(cells ~1, PairPiece(r = c(0.05, 0.1, 0.2))) # fit a stationary piecewise constant pairwise interaction process ppm(cells ~polynom(x,y,3), PairPiece(c(0.05, 0.1))) # nonstationary process with log-cubic polynomial trend
An object describing the Saturated Pairwise Interaction family of point process models
Advanced Use Only!
This structure would not normally be touched by the user. It describes the “saturated pairwise interaction” family of point process models.
If you need to create a specific interaction model for use in
spatial pattern analysis, use the function Saturated()
or the two existing implementations of models in this family,
Geyer()
and SatPiece()
.
Geyer (1999) introduced the “saturation process”, a modification of the
Strauss process in which the total contribution
to the potential from each point (from its pairwise interaction with all
other points) is trimmed to a maximum value .
This model is implemented in the function
Geyer()
.
The present class pairsat.family
is the
extension of this saturation idea to all pairwise interactions.
Note that the resulting models are no longer pairwise interaction
processes - they have interactions of infinite order.
pairsat.family
is an object of class "isf"
containing a function pairwise$eval
for
evaluating the sufficient statistics of any saturated pairwise interaction
point process model in which the original pair potentials
take an exponential family form.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79–140.
Geyer
to create the Geyer saturation process.
SatPiece
to create a saturated process with
piecewise constant pair potential.
Saturated
to create a more general saturation model.
Other families:
inforder.family
,
ord.family
,
pairwise.family
.
Creates an instance of a pairwise interaction point process model which can then be fitted to point pattern data.
Pairwise(pot, name, par, parnames, printfun)
Pairwise(pot, name, par, parnames, printfun)
pot |
An R language function giving the user-supplied pairwise interaction potential. |
name |
Character string. |
par |
List of numerical values for irregular parameters |
parnames |
Vector of names of irregular parameters |
printfun |
Do not specify this argument: for internal use only. |
This code constructs a member of the
pairwise interaction family pairwise.family
with arbitrary pairwise interaction potential given by
the user.
Each pair of points in the point pattern contributes a factor
to the probability density, where
is the distance
between the two points. The factor term
is
provided is finite,
where
is the coefficient vector in the model.
The function pot
must take as its first argument
a matrix of interpoint distances, and evaluate the
potential for each of these distances. The result must be
either a matrix with the same dimensions as its input,
or an array with its first two dimensions the same as its input
(the latter case corresponds to a vector-valued potential).
If irregular parameters are present, then the second argument
to pot
should be a vector of the same type as par
giving those parameter values.
The values returned by pot
may be finite numeric values,
or -Inf
indicating a hard core (that is, the corresponding
interpoint distance is forbidden). We define
if
.
Thus, a potential value of minus infinity is always interpreted
as corresponding to
, regardless of the sign
and magnitude of
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
ppm
,
pairwise.family
,
ppm.object
#This is the same as StraussHard(r=0.7,h=0.05) strpot <- function(d,par) { r <- par$r h <- par$h value <- (d <= r) value[d < h] <- -Inf value } mySH <- Pairwise(strpot, "StraussHard process", list(r=0.7,h=0.05), c("interaction distance r", "hard core distance h")) ppm(cells ~ 1, mySH, correction="isotropic") # Fiksel (1984) double exponential interaction # see Stoyan, Kendall, Mecke 1987 p 161 fikspot <- function(d, par) { r <- par$r h <- par$h zeta <- par$zeta value <- exp(-zeta * d) value[d < h] <- -Inf value[d > r] <- 0 value } Fiksel <- Pairwise(fikspot, "Fiksel double exponential process", list(r=3.5, h=1, zeta=1), c("interaction distance r", "hard core distance h", "exponential coefficient zeta")) fit <- ppm(unmark(spruces) ~1, Fiksel, rbord=3.5) fit plot(fitin(fit), xlim=c(0,4)) coef(fit) # corresponding values obtained by Fiksel (1984) were -1.9 and -6.0
#This is the same as StraussHard(r=0.7,h=0.05) strpot <- function(d,par) { r <- par$r h <- par$h value <- (d <= r) value[d < h] <- -Inf value } mySH <- Pairwise(strpot, "StraussHard process", list(r=0.7,h=0.05), c("interaction distance r", "hard core distance h")) ppm(cells ~ 1, mySH, correction="isotropic") # Fiksel (1984) double exponential interaction # see Stoyan, Kendall, Mecke 1987 p 161 fikspot <- function(d, par) { r <- par$r h <- par$h zeta <- par$zeta value <- exp(-zeta * d) value[d < h] <- -Inf value[d > r] <- 0 value } Fiksel <- Pairwise(fikspot, "Fiksel double exponential process", list(r=3.5, h=1, zeta=1), c("interaction distance r", "hard core distance h", "exponential coefficient zeta")) fit <- ppm(unmark(spruces) ~1, Fiksel, rbord=3.5) fit plot(fitin(fit), xlim=c(0,4)) coef(fit) # corresponding values obtained by Fiksel (1984) were -1.9 and -6.0
An object describing the family of all pairwise interaction Gibbs point processes.
Advanced Use Only!
This structure would not normally be touched by the user. It describes the pairwise interaction family of point process models.
If you need to create a specific pairwise interaction model for use in
modelling, use the function Pairwise
or one of the existing
functions listed below.
Anyway, pairwise.family
is an object of class "isf"
containing a function pairwise.family$eval
for
evaluating the sufficient statistics of any pairwise interaction
point process model taking an exponential family form.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Other families:
pairsat.family
,
ord.family
,
inforder.family
.
Pairwise interactions:
Poisson
,
Pairwise
,
PairPiece
,
Fiksel
,
Hardcore
,
LennardJones
,
MultiHard
,
MultiStrauss
,
MultiStraussHard
,
Strauss
,
StraussHard
,
Softcore
.
Other interactions:
AreaInter
,
Geyer
,
Saturated
,
Ord
,
OrdThresh
.
Given a fitted cluster process or Cox process model, calculate a diagnostic which compares nonparametric and parametric estimates of the Palm intensity.
palmdiagnose(object, ..., breaks = 30, trim = 30, rmax=Inf)
palmdiagnose(object, ..., breaks = 30, trim = 30, rmax=Inf)
object |
Fitted model (object of class |
... |
Optional.
Additional arguments which are fitted models of class |
breaks |
Optional argument passed to |
trim |
Optional. Maximum value of the translation edge correction weight. |
rmax |
Optional. Maximum interpoint distance |
This function computes the diagnostic proposed by Tanaka, Ogata and Stoyan (2008, Section 2.3) for assessing goodness-of-fit of a Neyman-Scott cluster process model to a point pattern dataset.
The fitted model object
should be an object of class "kppm"
representing a Neyman-Scott cluster process model or a Cox process
model. In the current implementation, the model must be stationary.
The code computes parametric and non-parametric estimates of the
Palm intensity , loosely speaking,
the intensity of the point process given that there is a point at the origin.
The parametric estimate is obtained from the fitted model by
substituting the fitted parameter estimates into
expressions for the pair correlation and the intensity.
The non-parametric estimate is obtained by considering all pairs of
data points, dividing the range of interpoint distances into
several equally-spaced bands (determined by the argument
breaks
), counting the number of pairs of points whose
interpoint distances fall in each band, and numerically adjusting for
edge effects. Tanaka, Ogata and Stoyan (2008) used the
periodic (toroidal) edge correction; our code uses the
translation edge correction so that the method can be applied to
data in any window.
The result is a function value table (object of class "fv"
)
containing the nonparametric and parametric estimates of the Palm
intensity. The result also belongs to the class "palmdiag"
which has a method for plot
. The default behaviour of
plot.palmdiag
is to plot the model fit as a curve,
and to display the nonparametric estimates as dots; this is the plot style
proposed by Tanaka, Ogata and Stoyan (2008). Alternative display
styles are also supported by plot.palmdiag
.
For computational efficiency, the argument rmax
specifies the maximum value of interpoint distance
for which estimates of
shall be computed.
The default
rmax = Inf
implies
there is no constraint on interpoint distance,
and the resulting function object contains estimates of
up to
the maximum distance that would have been observable
in the window containing the original point pattern data.
If there are additional arguments ...
which are fitted models
of class "kppm"
, or if object
is a list of fitted models
of class "kppm"
, then the parametric estimates for each
of the fitted models will be included in the resulting function object.
If names are attached to these fitted models, the names will be used
in the resulting function object.
Function value table (object of class "fv"
) containing the
nonparametric and parametric estimates of the Palm intensity.
Also belongs to the class "palmdiag"
which has a plot
method.
Adrian Baddeley [email protected].
Tanaka, U., Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott Point Processes. Biometrical Journal 50, 1, 43–57.
fitK <- kppm(redwood) R <- palmdiagnose(fitK) plot(R) fitg <- kppm(redwood, statistic="pcf") R2 <- palmdiagnose(A=fitK, B=fitg) plot(R2)
fitK <- kppm(redwood) R <- palmdiagnose(fitK) plot(R) fitg <- kppm(redwood, statistic="pcf") R2 <- palmdiagnose(A=fitK, B=fitg) plot(R2)
These functions can be passed to pairs
or
coplot
to determine what kind of plotting is done in each panel
of a multi-panel graphical display.
panel.contour(x, y, ..., sigma = NULL) panel.image(x, y, ..., sigma = NULL) panel.histogram(x, ...)
panel.contour(x, y, ..., sigma = NULL) panel.image(x, y, ..., sigma = NULL) panel.histogram(x, ...)
x , y
|
Coordinates of points in a scatterplot. |
... |
Extra graphics arguments, passed to |
sigma |
Bandwidth of kernel smoother, on a scale where
|
These functions can serve as one of the arguments panel
,
lower.panel
, upper.panel
, diag.panel
passed to graphics commands like
pairs
or coplot
,
to determine what kind of plotting is done in each panel
of a multi-panel graphical display. In particular they work
with pairs.im
.
The functions panel.contour
and panel.contour
are suitable for the off-diagonal plots which involve
two datasets x
and y
.
They first rescale x
and y
to the unit square,
then apply kernel smoothing with bandwidth sigma
using density.ppp
.
Then panel.contour
draws a contour plot
while panel.image
draws a colour image.
The function panel.histogram
is suitable for the
diagonal plots which involve a single dataset x
.
It displays a histogram of the data.
Null.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
pairs.im
,
pairs.default
,
panel.smooth
pairs(bei.extra, panel = panel.contour, diag.panel = panel.histogram) with(bei.extra, pairs(grad, elev, panel = panel.image, diag.panel = panel.histogram)) pairs(marks(finpines), panel=panel.contour, diag.panel=panel.histogram)
pairs(bei.extra, panel = panel.contour, diag.panel = panel.histogram) with(bei.extra, pairs(grad, elev, panel = panel.image, diag.panel = panel.histogram)) pairs(marks(finpines), panel=panel.contour, diag.panel=panel.histogram)
Given a cluster process model, calculate the probability that a point of the process has any siblings.
panysib(object)
panysib(object)
object |
Fitted cluster process model (object of class |
In a Poisson cluster process, two points are called siblings
if they belong to the same cluster, that is, if they had the same
parent point. This function computes the probability that a
given random point has any
siblings.
If object
is a stationary point process, the result is a single
number, which is the probability that a typical point of the process
has any siblings.
If this number is small,
then the process is approximately a homogeneous Poisson process
(complete spatial randomness). The converse is not true (Baddeley et al, 2022).
Otherwise, the result is a pixel image, in which the value at any
location u
is the conditional probability,
given there is a point of the
process at u
, that this point has any siblings.
If the pixel values are all small,
then the process is approximately an inhomogeneous Poisson process.
This concept was proposed by Baddeley et al (2022).
A single number (if object
is a stationary point process)
or a pixel image (otherwise).
Adrian Baddeley [email protected].
Baddeley, A., Davies, T.M., Hazelton, M.L., Rakshit, S. and Turner, R.
(2022)
Fundamental problems in fitting spatial cluster process models.
Spatial Statistics 52, 100709.
DOI: 10.1016/j.spasta.2022.100709
fit <- kppm(redwood ~ polynom(x,y,2)) plot(panysib(fit))
fit <- kppm(redwood ~ polynom(x,y,2)) plot(panysib(fit))
Given a fitted model of some kind, this function extracts all the parameters needed to specify the model, and returns them as a list.
parameters(model, ...) ## S3 method for class 'dppm' parameters(model, ...) ## S3 method for class 'kppm' parameters(model, ...) ## S3 method for class 'slrm' parameters(model, ...) ## S3 method for class 'ppm' parameters(model, ...) ## S3 method for class 'profilepl' parameters(model, ...) ## S3 method for class 'fii' parameters(model, ...) ## S3 method for class 'interact' parameters(model, ...)
parameters(model, ...) ## S3 method for class 'dppm' parameters(model, ...) ## S3 method for class 'kppm' parameters(model, ...) ## S3 method for class 'slrm' parameters(model, ...) ## S3 method for class 'ppm' parameters(model, ...) ## S3 method for class 'profilepl' parameters(model, ...) ## S3 method for class 'fii' parameters(model, ...) ## S3 method for class 'interact' parameters(model, ...)
model |
A fitted model of some kind. |
... |
Arguments passed to methods. |
The argument model
should be a fitted model of some kind.
This function extracts all the parameters that would be needed to
specify the model, and returns them as a list.
The function parameters
is generic, with methods
for class "ppm"
, "kppm"
, "dppm"
and "profilepl"
and other classes.
A named list, whose format depends on the fitted model.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
parameters(Strauss(0.1)) fit1 <- ppm(cells ~ x, Strauss(0.1)) parameters(fit1) fit2 <- kppm(redwood ~ x, "Thomas") parameters(fit2)
parameters(Strauss(0.1)) fit1 <- ppm(cells ~ x, Strauss(0.1)) parameters(fit1) fit2 <- kppm(redwood ~ x, "Thomas") parameters(fit2)
Computes the smoothed partial residuals, a diagnostic for transformation of a covariate in a Poisson point process model.
parres(model, covariate, ..., smooth.effect=FALSE, subregion=NULL, bw = "nrd0", adjust=1, from = NULL, to = NULL, n = 512, bw.input = c("points", "quad"), bw.restrict=FALSE, covname)
parres(model, covariate, ..., smooth.effect=FALSE, subregion=NULL, bw = "nrd0", adjust=1, from = NULL, to = NULL, n = 512, bw.input = c("points", "quad"), bw.restrict=FALSE, covname)
model |
Fitted point process model (object of class |
covariate |
The covariate of interest.
Either a character string matching the name of one of
the canonical covariates in the model, or one of the names
|
smooth.effect |
Logical. Determines the choice of algorithm. See Details. |
subregion |
Optional. A window (object of class |
bw |
Smoothing bandwidth or bandwidth rule
(passed to |
adjust |
Smoothing bandwidth adjustment factor
(passed to |
n , from , to
|
Arguments passed to |
... |
Additional arguments passed to |
bw.input |
Character string specifying the input data used for automatic bandwidth selection. |
bw.restrict |
Logical value, specifying whether bandwidth selection is performed using
data from the entire spatial domain or from the |
covname |
Optional. Character string to use as the name of the covariate. |
This command computes the smoothed partial residual diagnostic (Baddeley, Chang, Song and Turner, 2012) for the transformation of a covariate in a Poisson point process model.
The argument model
must be a fitted Poisson point process model.
The diagnostic works in two different ways:
The argument covariate
may be a character string
which is the name of one of the canonical covariates in the
model.
The canonical covariates are the
functions that appear
in the expression for the Poisson point process intensity
at spatial location .
Type
names(coef(model))
to see the names of the
canonical covariates in model
.
If the selected covariate is , then
the diagnostic plot concerns the model term
. The plot shows a smooth
estimate of a function
that should replace this linear
term, that is,
should be
replaced by
. The linear function is
also plotted as a dotted line.
If the argument covariate
is a pixel image
(object of class "im"
) or a function(x,y)
,
it is assumed to provide the values of a covariate that is
not present in the model.
Alternatively covariate
can be the name of a
covariate that was supplied when the model was fitted
(i.e. in the call to ppm
)
but which does not feature in the model formula.
In either case we speak of a new covariate .
If the fitted model intensity is
then we consider modifying this to
where
is some function. The diagnostic plot shows
an estimate of
.
Warning: in this case the diagnostic is not theoretically
justified. This option is provided for research purposes.
Alternatively covariate
can be one of the character strings
"x"
or "y"
signifying the Cartesian coordinates.
The behaviour here depends on whether the coordinate was one of the
canonical covariates in the model.
If there is more than one canonical covariate in the model
that depends on the specified covariate
, then
the covariate effect is computed using all these canonical covariates.
For example in a log-quadratic model which includes the terms x
and
I(x^2)
, the quadratic effect involving both these terms
will be computed.
There are two choices for the algorithm.
If smooth.effect=TRUE
, the fitted covariate effect (according
to model
) is added to the point process residuals, then
smoothing is applied to these values. If smooth.effect=FALSE
,
the point process residuals are smoothed first, and then the fitted
covariate effect is added to the result.
The smoothing bandwidth is controlled by the arguments
bw
, adjust
, bw.input
and bw.restrict
.
If bw
is a numeric value, then
the bandwidth is taken to be adjust * bw
.
If bw
is a string representing a bandwidth selection rule
(recognised by density.default
)
then the bandwidth is selected by this rule.
The data used for automatic bandwidth selection are
specified by bw.input
and bw.restrict
.
If bw.input="points"
(the default) then bandwidth selection is
based on the covariate values at the points of the original point
pattern dataset to which the model was fitted.
If bw.input="quad"
then bandwidth selection is
based on the covariate values at every quadrature point used to
fit the model.
If bw.restrict=TRUE
then the bandwidth selection is performed
using only data from inside the subregion
.
A function value table (object of class "fv"
)
containing the values of the smoothed partial residual,
the estimated variance, and the fitted effect of the covariate.
Also belongs to the class "parres"
which has methods for print
and plot
.
In a large dataset, computation can be very slow if the default
settings are used, because the smoothing bandwidth is selected
automatically. To avoid this, specify a numerical value
for the bandwidth bw
. One strategy is to use a coarser
subset of the data to select bw
automatically.
The selected bandwidth can be read off the print output for
parres
.
Adrian Baddeley [email protected], Rolf Turner [email protected], Ya-Mei Chang and Yong Song.
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2013) Residual diagnostics for covariate effects in spatial point process models. Journal of Computational and Graphical Statistics, 22, 886–905.
X <- rpoispp(function(x,y){exp(3+x+2*x^2)}) model <- ppm(X ~x+y) tra <- parres(model, "x") plot(tra) tra plot(parres(model, "x", subregion=square(0.5))) model2 <- ppm(X ~x+I(x^2)+y) plot(parres(model2, "x")) Z <- setcov(owin()) plot(parres(model2, Z)) #' when the model involves only one covariate modelb <- ppm(bei ~ elev + I(elev^2), data=bei.extra) plot(parres(modelb))
X <- rpoispp(function(x,y){exp(3+x+2*x^2)}) model <- ppm(X ~x+y) tra <- parres(model, "x") plot(tra) tra plot(parres(model, "x", subregion=square(0.5))) model2 <- ppm(X ~x+I(x^2)+y) plot(parres(model2, "x")) Z <- setcov(owin()) plot(parres(model2, Z)) #' when the model involves only one covariate modelb <- ppm(bei ~ elev + I(elev^2), data=bei.extra) plot(parres(modelb))
Creates an instance of the Penttinen pairwise interaction point process model, which can then be fitted to point pattern data.
Penttinen(r)
Penttinen(r)
r |
circle radius |
Penttinen (1984, Example 2.1, page 18), citing Cormack (1979), described the pairwise interaction point process with interaction factor
between each pair of points separated by a distance $d$.
Here is the area of intersection between two discs
of radius
separated by a distance
, normalised so that
.
The scale of interaction is controlled by the disc radius :
two points interact if they are closer than
apart.
The strength of interaction is controlled by the
canonical parameter
, which
must be less than or equal to zero, or equivalently by the
parameter
,
which must lie between 0 and 1.
The potential is inhibitory, i.e.\ this model is only appropriate for
regular point patterns.
For the model is
a hard core process with hard core diameter
.
For
the model is a Poisson process.
The irregular parameter
must be given in the call to
Penttinen
, while the
regular parameter will be estimated.
This model can be considered as a pairwise approximation
to the area-interaction model AreaInter
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Cormack, R.M. (1979) Spatial aspects of competition between individuals. Pages 151–212 in Spatial and Temporal Analysis in Ecology, eds. R.M. Cormack and J.K. Ord, International Co-operative Publishing House, Fairland, MD, USA.
Penttinen, A. (1984) Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method. Jyvaskyla Studies in Computer Science, Economics and Statistics 7, University of Jyvaskyla, Finland.
ppm
,
ppm.object
,
Pairwise
,
AreaInter
.
fit <- ppm(cells ~ 1, Penttinen(0.07)) fit reach(fit) # interaction range is circle DIAMETER
fit <- ppm(cells ~ 1, Penttinen(0.07)) fit reach(fit) # interaction range is circle DIAMETER
Plots a fitted determinantal point process model, displaying the fitted intensity and the fitted summary function.
## S3 method for class 'dppm' plot(x, ..., what=c("intensity", "statistic"))
## S3 method for class 'dppm' plot(x, ..., what=c("intensity", "statistic"))
x |
Fitted determinantal point process model.
An object of class |
... |
Arguments passed to |
what |
Character vector determining what will be plotted. |
This is a method for the generic function
plot
for the class "dppm"
of fitted
determinantal point process models.
The argument x
should be a determinantal point process model
(object of class "dppm"
) obtained using
the function dppm
.
The choice of plots (and the order in which they are
displayed) is controlled by the argument what
.
The options (partially matched) are "intensity"
and
"statistic"
.
This command is capable of producing two different plots:
specifies the fitted intensity of the model,
which is plotted using plot.ppm
. By default this plot
is not produced for stationary models.
specifies the empirical and fitted summary
statistics, which are plotted using plot.fv
. This is
only meaningful if the model has been fitted using the Method of
Minimum Contrast, and it is turned off otherwise.
Null.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") plot(fit)
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") plot(fit)
Plots an influence measure that has been
computed by influence.ppm
.
## S3 method for class 'influence.ppm' plot(x, ..., multiplot=TRUE)
## S3 method for class 'influence.ppm' plot(x, ..., multiplot=TRUE)
x |
Influence measure (object of class |
... |
Arguments passed to |
multiplot |
Logical value indicating whether it is permissible to plot more than one panel. This happens if the original point process model is multitype. |
This is the plot method for objects of class "influence.ppm"
.
These objects are computed by the command influence.ppm
.
For a point process model fitted by maximum likelihood or
maximum pseudolikelihood (the default), influence values are
associated with the data points.
The display shows circles centred at the data points
with radii proportional to the influence values.
If the original data were a multitype point pattern, then
if multiplot=TRUE
(the default),
there is one such display for each possible type of point,
while if multiplot=FALSE
there is a single plot
combining all data points regardless of type.
For a model fitted by logistic composite likelihood
(method="logi"
in ppm
) influence values
are associated with the data points and also with the
dummy points used to fit the model. The display consist of two
panels, for the data points and dummy points respectively,
showing circles with radii proportional to the influence values.
If the original data were a multitype point pattern, then
if multiplot=TRUE
(the default),
there is one pair of panels for each possible type of point,
while if multiplot=FALSE
there is a single plot
combining all data and dummy points regardless of type.
Use the argument clipwin
to restrict the plot to a subset
of the full data.
None.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) plot(influence(fit))
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) plot(influence(fit))
Plots a fitted cluster point process model,
displaying the fitted intensity and the fitted -function.
## S3 method for class 'kppm' plot(x, ..., what=c("intensity", "statistic", "cluster"), pause=interactive(), xname)
## S3 method for class 'kppm' plot(x, ..., what=c("intensity", "statistic", "cluster"), pause=interactive(), xname)
x |
Fitted cluster point process model.
An object of class |
... |
Arguments passed to |
what |
Character vector determining what will be plotted. |
pause |
Logical value specifying whether to pause between plots. |
xname |
Optional. Character string. The name of the object |
This is a method for the generic function
plot
for the class "kppm"
of fitted
cluster point process models.
The argument x
should be a cluster point process model
(object of class "kppm"
) obtained using
the function kppm
.
The choice of plots (and the order in which they are
displayed) is controlled by the argument what
.
The options (partially matched) are "intensity"
,
"statistic"
and "cluster"
.
This command is capable of producing three different plots:
specifies the fitted intensity of the model,
which is plotted using plot.ppm
. By default this plot
is not produced for stationary models.
specifies the empirical and fitted summary
statistics, which are plotted using plot.fv
. This is
only meaningful if the model has been fitted using the Method of
Minimum Contrast, and it is turned off otherwise.
specifies a fitted cluster, which is computed
by clusterfield
and plotted by
plot.im
. It is only meaningful for Poisson cluster
(incl. Neyman-Scott) processes, and it is turned off for
log-Gaussian Cox processes (LGCP). If the model is stationary (and
non-LGCP) this option is turned on by default and shows a fitted
cluster positioned at the centroid of the observation window. For
non-stationary (and non-LGCP) models this option is only invoked if
explicitly told so, and in that case an additional argument
locations
(see clusterfield
) must be given to
specify where to position the parent point(s) .
Alternatively what="all"
selects all available options.
Null.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- kppm(redwood~1, "Thomas") plot(fit)
fit <- kppm(redwood~1, "Thomas") plot(fit)
Generate a pixel image plot, or a contour plot, or a perspective plot,
of a leverage function that has been computed by leverage.ppm
.
## S3 method for class 'leverage.ppm' plot(x, ..., what=c("smooth", "nearest", "exact"), showcut=TRUE, args.cut=list(drawlabels=FALSE), multiplot=TRUE) ## S3 method for class 'leverage.ppm' contour(x, ..., what=c("smooth", "nearest"), showcut=TRUE, args.cut=list(col=3, lwd=3, drawlabels=FALSE), multiplot=TRUE) ## S3 method for class 'leverage.ppm' persp(x, ..., what=c("smooth", "nearest"), main, zlab="leverage")
## S3 method for class 'leverage.ppm' plot(x, ..., what=c("smooth", "nearest", "exact"), showcut=TRUE, args.cut=list(drawlabels=FALSE), multiplot=TRUE) ## S3 method for class 'leverage.ppm' contour(x, ..., what=c("smooth", "nearest"), showcut=TRUE, args.cut=list(col=3, lwd=3, drawlabels=FALSE), multiplot=TRUE) ## S3 method for class 'leverage.ppm' persp(x, ..., what=c("smooth", "nearest"), main, zlab="leverage")
x |
Leverage function (object of class |
... |
Arguments passed to |
what |
Character string (partially matched) specifying the values to be plotted. See Details. |
showcut |
Logical. If |
args.cut |
Optional list of arguments passed to
|
multiplot |
Logical value indicating whether it is permissible to display several plot panels. |
main |
Optional main title. A character string or character vector. |
zlab |
Label for the |
These functions are the plot
, contour
and persp
methods
for objects of class "leverage.ppm"
.
Such objects are computed by the command leverage.ppm
.
The plot
method displays the leverage function
as a colour pixel image using plot.im
,
and draws a single contour line at the mean leverage value
using contour.default
.
Use the argument clipwin
to restrict the plot to a subset
of the full data.
The contour
method displays the leverage function as a contour
plot, and also draws a single contour line at the mean leverage value,
using contour.im
.
The persp
method displays the leverage function as a surface
in perspective view, using persp.im
.
Since the exact values of leverage are computed only at a finite set of quadrature locations, there are several options for these plots:
what="smooth"
:(the default) an image plot showing a smooth function, obtained by applying kernel smoothing to the exact leverage values;
what="nearest"
:an image plot showing a piecewise-constant function, obtained by taking the exact leverage value at the nearest quadrature point;
what="exact"
:a symbol plot showing the exact values of leverage as circles, centred at the quadrature points, with diameters proportional to leverage.
The pixel images are already contained in the object x
and were computed by leverage.ppm
;
the resolution of these images is controlled
by arguments to leverage.ppm
.
Same as for plot.im
, contour.im
and persp.im
respectively.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point process models. Scandinavian Journal of Statistics 40, 86–104.
if(offline <- !interactive()) op <- spatstat.options(npixel=32, ndummy.min=16) X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) lef <- leverage(fit) plot(lef) contour(lef) persp(lef) if(offline) spatstat.options(op)
if(offline <- !interactive()) op <- spatstat.options(npixel=32, ndummy.min=16) X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~x+y) lef <- leverage(fit) plot(lef) contour(lef) persp(lef) if(offline) spatstat.options(op)
Given a point process model fitted to multiple point patterns
by mppm
,
compute spatial trend or conditional intensity surface of the model,
in a form suitable for plotting, and (optionally) plot this
surface.
## S3 method for class 'mppm' plot(x, ..., trend=TRUE, cif=FALSE, se=FALSE, how=c("image", "contour", "persp"), main)
## S3 method for class 'mppm' plot(x, ..., trend=TRUE, cif=FALSE, se=FALSE, how=c("image", "contour", "persp"), main)
x |
A point process model fitted to multiple point patterns,
typically obtained from
the model-fitting algorithm |
... |
Arguments passed to |
trend |
Logical value indicating whether to plot the fitted trend. |
cif |
Logical value indicating whether to plot the fitted conditional intensity. |
se |
Logical value indicating whether to plot the standard error of the fitted trend. |
how |
Single character string indicating the style of plot to be performed. |
main |
Character string for the main title of the plot. |
This is the plot
method for the class "mppm"
of point process models fitted to multiple point patterns
(see mppm
).
It invokes subfits
to compute the fitted model for
each individual point pattern dataset, then calls
plot.ppm
to plot these individual models. These
individual plots are displayed using plot.anylist
,
which generates either a series of separate plot frames or an
array of plot panels on a single page.
NULL
.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
# Synthetic data from known model n <- 9 H <- hyperframe(V=1:n, U=runif(n, min=-1, max=1)) H$Z <- setcov(square(1)) H$U <- with(H, as.im(U, as.rectangle(Z))) H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z)))) fit <- mppm(Y ~Z + U + V, data=H) plot(fit)
# Synthetic data from known model n <- 9 H <- hyperframe(V=1:n, U=runif(n, min=-1, max=1)) H$Z <- setcov(square(1)) H$U <- with(H, as.im(U, as.rectangle(Z))) H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z)))) fit <- mppm(Y ~Z + U + V, data=H) plot(fit)
Plot a signed measure or vector-valued measure.
## S3 method for class 'msr' plot(x, ..., add = FALSE, how = c("image", "contour", "imagecontour"), main = NULL, do.plot = TRUE, multiplot = TRUE, massthresh = 0, equal.markscale = FALSE, equal.ribbon = FALSE)
## S3 method for class 'msr' plot(x, ..., add = FALSE, how = c("image", "contour", "imagecontour"), main = NULL, do.plot = TRUE, multiplot = TRUE, massthresh = 0, equal.markscale = FALSE, equal.ribbon = FALSE)
x |
The signed or vector measure to be plotted.
An object of class |
... |
Extra arguments passed to |
add |
Logical flag; if |
how |
String indicating how to display the continuous density component. |
main |
String. Main title for the plot. |
do.plot |
Logical value determining whether to actually perform the plotting. |
multiplot |
Logical value indicating whether it is permissible to display a plot with multiple panels (representing different components of a vector-valued measure, or different types of points in a multitype measure.) |
massthresh |
Threshold for plotting atoms.
A single numeric value or |
equal.markscale |
Logical value indicating whether different panels should use the same symbol map (to represent the masses of atoms of the measure). |
equal.ribbon |
Logical value indicating whether different panels should use the same colour map (to represent the density values in the diffuse component of the measure). |
This is the plot
method for the class "msr"
.
The continuous density component of x
is interpolated
from the existing data by Smooth.ppp
,
and then displayed as a colour image by plot.im
.
The discrete atomic component of x
is then superimposed on this
image by plotting the atoms as circles (for positive mass)
or squares (for negative mass) by plot.ppp
.
By default, atoms with zero mass are not plotted at all.
To smooth both the discrete and continuous components,
use Smooth.msr
.
Use the argument clipwin
to restrict the plot to a subset
of the full data.
To remove atoms with tiny masses, use the argument massthresh
.
(Invisible) colour map (object of class "colourmap"
) for the
colour image.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
msr
,
Smooth.ppp
,
Smooth.msr
,
plot.im
,
plot.ppp
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") plot(rp) plot(rs) plot(rs, how="contour")
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") plot(rp) plot(rs) plot(rs, how="contour")
Plot the Palm intensity diagnostic for a fitted cluster process or Cox process model.
## S3 method for class 'palmdiag' plot(x, ..., style = c("intervals", "dots", "bands"), args.dots = list(pch = 16), args.intervals = list(), xlim=NULL, main)
## S3 method for class 'palmdiag' plot(x, ..., style = c("intervals", "dots", "bands"), args.dots = list(pch = 16), args.intervals = list(), xlim=NULL, main)
x |
Object of class |
... |
Additional arguments passed to |
style |
Character string specifying the style of plot for the nonparametric estimates. See Details. |
args.dots |
Arguments passed to |
args.intervals |
Arguments passed to |
xlim |
Optional range of distances plotted along the horizontal axis. A numeric vector of length 2. |
main |
Optional main title for plot. |
This function plots the diagnostic proposed by
Tanaka, Ogata and Stoyan (2008, Section 2.3)
for assessing goodness-of-fit of a Neyman-Scott cluster process model
to a point pattern dataset. The diagnostic is computed by
the function palmdiagnose
.
First the Palm intensity of the fitted model is plotted as a
function of interpoint distance using
plot.fv
. Then the nonparametric estimates
of the Palm intensity are plotted on the same graph as follows:
if style="dots"
, the nonparametric
estimate for each band of distances is plotted as a dot,
with horizontal coordinate at the middle of the band.
This is the style proposed by Tanaka et al (2008).
if style="intervals"
(the default), each nonparametric estimate
is plotted as a dot, and a 95% confidence interval is
plotted as a vertical line segment, centred on the dot.
The confidence interval is based on the Poisson approximation.
if style="bands"
, the nonparametric estimates
are drawn as a continuous curve which is flat on each band of
distances. The 95% confidence intervals are drawn as grey shading.
Null.
Adrian Baddeley [email protected].
Tanaka, U., Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for Neyman-Scott Point Processes. Biometrical Journal 50, 1, 43–57.
fit <- kppm(redwood) R <- palmdiagnose(fit) plot(R, style="d") plot(R) plot(R, style="b")
fit <- kppm(redwood) R <- palmdiagnose(fit) plot(R, style="d") plot(R) plot(R, style="b")
The function plot.ppm produces objects which specify plots of fitted point process models. The function plot.plotppm carries out the actual plotting of these objects.
## S3 method for class 'plotppm' plot(x, data = NULL, trend = TRUE, cif = TRUE, se = TRUE, pause = interactive(), how = c("persp", "image", "contour"), ..., pppargs)
## S3 method for class 'plotppm' plot(x, data = NULL, trend = TRUE, cif = TRUE, se = TRUE, pause = interactive(), how = c("persp", "image", "contour"), ..., pppargs)
x |
An object of class |
.
data |
The point pattern (an object of class |
trend |
Logical scalar; should the trend component of the fitted model be plotted? |
cif |
Logical scalar; should the complete conditional intensity of the fitted model be plotted? |
se |
Logical scalar; should the estimated standard error of the fitted intensity be plotted? |
pause |
Logical scalar indicating whether to pause with a prompt
after each plot. Set |
how |
Character string or character vector indicating the style or styles of plots to be performed. |
... |
Extra arguments to the plotting functions
|
pppargs |
List of extra arguments passed to |
If argument data
is supplied then the point pattern will
be superimposed on the image and contour plots.
Sometimes a fitted model does not have a trend component, or the
trend component may constitute all of the conditional intensity (if
the model is Poisson). In such cases the object x
will not
contain a trend component, or will contain only a trend component.
This will also be the case if one of the arguments trend
and cif
was set equal to FALSE
in the call to
plot.ppm()
which produced x
. If this is so then
only the item which is present will be plotted. Explicitly setting
trend=TRUE
, or cif=TRUE
, respectively, will then give
an error.
None.
Arguments which are passed to persp
, image
, and
contour
via the ... argument get passed to any of the
other functions listed in the how
argument, and won't be
recognized by them. This leads to a lot of annoying but harmless
warning messages. Arguments to persp
may be supplied via
spatstat.options()
which alleviates the warning
messages in this instance.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
plot.ppm()
if(interactive()) { m <- ppm(cells ~ 1, Strauss(0.05)) mpic <- plot(m) # Perspective plot only, with altered parameters: plot(mpic,how="persp", theta=-30,phi=40,d=4) # All plots, with altered parameters for perspective plot: op <- spatstat.options(par.persp=list(theta=-30,phi=40,d=4)) plot(mpic) # Revert spatstat.options(op) }
if(interactive()) { m <- ppm(cells ~ 1, Strauss(0.05)) mpic <- plot(m) # Perspective plot only, with altered parameters: plot(mpic,how="persp", theta=-30,phi=40,d=4) # All plots, with altered parameters for perspective plot: op <- spatstat.options(par.persp=list(theta=-30,phi=40,d=4)) plot(mpic) # Revert spatstat.options(op) }
Given a fitted point process model obtained by ppm
,
create spatial trend and conditional intensity surfaces of the model,
in a form suitable for plotting, and (optionally) plot these
surfaces.
## S3 method for class 'ppm' plot(x, ngrid = c(40,40), superimpose = TRUE, trend = TRUE, cif = TRUE, se = TRUE, pause = interactive(), how=c("persp","image", "contour"), plot.it = TRUE, locations = NULL, covariates=NULL, ...)
## S3 method for class 'ppm' plot(x, ngrid = c(40,40), superimpose = TRUE, trend = TRUE, cif = TRUE, se = TRUE, pause = interactive(), how=c("persp","image", "contour"), plot.it = TRUE, locations = NULL, covariates=NULL, ...)
x |
A fitted point process model, typically obtained from
the model-fitting algorithm |
ngrid |
The dimensions for a grid on which to evaluate,
for plotting, the spatial trend and conditional intensity.
A vector of 1 or 2 integers. If it is of length 1,
|
superimpose |
logical flag; if |
trend |
logical flag; if |
cif |
logical flag; if |
se |
logical flag; if |
pause |
logical flag indicating whether to pause with a prompt
after each plot. Set |
how |
character string or character vector indicating the style or styles of
plots to be performed. Ignored if |
plot.it |
logical scalar; should a plot be produced immediately? |
locations |
If present, this determines the locations of the pixels
at which predictions are computed. It must be a binary pixel image
(an object of class |
covariates |
Values of external covariates required by the fitted model.
Passed to |
... |
extra arguments to the plotting functions |
This is the plot
method for the class "ppm"
(see ppm.object
for details of this class).
It invokes predict.ppm
to compute the spatial
trend and conditional intensity of the fitted point process model.
See predict.ppm
for more explanation about spatial trend
and conditional intensity.
The default action is to create a rectangular grid
of points in (the bounding box of) the observation window of
the data point pattern, and evaluate the spatial trend and
conditional intensity of the fitted spatial point process model
x
at these locations. If the argument locations=
is supplied, then the spatial trend
and conditional intensity are calculated at the grid of points
specified by this argument.
The argument locations
, if present, should be a
binary image mask (an object of class "owin"
and type "mask"
). This determines a rectangular grid
of locations, or a subset of such a grid, at which predictions
will be computed. Binary image masks
are conveniently created using as.mask
.
The argument covariates
gives the values of any spatial covariates
at the prediction locations.
If the trend formula in the fitted model
involves spatial covariates (other than
the Cartesian coordinates x
, y
)
then covariates
is required.
The argument covariates
has the same format and interpretation
as in predict.ppm
. It may be
either a data frame (the number of whose rows must match
the number of pixels in locations
multiplied by the number of
possible marks in the point pattern), or a list of images.
If argument locations
is not supplied, and covariates
is supplied, then
it must be a list of images.
If the fitted model was a marked (multitype) point process, then predictions are made for each possible mark value in turn.
If the fitted model had no spatial trend, then the default is
to omit calculating this (flat) surface, unless trend=TRUE
is set explicitly.
If the fitted model was Poisson, so that there were no spatial interactions,
then the conditional intensity and spatial trend are identical, and the
default is to omit the conditional intensity, unless cif=TRUE
is set
explicitly.
If plot.it=TRUE
then plot.plotppm()
is called
upon to plot the class plotppm
object which is produced.
(That object is also returned, silently.)
Plots are produced successively using persp
,
image
and contour
(or only a
selection of these three, if how
is given). Extra
graphical parameters controlling the display may be passed
directly via the arguments ...
or indirectly reset using
spatstat.options
.
An object of class plotppm
. Such objects may be plotted by
plot.plotppm()
.
This is a list with components named trend
and cif
,
either of which may
be missing. They will be missing if the corresponding component
does not make sense for the model, or if the corresponding
argument was set equal to FALSE
.
Both trend
and cif
are lists of images.
If the model is an unmarked point process, then they are lists of
length 1, so that trend[[1]]
is an image of the spatial trend
and cif[[1]]
is an image of the conditional intensity.
If the model is a marked point process, then trend[[i]]
is an image of the spatial trend for the mark m[i]
,
and cif[[1]]
is an image of the conditional intensity
for the mark m[i]
, where m
is the vector of levels
of the marks.
See warnings in predict.ppm
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
plot.plotppm
,
ppm
,
ppm.object
,
predict.ppm
,
print.ppm
,
persp
,
image
,
contour
,
plot
,
spatstat.options
m <- ppm(cells ~1, Strauss(0.05)) pm <- plot(m) # The object ``pm'' will be plotted as well as saved # for future plotting. pm
m <- ppm(cells ~1, Strauss(0.05)) pm <- plot(m) # The object ``pm'' will be plotted as well as saved # for future plotting. pm
Plot the profile (pseudo) likelihood against the irregular parameters, for a model that was fitted by maximum profile (pseudo)likelihood.
## S3 method for class 'profilepl' plot(x, ..., add = FALSE, main = NULL, tag = TRUE, coeff = NULL, xvariable = NULL, col = 1, lty = 1, lwd = 1, col.opt = "green", lty.opt = 3, lwd.opt = 1)
## S3 method for class 'profilepl' plot(x, ..., add = FALSE, main = NULL, tag = TRUE, coeff = NULL, xvariable = NULL, col = 1, lty = 1, lwd = 1, col.opt = "green", lty.opt = 3, lwd.opt = 1)
x |
A point process model fitted by maximum profile (pseudo)likelihood.
Object of class |
... |
Additional plot arguments passed to |
add |
Logical. If |
main |
Optional. Main title for the plot. A character string or character vector. |
tag |
Logical value. If |
coeff |
Optional. If this is given, it should be a character string matching the name of one of the fitted model coefficients. This coefficient will then be plotted on the vertical axis. |
xvariable |
Optional. The name of the irregular parameter that should be plotted along the horizontal axis. The default is the first irregular parameter. |
col , lty , lwd
|
Graphical parameters (colour, line type, line width) for the curves on the plot. |
col.opt , lty.opt , lwd.opt
|
Graphical parameters for indicating the optimal parameter value. |
This is the plot
method for the
class "profilepl"
of fitted point process models obtained
by maximising the profile likelihood or profile pseudolikelihood.
The default behaviour is to plot the profile likelihood or profile pseudolikelihood on the vertical axis, against the value of the irregular parameter on the horizontal axis.
If there are several irregular parameters, then one of them
is plotted on the horizontal axis, and the plot consists of many different
curves, corresponding to different values of the other parameters.
The parameter to be plotted on the horizontal axis is specified by the
argument xvariable
; the default is to use the parameter that was
listed first in the original call to profilepl
.
If coeff
is given, it should be the name of one of the
fitted model coefficients names(coef(as.ppm(x)))
.
The fitted value of that coefficient is plotted on the vertical axis.
Null.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
live <- interactive() nr <- if(live) 20 else 3 # one irregular parameter rr <- data.frame(r=seq(0.05,0.15, length=nr)) ps <- profilepl(rr, Strauss, cells) plot(ps) # profile pseudolikelihood plot(ps, coeff="Interaction") # fitted interaction coefficient log(gamma) # two irregular parameters smax <- if(live) 3 else 2 rs <- expand.grid(r=seq(0.05,0.15, length=nr), sat=1:smax) pg <- profilepl(rs, Geyer, cells) plot(pg) # profile pseudolikelihood against r for each value of 'sat' plot(pg, coeff="Interaction") plot(pg, xvariable="sat", col=ifelse(r < 0.1, "red", "green"))
live <- interactive() nr <- if(live) 20 else 3 # one irregular parameter rr <- data.frame(r=seq(0.05,0.15, length=nr)) ps <- profilepl(rr, Strauss, cells) plot(ps) # profile pseudolikelihood plot(ps, coeff="Interaction") # fitted interaction coefficient log(gamma) # two irregular parameters smax <- if(live) 3 else 2 rs <- expand.grid(r=seq(0.05,0.15, length=nr), sat=1:smax) pg <- profilepl(rs, Geyer, cells) plot(pg) # profile pseudolikelihood against r for each value of 'sat' plot(pg, coeff="Interaction") plot(pg, xvariable="sat", col=ifelse(r < 0.1, "red", "green"))
Given a model which has been fitted to point pattern data by recursive partitioning, plot the partition tree or the fitted intensity.
## S3 method for class 'rppm' plot(x, ..., what = c("tree", "spatial"), treeplot=NULL)
## S3 method for class 'rppm' plot(x, ..., what = c("tree", "spatial"), treeplot=NULL)
x |
Fitted point process model of class |
what |
Character string (partially matched) specifying whether to plot the partition tree or the fitted intensity. |
... |
Arguments passed to |
treeplot |
Optional. A function to be used to plot and label the partition tree,
replacing the two functions |
If what="tree"
(the default), the partition tree will be plotted
using plot.rpart
, and labelled using
text.rpart
.
If the argument treeplot
is
given, then plotting and labelling will be performed by
treeplot
instead. A good choice is the function
prp
in package rpart.plot.
If what="spatial"
, the predicted intensity
will be computed using predict.rppm
, and
this intensity will be plotted as an image using plot.im
.
If what="tree"
, a list containing x
and y
coordinates of the plotted nodes of the tree.
If what="spatial"
, the return value of plot.im
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
# Murchison gold data mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) # fit <- rppm(gold ~ dfault + greenstone, data=mur) # opa <- par(mfrow=c(1,2)) plot(fit) plot(fit, what="spatial") par(opa)
# Murchison gold data mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) # fit <- rppm(gold ~ dfault + greenstone, data=mur) # opa <- par(mfrow=c(1,2)) plot(fit) plot(fit, what="spatial") par(opa)
Plots a fitted Spatial Logistic Regression model.
## S3 method for class 'slrm' plot(x, ..., type = "intensity")
## S3 method for class 'slrm' plot(x, ..., type = "intensity")
x |
a fitted spatial logistic regression model.
An object of class |
... |
Extra arguments passed to |
type |
Character string (partially) matching one of
|
This is a method for plot
for fitted spatial logistic
regression models (objects of class "slrm"
, usually obtained
from the function slrm
).
This function plots the result of predict.slrm
.
None.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
X <- copper$SouthPoints Y <- copper$SouthLines Z <- distmap(Y) fit <- slrm(X ~ Z) plot(fit) plot(fit, type="link")
X <- copper$SouthPoints Y <- copper$SouthLines Z <- distmap(Y) fit <- slrm(X ~ Z) plot(fit) plot(fit, type="link")
Creates an instance of the Poisson point process model which can then be fitted to point pattern data.
Poisson()
Poisson()
The function ppm
, which fits point process models to
point pattern data, requires an argument interaction
of class "interact"
describing the interpoint interaction structure
of the model to be fitted.
The appropriate description of the Poisson process is
provided by the value of the function Poisson
.
This works for all types of Poisson processes including multitype and nonstationary Poisson processes.
An object of class "interact"
describing the interpoint interaction
structure of the Poisson point process
(namely, there are no interactions).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
ppm(nztrees ~1, Poisson()) # fit the stationary Poisson process to 'nztrees' # no edge correction needed lon <- longleaf longadult <- unmark(subset(lon, marks >= 30)) ppm(longadult ~ x, Poisson()) # fit the nonstationary Poisson process # with intensity lambda(x,y) = exp( a + bx) # trees marked by species lans <- lansing ppm(lans ~ marks, Poisson()) # fit stationary marked Poisson process # with different intensity for each species ppm(lansing ~ marks * polynom(x,y,3), Poisson()) # fit nonstationary marked Poisson process # with different log-cubic trend for each species
ppm(nztrees ~1, Poisson()) # fit the stationary Poisson process to 'nztrees' # no edge correction needed lon <- longleaf longadult <- unmark(subset(lon, marks >= 30)) ppm(longadult ~ x, Poisson()) # fit the nonstationary Poisson process # with intensity lambda(x,y) = exp( a + bx) # trees marked by species lans <- lansing ppm(lans ~ marks, Poisson()) # fit stationary marked Poisson process # with different intensity for each species ppm(lansing ~ marks * polynom(x,y,3), Poisson()) # fit nonstationary marked Poisson process # with different log-cubic trend for each species
This function is used to represent a polynomial term in a model
formula.
It computes the homogeneous terms in the polynomial of degree n
in one variable x
or two variables x,y
.
polynom(x, ...)
polynom(x, ...)
x |
A numerical vector. |
... |
Either a single integer |
This function is typically used inside a model formula
in order to specify the most general possible polynomial
of order n
involving one numerical variable x
or two numerical variables x,y
.
It is equivalent to poly(, raw=TRUE)
.
If only one numerical vector argument x
is given,
the function computes the vectors x^k
for
k = 1, 2, ..., n
. These vectors are combined into a matrix
with n
columns.
If two numerical vector arguments x,y
are given,
the function computes the vectors x^k * y^m
for
k >= 0
and m >= 0
satisfying
0 < k + m <= n
. These vectors are combined into a matrix
with one column for each homogeneous term.
A numeric matrix, with rows corresponding to the entries of x
,
and columns corresponding to the terms in the polynomial.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
x <- 1:4 y <- 10 * (0:3) polynom(x, 3) polynom(x, y, 3)
x <- 1:4 y <- 10 * (0:3) polynom(x, 3) polynom(x, y, 3)
Fits a point process model to an observed point pattern.
ppm(Q, ...) ## S3 method for class 'formula' ppm(Q, interaction=NULL, ..., data=NULL, subset)
ppm(Q, ...) ## S3 method for class 'formula' ppm(Q, interaction=NULL, ..., data=NULL, subset)
Q |
A |
interaction |
An object of class |
... |
Arguments passed to |
data |
Optional. The values of spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list whose entries are images, functions, windows, tessellations or single numbers. See Details. |
subset |
Optional.
An expression (which may involve the names of the
Cartesian coordinates |
This function fits a point process model to an observed point pattern. The model may include spatial trend, interpoint interaction, and dependence on covariates.
The model fitted by ppm
is either a Poisson point process (in which different points
do not interact with each other) or a Gibbs point process (in which
different points typically inhibit each other).
For clustered point process models, use kppm
.
The function ppm
is generic, with methods for
the classes formula
, ppp
and quad
.
This page describes the method for a formula
.
The first argument is a formula
in the R language
describing the spatial trend model to be fitted. It has the general form
pattern ~ trend
where the left hand side pattern
is usually
the name of a spatial point pattern (object of class "ppp"
)
to which the model should be fitted, or an expression which evaluates
to a point pattern;
and the right hand side trend
is an expression specifying the
spatial trend of the model.
Systematic effects (spatial trend and/or dependence on
spatial covariates) are specified by the
trend
expression on the right hand side of the formula.
The trend may involve
the Cartesian coordinates x
, y
,
the marks marks
,
the names of entries in the argument data
(if supplied),
or the names of objects that exist in the R session.
The trend formula specifies the logarithm of the
intensity of a Poisson process, or in general, the logarithm of
the first order potential of the Gibbs process.
The formula should not use any names beginning with .mpl
as these are reserved for internal use.
If the formula is pattern~1
, then
the model to be fitted is stationary (or at least, its first order
potential is constant).
The symbol .
in the trend expression stands for
all the covariates supplied in the argument data
.
For example the formula pattern ~ .
indicates an additive
model with a main effect for each covariate in data
.
Stochastic interactions between random points of the point process
are defined by the argument interaction
. This is an object of
class "interact"
which is initialised in a very similar way to the
usage of family objects in glm
and gam
.
The interaction models currently available are:
AreaInter
, BadGey
, Concom
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, HierHard
, HierStrauss
, HierStraussHard
, Hybrid
, LennardJones
, MultiHard
, MultiStrauss
, MultiStraussHard
, OrdThresh
, Ord
, Pairwise
, PairPiece
, Penttinen
, Poisson
, Saturated
, SatPiece
, Softcore
, Strauss
, StraussHard
and Triplets
.
See the examples below.
Note that it is possible to combine several interactions
using Hybrid
.
If interaction
is missing or NULL
,
then the model to be fitted
has no interpoint interactions, that is, it is a Poisson process
(stationary or nonstationary according to trend
). In this case
the methods of maximum pseudolikelihood and maximum logistic likelihood
coincide with maximum likelihood.
The fitted point process model returned by this function can be printed
(by the print method print.ppm
)
to inspect the fitted parameter values.
If a nonparametric spatial trend was fitted, this can be extracted using
the predict method predict.ppm
.
To fit a model involving spatial covariates
other than the Cartesian coordinates and
,
the values of the covariates should either be supplied in the
argument
data
, or should be stored in objects that exist
in the R session.
Note that it is not sufficient to have observed
the covariate only at the points of the data point pattern;
the covariate must also have been observed at other
locations in the window.
If it is given, the argument data
is typically
a list, with names corresponding to variables in the trend
formula.
Each entry in the list is either
giving the values of a spatial covariate at
a fine grid of locations. It should be an object of
class "im"
, see im.object
.
which can be evaluated
at any location (x,y)
to obtain the value of the spatial
covariate. It should be a function(x, y)
or function(x, y, ...)
in the R language.
For marked point pattern data, the covariate can be a
function(x, y, marks)
or function(x, y, marks, ...)
.
The first two arguments of the function should be the
Cartesian coordinates and
.
The function may have additional arguments;
if the function does not have default
values for these additional arguments, then the user must
supply values for them, in
covfunargs
.
See the Examples.
interpreted as a logical variable
which is TRUE
inside the window and FALSE
outside
it. This should be an object of class "owin"
.
interpreted as a factor covariate.
For each spatial location, the factor value indicates
which tile of the tessellation it belongs to.
This should be an object of class "tess"
.
(To make a covariate in which each tile of the tessellation
has a numerical value, convert the tessellation to a function(x,y)
using as.function.tess
.)
indicating a covariate that is constant in this dataset.
The software will look up the values of each covariate at the required locations (quadrature points).
Note that, for covariate functions, only the name of the function appears in the trend formula. A covariate function is treated as if it were a single variable. The function arguments do not appear in the trend formula. See the Examples.
If data
is a list,
the list entries should have names corresponding to
(some of) the names of covariates in the model formula trend
.
The variable names x
, y
and marks
are reserved for the Cartesian
coordinates and the mark values,
and these should not be used for variables in data
.
Alternatively, data
may be a data frame
giving the values of the covariates at specified locations.
Then pattern
should be a quadrature scheme (object of class
"quad"
) giving the corresponding locations.
See ppm.quad
for details.
An object of class "ppm"
describing a fitted point process
model.
See ppm.object
for details of the format of this object
and methods available for manipulating it.
Apart from the Poisson model, every point process model fitted by
ppm
has parameters that determine the strength and
range of ‘interaction’ or dependence between points.
These parameters are of two types:
A parameter is called regular
if the log likelihood is a linear function of
where
is some transformation of
. [Then
is called the canonical
parameter.]
Other parameters are called irregular.
Typically, regular parameters determine the ‘strength’
of the interaction, while irregular parameters determine the
‘range’ of the interaction. For example, the Strauss process
has a regular parameter controlling the strength
of interpoint inhibition, and an irregular parameter
determining the range of interaction.
The ppm
command is only designed to estimate regular
parameters of the interaction.
It requires the values of any irregular parameters of the interaction
to be fixed. For example, to fit a Strauss process model to the cells
dataset, you could type ppm(cells ~ 1, Strauss(r=0.07))
.
Note that the value of the irregular parameter r
must be given.
The result of this command will be a fitted model in which the
regular parameter has been estimated.
To determine the irregular parameters, there are several
practical techniques, but no general statistical theory available.
Useful techniques include maximum profile pseudolikelihood, which
is implemented in the command profilepl
,
and Newton-Raphson maximisation, implemented in the
experimental command ippm
.
Some irregular parameters can be estimated directly from data:
the hard-core radius in the model Hardcore
and the matrix of hard-core radii in MultiHard
can be
estimated easily from data. In these cases, ppm
allows the user
to specify the interaction without giving
the value of the irregular parameter. The user can give the
hard core interaction as interaction=Hardcore()
or even interaction=Hardcore
, and
the hard core radius will then be estimated from the data.
See ppm.ppp
for some technical warnings about the
weaknesses of the algorithm, and explanation of some common error messages.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 283–322.
Berman, M. and Turner, T.R. (1992) Approximating point process likelihoods with GLIM. Applied Statistics 41, 31–38.
Besag, J. (1975) Statistical analysis of non-lattice data. The Statistician 24, 179-195.
Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994) On parameter estimation for pairwise interaction processes. International Statistical Review 62, 99-117.
Huang, F. and Ogata, Y. (1999) Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. Journal of Computational and Graphical Statistics 8, 510–530.
Jensen, J.L. and Moeller, M. (1991) Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability 1, 445–461.
Jensen, J.L. and Kuensch, H.R. (1994) On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46, 475–486.
ppm.ppp
and ppm.quad
for
more details on the fitting technique and edge correction.
ppm.object
for details of how to
print, plot and manipulate a fitted model.
ppp
and quadscheme
for constructing data.
Interactions: AreaInter
, BadGey
, Concom
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, HierHard
, HierStrauss
, HierStraussHard
, Hybrid
, LennardJones
, MultiHard
, MultiStrauss
, MultiStraussHard
, OrdThresh
, Ord
, Pairwise
, PairPiece
, Penttinen
, Poisson
, Saturated
, SatPiece
, Softcore
, Strauss
, StraussHard
and Triplets
.
See profilepl
for advice on
fitting nuisance parameters in the interaction,
and ippm
for irregular parameters in the trend.
See valid.ppm
and project.ppm
for
ensuring the fitted model is a valid point process.
See kppm
for fitting Cox point process models
and cluster point process models, and dppm
for fitting
determinantal point process models.
online <- interactive() if(!online) { # reduce grid sizes for efficiency in tests spatstat.options(npixel=32, ndummy.min=16) } # fit the stationary Poisson process # to point pattern 'nztrees' ppm(nztrees ~ 1) if(online) { Q <- quadscheme(nztrees) ppm(Q ~ 1) # equivalent. } fit1 <- ppm(nztrees ~ x) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx) # where x,y are the Cartesian coordinates # and a,b are parameters to be estimated fit1 coef(fit1) coef(summary(fit1)) ppm(nztrees ~ polynom(x,2)) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx + cx^2) if(online) { library(splines) ppm(nztrees ~ bs(x,df=3)) } # Fits the nonstationary Poisson process # with intensity function lambda(x,y) = exp(B(x)) # where B is a B-spline with df = 3 ppm(nztrees ~ 1, Strauss(r=10), rbord=10) # Fit the stationary Strauss process with interaction range r=10 # using the border method with margin rbord=10 ppm(nztrees ~ x, Strauss(13), correction="periodic") # Fit the nonstationary Strauss process with interaction range r=13 # and exp(first order potential) = activity = beta(x,y) = exp(a+bx) # using the periodic correction. # Compare Maximum Pseudolikelihood, Huang-Ogata and Variational Bayes fits: if(online) ppm(swedishpines ~ 1, Strauss(9)) ppm(swedishpines ~ 1, Strauss(9), method="VBlogi") ppm(swedishpines ~ 1, Strauss(9), improve.type="ho", nsim=if(!online) 8 else 99) # Elastic net fit: if(require(glmnet)) { ppm(swedishpines ~ x+y, Strauss(9), improve.type="enet") } # COVARIATES # X <- rpoispp(20) weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) } # # (a) covariate values as function ppm(X ~ y + weirdfunction) # # (b) covariate values in pixel image Zimage <- as.im(weirdfunction, unit.square()) ppm(X ~ y + Z, covariates=list(Z=Zimage)) # # (c) covariate values in data frame Q <- quadscheme(X) xQ <- x.quad(Q) yQ <- y.quad(Q) Zvalues <- weirdfunction(xQ,yQ) ppm(Q ~ y + Z, data=data.frame(Z=Zvalues)) # Note Q not X # COVARIATE FUNCTION WITH EXTRA ARGUMENTS # f <- function(x,y,a){ y - a } ppm(X ~ x + f, covfunargs=list(a=1/2)) # COVARIATE: logical value TRUE inside window, FALSE outside b <- owin(c(0.1, 0.6), c(0.1, 0.9)) ppm(X ~ b) ## MULTITYPE POINT PROCESSES ### # fit stationary marked Poisson process # with different intensity for each species if(online) { ppm(lansing ~ marks, Poisson()) } else { ama <- amacrine[square(0.7)] a <- ppm(ama ~ marks, Poisson(), nd=16) } # fit nonstationary marked Poisson process # with different log-cubic trend for each species if(online) { ppm(lansing ~ marks * polynom(x,y,3), Poisson()) } else { b <- ppm(ama ~ marks * polynom(x,y,2), Poisson(), nd=16) }
online <- interactive() if(!online) { # reduce grid sizes for efficiency in tests spatstat.options(npixel=32, ndummy.min=16) } # fit the stationary Poisson process # to point pattern 'nztrees' ppm(nztrees ~ 1) if(online) { Q <- quadscheme(nztrees) ppm(Q ~ 1) # equivalent. } fit1 <- ppm(nztrees ~ x) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx) # where x,y are the Cartesian coordinates # and a,b are parameters to be estimated fit1 coef(fit1) coef(summary(fit1)) ppm(nztrees ~ polynom(x,2)) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx + cx^2) if(online) { library(splines) ppm(nztrees ~ bs(x,df=3)) } # Fits the nonstationary Poisson process # with intensity function lambda(x,y) = exp(B(x)) # where B is a B-spline with df = 3 ppm(nztrees ~ 1, Strauss(r=10), rbord=10) # Fit the stationary Strauss process with interaction range r=10 # using the border method with margin rbord=10 ppm(nztrees ~ x, Strauss(13), correction="periodic") # Fit the nonstationary Strauss process with interaction range r=13 # and exp(first order potential) = activity = beta(x,y) = exp(a+bx) # using the periodic correction. # Compare Maximum Pseudolikelihood, Huang-Ogata and Variational Bayes fits: if(online) ppm(swedishpines ~ 1, Strauss(9)) ppm(swedishpines ~ 1, Strauss(9), method="VBlogi") ppm(swedishpines ~ 1, Strauss(9), improve.type="ho", nsim=if(!online) 8 else 99) # Elastic net fit: if(require(glmnet)) { ppm(swedishpines ~ x+y, Strauss(9), improve.type="enet") } # COVARIATES # X <- rpoispp(20) weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) } # # (a) covariate values as function ppm(X ~ y + weirdfunction) # # (b) covariate values in pixel image Zimage <- as.im(weirdfunction, unit.square()) ppm(X ~ y + Z, covariates=list(Z=Zimage)) # # (c) covariate values in data frame Q <- quadscheme(X) xQ <- x.quad(Q) yQ <- y.quad(Q) Zvalues <- weirdfunction(xQ,yQ) ppm(Q ~ y + Z, data=data.frame(Z=Zvalues)) # Note Q not X # COVARIATE FUNCTION WITH EXTRA ARGUMENTS # f <- function(x,y,a){ y - a } ppm(X ~ x + f, covfunargs=list(a=1/2)) # COVARIATE: logical value TRUE inside window, FALSE outside b <- owin(c(0.1, 0.6), c(0.1, 0.9)) ppm(X ~ b) ## MULTITYPE POINT PROCESSES ### # fit stationary marked Poisson process # with different intensity for each species if(online) { ppm(lansing ~ marks, Poisson()) } else { ama <- amacrine[square(0.7)] a <- ppm(ama ~ marks, Poisson(), nd=16) } # fit nonstationary marked Poisson process # with different log-cubic trend for each species if(online) { ppm(lansing ~ marks * polynom(x,y,3), Poisson()) } else { b <- ppm(ama ~ marks * polynom(x,y,2), Poisson(), nd=16) }
A class ppm
to represent a fitted stochastic model
for a point process. The output of ppm
.
An object of class ppm
represents a stochastic point process
model that has been fitted to a point pattern dataset.
Typically it is the output of the model fitter,
ppm
.
The class ppm
has methods for the following
standard generic functions:
generic | method | description |
print |
print.ppm
|
print details |
plot |
plot.ppm
|
plot fitted model |
predict |
predict.ppm
|
fitted intensity and conditional intensity |
fitted |
fitted.ppm
|
fitted intensity |
coef |
coef.ppm
|
fitted coefficients of model |
anova |
anova.ppm
|
Analysis of Deviance |
formula |
formula.ppm
|
Extract model formula |
terms |
terms.ppm
|
Terms in the model formula |
labels |
labels.ppm
|
Names of estimable terms in the model formula |
residuals |
residuals.ppm
|
Point process residuals |
simulate |
simulate.ppm
|
Simulate the fitted model |
update |
update.ppm
|
Change or refit the model |
vcov |
vcov.ppm
|
Variance/covariance matrix of parameter estimates |
model.frame |
model.frame.ppm
|
Model frame |
model.matrix |
model.matrix.ppm
|
Design matrix |
logLik |
logLik.ppm
|
log pseudo likelihood |
extractAIC |
extractAIC.ppm
|
pseudolikelihood counterpart of AIC |
nobs |
nobs.ppm
|
number of observations |
Objects of class ppm
can also be handled by the
following standard functions, without requiring a special method:
name | description |
confint |
Confidence intervals for parameters |
step |
Stepwise model selection |
drop1 |
One-step model improvement |
add1 |
One-step model improvement |
The class ppm
also has methods for the following
generic functions defined in the spatstat package:
generic | method | description |
as.interact |
as.interact.ppm
|
Interpoint interaction structure |
as.owin |
as.owin.ppm
|
Observation window of data |
berman.test |
berman.test.ppm
|
Berman's test |
envelope |
envelope.ppm
|
Simulation envelopes |
fitin |
fitin.ppm
|
Fitted interaction |
is.marked |
is.marked.ppm
|
Determine whether the model is marked |
is.multitype |
is.multitype.ppm
|
Determine whether the model is multitype |
is.poisson |
is.poisson.ppm
|
Determine whether the model is Poisson |
is.stationary |
is.stationary.ppm
|
Determine whether the model is stationary |
cdf.test |
cdf.test.ppm
|
Spatial distribution test |
quadrat.test |
quadrat.test.ppm
|
Quadrat counting test |
reach |
reach.ppm
|
Interaction range of model |
rmhmodel |
rmhmodel.ppm
|
Model in a form that can be simulated |
rmh |
rmh.ppm
|
Perform simulation |
unitname |
unitname.ppm
|
Name of unit of length |
Information about the data (to which the model was fitted)
can be extracted using data.ppm
, dummy.ppm
and quad.ppm
.
If you really need to get at the internals,
a ppm
object contains at least the following entries:
coef |
the fitted regular parameters (as returned by
glm ) |
trend |
the trend formula or NULL |
interaction |
the point process interaction family
(an object of class "interact" )
or NULL |
Q |
the quadrature scheme used |
maxlogpl |
the maximised value of log pseudolikelihood |
correction |
name of edge correction method used |
See ppm
for explanation of these concepts.
The irregular parameters (e.g. the interaction radius of the
Strauss process) are encoded in the interaction
entry.
However see the Warnings.
The internal representation of ppm
objects
may change slightly between releases of the spatstat package.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
ppm
,
coef.ppm
,
fitted.ppm
,
print.ppm
,
predict.ppm
,
plot.ppm
.
fit <- ppm(cells ~ x, Strauss(0.1), correction="periodic") fit coef(fit) pred <- predict(fit) pred <- predict(fit, ngrid=20, type="trend") if(interactive()) { plot(fit) }
fit <- ppm(cells ~ x, Strauss(0.1), correction="periodic") fit coef(fit) pred <- predict(fit) pred <- predict(fit, ngrid=20, type="trend") if(interactive()) { plot(fit) }
Fits a point process model to an observed point pattern.
## S3 method for class 'ppp' ppm(Q, trend=~1, interaction=Poisson(), ..., covariates=data, data=NULL, covfunargs = list(), subset, clipwin, correction="border", rbord=reach(interaction), use.gam=FALSE, method=c("mpl", "logi", "VBlogi"), forcefit=FALSE, improve.type = c("none", "ho", "enet"), improve.args=list(), emend=project, project=FALSE, prior.mean = NULL, prior.var = NULL, nd = NULL, eps = NULL, quad.args=list(), gcontrol=list(), nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh), verb=TRUE, callstring=NULL) ## S3 method for class 'quad' ppm(Q, trend=~1, interaction=Poisson(), ..., covariates=data, data=NULL, covfunargs = list(), subset, clipwin, correction="border", rbord=reach(interaction), use.gam=FALSE, method=c("mpl", "logi", "VBlogi"), forcefit=FALSE, improve.type = c("none", "ho", "enet"), improve.args=list(), emend=project, project=FALSE, prior.mean = NULL, prior.var = NULL, nd = NULL, eps = NULL, quad.args=list(), gcontrol=list(), nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh), verb=TRUE, callstring=NULL)
## S3 method for class 'ppp' ppm(Q, trend=~1, interaction=Poisson(), ..., covariates=data, data=NULL, covfunargs = list(), subset, clipwin, correction="border", rbord=reach(interaction), use.gam=FALSE, method=c("mpl", "logi", "VBlogi"), forcefit=FALSE, improve.type = c("none", "ho", "enet"), improve.args=list(), emend=project, project=FALSE, prior.mean = NULL, prior.var = NULL, nd = NULL, eps = NULL, quad.args=list(), gcontrol=list(), nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh), verb=TRUE, callstring=NULL) ## S3 method for class 'quad' ppm(Q, trend=~1, interaction=Poisson(), ..., covariates=data, data=NULL, covfunargs = list(), subset, clipwin, correction="border", rbord=reach(interaction), use.gam=FALSE, method=c("mpl", "logi", "VBlogi"), forcefit=FALSE, improve.type = c("none", "ho", "enet"), improve.args=list(), emend=project, project=FALSE, prior.mean = NULL, prior.var = NULL, nd = NULL, eps = NULL, quad.args=list(), gcontrol=list(), nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh), verb=TRUE, callstring=NULL)
Q |
A data point pattern (of class |
trend |
An R formula object specifying the spatial trend to be fitted.
The default formula, |
interaction |
An object of class |
... |
Ignored. |
data , covariates
|
The values of any spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list whose entries are images, functions, windows, tessellations or single numbers. See Details. |
subset |
Optional.
An expression (which may involve the names of the
Cartesian coordinates |
clipwin |
Optional. A spatial window (object of class |
covfunargs |
A named list containing the values of any additional arguments required by covariate functions. |
correction |
The name of the edge correction to be used. The default
is |
rbord |
If |
use.gam |
Logical flag; if |
method |
String (partially matched) specifying the method used to fit the
model. Options are
|
forcefit |
Logical flag for internal use.
If |
improve.type |
String (partially matched) specifying a method for improving the
initial fit.
If |
improve.args |
Arguments used to control the algorithm for improving the initial fit. See Details. |
emend , project
|
(These are equivalent: |
prior.mean |
Optional vector of prior means for canonical parameters (for
|
prior.var |
Optional prior variance covariance matrix for canonical parameters (for |
nd |
Optional. Integer or pair of integers.
The dimension of the grid of dummy points ( |
eps |
Optional.
A positive number, or a vector of two positive numbers, giving the
horizontal and vertical spacing, respectively, of the grid of
dummy points. Incompatible with |
quad.args |
Arguments controlling the construction of the quadrature scheme,
when |
gcontrol |
Optional. List of parameters passed to |
nsim |
Number of simulated realisations
to generate (for |
nrmh |
Number of Metropolis-Hastings iterations
for each simulated realisation (for |
start , control
|
Arguments passed to |
verb |
Logical flag indicating whether to print progress reports
(for |
callstring |
Internal use only. |
NOTE: This help page describes the old syntax of the
function ppm
, described in many older documents.
This old syntax is still supported. However, if you are learning about
ppm
for the first time, we recommend you use the
new syntax described in the help file for ppm
.
This function fits a point process model to an observed point pattern. The model may include spatial trend, interpoint interaction, and dependence on covariates.
In basic use, Q
is a point pattern dataset
(an object of class "ppp"
) to which we wish to fit a model.
The syntax of ppm()
is closely analogous to the R functions
glm
and gam
.
The analogy is:
glm | ppm |
formula |
trend |
family |
interaction
|
The point process model to be fitted is specified by the
arguments trend
and interaction
which are respectively analogous to
the formula
and family
arguments of glm().
Systematic effects (spatial trend and/or dependence on
spatial covariates) are specified by the argument
trend
. This is an R formula object, which may be expressed
in terms of the Cartesian coordinates x
, y
,
the marks marks
,
or the variables in covariates
(if supplied), or both.
It specifies the logarithm of the first order potential
of the process.
The formula should not use any names beginning with .mpl
as these are reserved for internal use.
If trend
is absent or equal to the default, ~1
, then
the model to be fitted is stationary (or at least, its first order
potential is constant).
The symbol .
in the trend expression stands for
all the covariates supplied in the argument data
.
For example the formula ~ .
indicates an additive
model with a main effect for each covariate in data
.
Stochastic interactions between random points of the point process
are defined by the argument interaction
. This is an object of
class "interact"
which is initialised in a very similar way to the
usage of family objects in glm
and gam
.
The models currently available are:
AreaInter
, BadGey
, Concom
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, HierHard
, HierStrauss
, HierStraussHard
, Hybrid
, LennardJones
, MultiHard
, MultiStrauss
, MultiStraussHard
, OrdThresh
, Ord
, Pairwise
, PairPiece
, Penttinen
, Poisson
, Saturated
, SatPiece
, Softcore
, Strauss
, StraussHard
and Triplets
.
See the examples below.
It is also possible to combine several interactions
using Hybrid
.
If interaction
is missing or NULL
,
then the model to be fitted
has no interpoint interactions, that is, it is a Poisson process
(stationary or nonstationary according to trend
). In this case
the methods of maximum pseudolikelihood and maximum logistic likelihood
coincide with maximum likelihood.
The fitted point process model returned by this function can be printed
(by the print method print.ppm
)
to inspect the fitted parameter values.
If a nonparametric spatial trend was fitted, this can be extracted using
the predict method predict.ppm
.
To fit a model involving spatial covariates
other than the Cartesian coordinates and
,
the values of the covariates should be supplied in the
argument
covariates
.
Note that it is not sufficient to have observed
the covariate only at the points of the data point pattern;
the covariate must also have been observed at other
locations in the window.
Typically the argument covariates
is a list,
with names corresponding to variables in the trend
formula.
Each entry in the list is either
giving the values of a spatial covariate at
a fine grid of locations. It should be an object of
class "im"
, see im.object
.
which can be evaluated
at any location (x,y)
to obtain the value of the spatial
covariate. It should be a function(x, y)
or function(x, y, ...)
in the R language.
For marked point pattern data, the covariate can be a
function(x, y, marks)
or function(x, y, marks, ...)
.
The first two arguments of the function should be the
Cartesian coordinates and
. The function may have
additional arguments; if the function does not have default
values for these additional arguments, then the user must
supply values for them, in
covfunargs
.
See the Examples.
interpreted as a logical variable
which is TRUE
inside the window and FALSE
outside
it. This should be an object of class "owin"
.
interpreted as a factor covariate.
For each spatial location, the factor value indicates
which tile of the tessellation it belongs to.
This should be an object of class "tess"
.
indicating a covariate that is constant in this dataset.
The software will look up the values of each covariate at the required locations (quadrature points).
Note that, for covariate functions, only the name of the function appears in the trend formula. A covariate function is treated as if it were a single variable. The function arguments do not appear in the trend formula. See the Examples.
If covariates
is a list,
the list entries should have names corresponding to
the names of covariates in the model formula trend
.
The variable names x
, y
and marks
are reserved for the Cartesian
coordinates and the mark values,
and these should not be used for variables in covariates
.
If covariates
is a data frame, Q
must be a
quadrature scheme (see under Quadrature Schemes below).
Then covariates
must have
as many rows as there are points in Q
.
The th row of
covariates
should contain the values of
spatial variables which have been observed
at the th point of
Q
.
In advanced use, Q
may be a ‘quadrature scheme’.
This was originally just a technicality but it has turned out
to have practical uses, as we explain below.
Quadrature schemes are required for our implementation of the method of maximum pseudolikelihood. The definition of the pseudolikelihood involves an integral over the spatial window containing the data. In practice this integral must be approximated by a finite sum over a set of quadrature points. We use the technique of Baddeley and Turner (2000), a generalisation of the Berman-Turner (1992) device. In this technique the quadrature points for the numerical approximation include all the data points (points of the observed point pattern) as well as additional ‘dummy’ points.
Quadrature schemes are also required for the method of maximum logistic likelihood, which combines the data points with additional ‘dummy’ points.
A quadrature scheme is an object of class "quad"
(see quad.object
)
which specifies both the data point pattern and the dummy points
for the quadrature scheme, as well as the quadrature weights
associated with these points.
If Q
is simply a point pattern
(of class "ppp"
, see ppp.object
)
then it is interpreted as specifying the
data points only; a set of dummy points specified
by default.dummy()
is added,
and the default weighting rule is
invoked to compute the quadrature weights.
Finer quadrature schemes (i.e. those with more dummy points) generally yield a better approximation, at the expense of higher computational load.
An easy way to fit models using a finer quadrature scheme
is to let Q
be the original point pattern data,
and use the argument nd
to determine the number of dummy points in the quadrature scheme.
Complete control over the quadrature scheme is possible.
See quadscheme
for an overview.
Use quadscheme(X, D, method="dirichlet")
to compute
quadrature weights based on the Dirichlet tessellation,
or quadscheme(X, D, method="grid")
to compute
quadrature weights by counting points in grid squares,
where X
and D
are the patterns of data points
and dummy points respectively.
Alternatively use pixelquad
to make a quadrature
scheme with a dummy point at every pixel in a pixel image.
The argument quad.args
can be used to control the
construction of the quadrature scheme. For example
quad.args=list(quasi=TRUE, method="dirichlet", eps=0.1)
would
create dummy points according to a quasirandom pattern, with a
typical spacing of 0.1 units between dummy points,
and compute quadrature weights based on the Dirichlet tessellation.
A practical advantage of quadrature schemes arises when we want to fit
a model involving covariates (e.g. soil pH). Suppose we have only been
able to observe the covariates at a small number of locations.
Suppose cov.dat
is a data frame containing the values of
the covariates at the data points (i.e.\ cov.dat[i,]
contains the observations for the i
th data point)
and cov.dum
is another data frame (with the same columns as
cov.dat
) containing the covariate values at another
set of points whose locations are given by the point pattern Y
.
Then setting Q = quadscheme(X,Y)
combines the data points
and dummy points into a quadrature scheme, and
covariates = rbind(cov.dat, cov.dum)
combines the covariate
data frames. We can then fit the model by calling
ppm(Q, ..., covariates)
.
There are several choices for the technique used to fit the model.
(the default):
the model will be fitted by maximising the
pseudolikelihood (Besag, 1975) using the
Berman-Turner computational approximation
(Berman and Turner, 1992; Baddeley and Turner, 2000).
Maximum pseudolikelihood is equivalent to maximum likelihood
if the model is a Poisson process.
Maximum pseudolikelihood is biased if the
interpoint interaction is very strong, unless there
is a large number of dummy points.
The default settings for method='mpl'
specify a moderately large number of dummy points,
striking a compromise between speed and accuracy.
the model will be fitted by maximising the
logistic likelihood (Baddeley et al, 2014).
This technique is roughly equivalent in speed to
maximum pseudolikelihood, but is
believed to be less biased. Because it is less biased,
the default settings for method='logi'
specify a relatively small number of dummy points,
so that this method is the fastest, in practice.
the model will be fitted in a Bayesian setup by maximising the
posterior probability density for the canonical model
parameters. This uses the variational Bayes approximation to
the posterior derived from the logistic likelihood as described
in Rajala (2014). The prior is assumed to be multivariate
Gaussian with mean vector prior.mean
and variance-covariance
matrix prior.var
.
Note that method='logi'
and method='VBlogi'
involve randomisation, so that the results are
subject to random variation.
After this initial fit, there are several ways to improve the fit:
No further improvement is performed.
the model will be re-fitted by applying the approximate maximum likelihood method of Huang and Ogata (1999). See below. The Huang-Ogata method is slower than the other options, but has better statistical properties. This method involves randomisation, so the results are subject to random variation.
The model will be re-fitted using a regularized version of the composite likelihood. See below.
If improve.type="ho"
then the model will be fitted using
the Huang-Ogata (1999) approximate maximum likelihood method.
First the model is fitted by maximum pseudolikelihood as
described above, yielding an initial estimate of the parameter
vector .
From this initial model,
nsim
simulated
realisations are generated. The score and Fisher information of
the model at
are estimated from the simulated realisations. Then one step
of the Fisher scoring algorithm is taken, yielding an updated
estimate
. The corresponding model is
returned.
Simulated realisations are generated using rmh
.
The iterative behaviour of the Metropolis-Hastings algorithm
is controlled by the arguments start
and control
which are passed to rmh
.
As a shortcut, the argument
nrmh
determines the number of Metropolis-Hastings
iterations run to produce one simulated realisation (if
control
is absent). Also
if start
is absent or equal to NULL
, it defaults to
list(n.start=N)
where N
is the number of points
in the data point pattern.
This requires the package glmnet. Details to be written.
Edge correction should be applied to the sufficient statistics
of the model, to reduce bias.
The argument correction
is the name of an edge correction
method.
The default correction="border"
specifies the border correction,
in which the quadrature window (the domain of integration of the
pseudolikelihood) is obtained by trimming off a margin of width
rbord
from the observation window of the data pattern.
Not all edge corrections are implemented (or implementable)
for arbitrary windows.
Other options depend on the argument interaction
, but these
generally include correction="periodic"
(the periodic or toroidal edge
correction in which opposite edges of a rectangular window are
identified) and correction="translate"
(the translation correction,
see Baddeley 1998 and Baddeley and Turner 2000).
For pairwise interaction models
there is also Ripley's isotropic correction,
identified by correction="isotropic"
or "Ripley"
.
The arguments subset
and clipwin
specify that the
model should be fitted to a restricted subset of the available
data. These arguments are equivalent for Poisson point process models,
but different for Gibbs models.
If clipwin
is specified, then all the available data will
be restricted to this spatial region, and data outside this region
will be discarded, before the model is fitted.
If subset
is specified, then no data are deleted, but
the domain of integration of the likelihood or pseudolikelihood
is restricted to the subset
.
For Poisson models, these two arguments have the same effect;
but for a Gibbs model,
interactions between points inside and outside the subset
are taken into account, while
interactions between points inside and outside the clipwin
are ignored.
An object of class "ppm"
describing a fitted point process
model.
See ppm.object
for details of the format of this object
and methods available for manipulating it.
Apart from the Poisson model, every point process model fitted by
ppm
has parameters that determine the strength and
range of ‘interaction’ or dependence between points.
These parameters are of two types:
A parameter is called regular
if the log likelihood is a linear function of
where
is some transformation of
. [Then
is called the canonical
parameter.]
Other parameters are called irregular.
Typically, regular parameters determine the ‘strength’
of the interaction, while irregular parameters determine the
‘range’ of the interaction. For example, the Strauss process
has a regular parameter controlling the strength
of interpoint inhibition, and an irregular parameter
determining the range of interaction.
The ppm
command is only designed to estimate regular
parameters of the interaction.
It requires the values of any irregular parameters of the interaction
to be fixed. For example, to fit a Strauss process model to the cells
dataset, you could type ppm(cells, ~1, Strauss(r=0.07))
.
Note that the value of the irregular parameter r
must be given.
The result of this command will be a fitted model in which the
regular parameter has been estimated.
To determine the irregular parameters, there are several
practical techniques, but no general statistical theory available.
Useful techniques include maximum profile pseudolikelihood, which
is implemented in the command profilepl
,
and Newton-Raphson maximisation, implemented in the
experimental command ippm
.
Some irregular parameters can be estimated directly from data:
the hard-core radius in the model Hardcore
and the matrix of hard-core radii in MultiHard
can be
estimated easily from data. In these cases, ppm
allows the user
to specify the interaction without giving
the value of the irregular parameter. The user can give the
hard core interaction as interaction=Hardcore()
or even interaction=Hardcore
, and
the hard core radius will then be estimated from the data.
Some common error messages and warning messages are listed below, with explanations.
The fitted model coefficients do not define a valid point
process. This can occur because some of the fitted coefficients
are NA
(perhaps because the model formula included redundant
covariates so that the coefficients cannot be estimated), or
because the fitted interaction coefficients do not define
a valid point process (e.g. because a point process
model which always has inhibition between points was fitted to a
clustered point pattern). See valid.ppm
for detailed
information.
The software cannot calculate standard errors or confidence
intervals for the coefficients of the fitted model.
This requires the (asymptotic) variance-covariance
matrix, which is the inverse matrix of the Fisher information
matrix of the fitted model. The error message states
that the determinant of the Fisher information matrix is zero,
or close to zero, so that the matrix cannot be inverted.
This error is usually reported when the model is printed,
because the print
method calculates standard errors for the
fitted parameters. Singularity usually occurs because the spatial
coordinates in the original data were very large numbers
(e.g. expressed in metres) so that the fitted coefficients were
very small numbers. The simple remedy is to
rescale the data, for example, to convert from metres to
kilometres by X <- rescale(X, 1000)
, then re-fit the
model. Singularity can also occur if the covariate values are
very large numbers, or if the covariates are approximately
collinear.
The covariate data (typically a pixel image) did not provide values of the covariate at some of the spatial locations in the observation window of the point pattern. This means that the spatial domain of the pixel image does not completely cover the observation window of the point pattern. If the percentage is small, this warning can be ignored - typically it happens because of rounding effects which cause the pixel image to be one-pixel-width narrower than the observation window. However if more than a few percent of covariate values are undefined, it would be prudent to check that the pixel images are correct, and are correctly registered in their spatial relation to the observation window.
A problem has arisen when creating the quadrature scheme
used to fit the model. In the default rule for computing the
quadrature weights, space is divided into rectangular tiles,
and the number of quadrature points (data and dummy points) in
each tile is counted. It is possible for a tile with non-zero area
to contain no quadrature points; in this case, the quadrature
scheme will contribute a bias to the model-fitting procedure.
A small relative error (less than 2 percent) is not important.
Relative errors of a few percent can occur because of the shape of
the window.
If the relative error is greater than about 5 percent, we
recommend trying different parameters for the quadrature scheme,
perhaps setting a larger value of nd
to increase the number
of dummy points. A relative error greater than 10 percent
indicates a major problem with the input data: in this case,
extract the quadrature scheme by applying quad.ppm
to the fitted model, and inspect it.
(The most likely cause of this problem is that the spatial coordinates
of the original data were not handled correctly, for example,
coordinates of the locations and the window boundary were incompatible.)
It is not possible to estimate all the model parameters from this dataset. The error message gives a further explanation, such as “data pattern is empty”. Choose a simpler model, or check the data.
In a Gibbs model (i.e. with interaction between points), the conditional intensity may be zero at some spatial locations, indicating that the model forbids the presence of a point at these locations. However if the conditional intensity is zero at a data point, this means that the model is inconsistent with the data. Modify the interaction parameters so that the data point is not illegal (e.g. reduce the value of the hard core radius) or choose a different interaction.
The implementation of the Huang-Ogata method is experimental; several bugs were fixed in spatstat 1.19-0.
See the comments above about the possible inefficiency and bias of the maximum pseudolikelihood estimator.
The accuracy of the Berman-Turner approximation to the pseudolikelihood depends on the number of dummy points used in the quadrature scheme. The number of dummy points should at least equal the number of data points.
The parameter values of the fitted model
do not necessarily determine a valid point process.
Some of the point process models are only defined when the parameter
values lie in a certain subset. For example the Strauss process only
exists when the interaction parameter
is less than or equal to
,
corresponding to a value of
ppm()$theta[2]
less than or equal to 0
.
By default (if emend=FALSE
) the algorithm
maximises the pseudolikelihood
without constraining the parameters, and does not apply any checks for
sanity after fitting the model.
This is because the fitted parameter value
could be useful information for data analysis.
To constrain the parameters to ensure that the model is a valid
point process, set emend=TRUE
. See also the functions
valid.ppm
and emend.ppm
.
The trend
formula should not use any variable names
beginning with the prefixes .mpl
or Interaction
as these names are reserved
for internal use. The data frame covariates
should have as many rows
as there are points in Q
. It should not contain
variables called x
, y
or marks
as these names are reserved for the Cartesian coordinates
and the marks.
If the model formula involves one of the functions
poly()
, bs()
or ns()
(e.g. applied to spatial coordinates x
and y
),
the fitted coefficients can be misleading.
The resulting fit is not to the raw spatial variates
(x
, x^2
, x*y
, etc.)
but to a transformation of these variates. The transformation is implemented
by poly()
in order to achieve better numerical stability.
However the
resulting coefficients are appropriate for use with the transformed
variates, not with the raw variates.
This affects the interpretation of the constant
term in the fitted model, logbeta
.
Conventionally, is the background intensity, i.e. the
value taken by the conditional intensity function when all predictors
(including spatial or “trend” predictors) are set equal to
.
However the coefficient actually produced is the value that the
log conditional intensity takes when all the predictors,
including the transformed
spatial predictors, are set equal to
0
, which is not the same thing.
Worse still, the result of predict.ppm
can be
completely wrong if the trend formula contains one of the
functions poly()
, bs()
or ns()
. This is a weakness of the underlying
function predict.glm
.
If you wish to fit a polynomial trend,
we offer an alternative to poly()
,
namely polynom()
, which avoids the
difficulty induced by transformations. It is completely analogous
to poly
except that it does not orthonormalise.
The resulting coefficient estimates then have
their natural interpretation and can be predicted correctly.
Numerical stability may be compromised.
Values of the maximised pseudolikelihood are not comparable
if they have been obtained with different values of rbord
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31–38.
Besag, J. Statistical analysis of non-lattice data. The Statistician 24 (1975) 179-195.
Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. On parameter estimation for pairwise interaction processes. International Statistical Review 62 (1994) 99-117.
Huang, F. and Ogata, Y. Improvements of the maximum pseudo-likelihood estimators in various spatial statistical models. Journal of Computational and Graphical Statistics 8 (1999) 510-530.
Jensen, J.L. and Moeller, M. Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability 1 (1991) 445–461.
Jensen, J.L. and Kuensch, H.R. On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46 (1994) 475-486.
Rajala T. (2014) A note on Bayesian logistic regression for spatial exponential family Gibbs point processes, Preprint on ArXiv.org. https://arxiv.org/abs/1411.0539
ppm.object
for details of how to
print, plot and manipulate a fitted model.
ppp
and quadscheme
for constructing data.
Interactions:
AreaInter
, BadGey
, Concom
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, HierHard
, HierStrauss
, HierStraussHard
, Hybrid
, LennardJones
, MultiHard
, MultiStrauss
, MultiStraussHard
, OrdThresh
, Ord
, Pairwise
, PairPiece
, Penttinen
, Poisson
, Saturated
, SatPiece
, Softcore
, Strauss
, StraussHard
and Triplets
.
See profilepl
for advice on
fitting nuisance parameters in the interaction,
and ippm
for irregular parameters in the trend.
See valid.ppm
and emend.ppm
for
ensuring the fitted model is a valid point process.
# fit the stationary Poisson process # to point pattern 'nztrees' ppm(nztrees) ppm(nztrees ~ 1) # equivalent. Q <- quadscheme(nztrees) ppm(Q) # equivalent. fit1 <- ppm(nztrees, ~ x) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx) # where x,y are the Cartesian coordinates # and a,b are parameters to be estimated # For other examples, see help(ppm)
# fit the stationary Poisson process # to point pattern 'nztrees' ppm(nztrees) ppm(nztrees ~ 1) # equivalent. Q <- quadscheme(nztrees) ppm(Q) # equivalent. fit1 <- ppm(nztrees, ~ x) # fit the nonstationary Poisson process # with intensity function lambda(x,y) = exp(a + bx) # where x,y are the Cartesian coordinates # and a,b are parameters to be estimated # For other examples, see help(ppm)
Calculates all the leverage and
influence measures described in influence.ppm
,
leverage.ppm
and dfbetas.ppm
.
ppmInfluence(fit, what = c("leverage", "influence", "dfbetas"), ..., iScore = NULL, iHessian = NULL, iArgs = NULL, drop = FALSE, fitname = NULL)
ppmInfluence(fit, what = c("leverage", "influence", "dfbetas"), ..., iScore = NULL, iHessian = NULL, iArgs = NULL, drop = FALSE, fitname = NULL)
fit |
A fitted point process model of class |
what |
Character vector specifying which quantities are to be calculated. Default is to calculate all quantities. |
... |
Ignored. |
iScore , iHessian
|
Components of the score vector and Hessian matrix for the irregular parameters, if required. See Details. |
iArgs |
List of extra arguments for the functions |
drop |
Logical. Whether to include ( |
fitname |
Optional character string name for the fitted model |
This function calculates all the
leverage and influence measures
described in influence.ppm
, leverage.ppm
and dfbetas.ppm
.
When analysing large datasets, the user can
call ppmInfluence
to perform the calculations efficiently,
then extract the leverage and influence values as desired.
For example the leverage can be extracted either as
result$leverage
or leverage(result)
.
If the point process model trend has irregular parameters that were
fitted (using ippm
)
then the influence calculation requires the first and second
derivatives of the log trend with respect to the irregular parameters.
The argument iScore
should be a list,
with one entry for each irregular parameter, of R functions that compute the
partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular
parameter. The argument iHessian
should be a list,
with entries where
is the number of irregular
parameters, of R functions that compute the second order
partial derivatives of the
log trend with respect to each pair of irregular parameters.
A list containing the leverage and influence measures specified by
what
. The result also belongs to the class "ppmInfluence"
.
Adrian Baddeley [email protected]
leverage.ppm
,
influence.ppm
,
dfbetas.ppm
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~ x+y) fI <- ppmInfluence(fit) fitlev <- fI$leverage fitlev <- leverage(fI) fitinf <- fI$influence fitinf <- influence(fI) fitdfb <- fI$dfbetas fitdfb <- dfbetas(fI)
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X ~ x+y) fI <- ppmInfluence(fit) fitlev <- fI$leverage fitlev <- leverage(fI) fitinf <- fI$influence fitinf <- influence(fI) fitdfb <- fI$dfbetas fitdfb <- dfbetas(fI)
Given a fitted determinantal point process model, these functions compute the fitted intensity.
## S3 method for class 'dppm' fitted(object, ...) ## S3 method for class 'dppm' predict(object, ...)
## S3 method for class 'dppm' fitted(object, ...) ## S3 method for class 'dppm' predict(object, ...)
object |
Fitted determinantal point process model.
An object of class |
... |
Arguments passed to |
These functions are methods for the generic functions
fitted
and predict
.
The argument object
should be a determinantal point process model
(object of class "dppm"
) obtained using
the function dppm
.
The intensity of the fitted model
is computed, using fitted.ppm
or
predict.ppm
respectively.
The value of fitted.dppm
is a numeric vector
giving the fitted values at the quadrature points.
The value of predict.dppm
is usually a pixel image
(object of class "im"
), but see predict.ppm
for details.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
dppm
,
plot.dppm
,
fitted.ppm
,
predict.ppm
if(interactive()) { fit <- dppm(swedishpines ~ x + y, dppGauss()) } else { fit <- dppm(redwood ~ x, dppGauss()) } predict(fit)
if(interactive()) { fit <- dppm(swedishpines ~ x + y, dppGauss()) } else { fit <- dppm(redwood ~ x, dppGauss()) } predict(fit)
Given a fitted cluster point process model, these functions compute the fitted intensity.
## S3 method for class 'kppm' fitted(object, ...) ## S3 method for class 'kppm' predict(object, ...)
## S3 method for class 'kppm' fitted(object, ...) ## S3 method for class 'kppm' predict(object, ...)
object |
Fitted cluster point process model.
An object of class |
... |
Arguments passed to |
These functions are methods for the generic functions
fitted
and predict
.
The argument object
should be a cluster point process model
(object of class "kppm"
) obtained using
the function kppm
.
The intensity of the fitted model
is computed, using fitted.ppm
or
predict.ppm
respectively.
The value of fitted.kppm
is a numeric vector
giving the fitted values at the quadrature points.
The value of predict.kppm
is usually a pixel image
(object of class "im"
), but see predict.ppm
for details.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
kppm
,
plot.kppm
,
vcov.kppm
,
fitted.ppm
,
predict.ppm
fit <- kppm(redwood ~ x, "Thomas") predict(fit)
fit <- kppm(redwood ~ x, "Thomas") predict(fit)
Given a fitted multiple point process model obtained by mppm
,
evaluate the spatial trend and/or the conditional intensity of the
model. By default, predictions are evaluated over a grid of
locations, yielding pixel images of the trend and conditional intensity.
Alternatively predictions may be evaluated at specified
locations with specified values of the covariates.
## S3 method for class 'mppm' predict(object, ..., newdata = NULL, type = c("trend", "cif"), ngrid = 40, locations=NULL, verbose=FALSE)
## S3 method for class 'mppm' predict(object, ..., newdata = NULL, type = c("trend", "cif"), ngrid = 40, locations=NULL, verbose=FALSE)
object |
The fitted model. An object of class |
... |
Ignored. |
newdata |
Optional. New values of the covariates, for which the predictions should be computed. See Details. |
type |
Type of predicted values required. A character string or vector of
character strings. Options are |
ngrid |
Dimensions of the grid of spatial locations at which prediction will be
performed (if |
locations |
Optional. The locations at which
predictions should be performed. A list of point patterns, with one entry
for each row of |
verbose |
Logical flag indicating whether to print progress reports. |
This function computes the spatial trend and the conditional intensity of a spatial point process model that has been fitted to several spatial point patterns. See Chapter 16 of Baddeley, Rubak and Turner (2015) for explanation and examples.
Note that by “spatial trend” we mean the (exponentiated) first
order potential and not the intensity of the process. [For example
if we fit the stationary Strauss process with parameters
and
,
then the spatial trend is constant and equal to
.]
The conditional intensity
of the fitted
model is evaluated at each required spatial location u, with respect
to the response point pattern X.
If newdata=NULL
, predictions are computed for the original
values of the covariates, to which the model was fitted.
Otherwise newdata
should be a hyperframe
(see hyperframe
) containing columns of covariates
as required by the model. If type
includes "cif"
,
then newdata
must also include a column of spatial point
pattern responses, in order to compute the conditional intensity.
If locations=NULL
, then predictions are performed
at an ngrid
by ngrid
grid of locations in the window
for each response point pattern. The result will be a hyperframe
containing a column of images of the trend (if selected)
and a column of images of the conditional intensity (if selected).
The result can be plotted.
If locations
is given, then it should be a list of point
patterns (objects of class "ppp"
). Predictions are performed at these
points, and the results are returned as mark values attached to the
locations
. The result is a hyperframe containing columns
called trend
and/or cif
. The column called trend
contains marked point patterns in which the point locations are
the locations
and the mark value is the predicted trend.
The column called cif
contains marked point patterns in which the point locations are
the locations
and the mark value is the predicted conditional
intensity.
A hyperframe with columns named trend
and/or cif
.
If locations=NULL
, the entries of the hyperframe are
pixel images.
If locations
is not null, the entries are
marked point patterns constructed by attaching the predicted values
to the locations
point patterns.
The point process model that is described by an mppm
object
may be a different point process for each row of the original
hyperframe of data. This occurs if
the model formula includes the variable id
(representing
row number) or if the model has a different interpoint interaction on
each row.
If the point process model is different on each row of the original data, then either
newdata
is missing. Predictions are computed for
each row of the original data using the point process model
that applies on each row.
newdata
must have the same number of rows
as the original data. Each row of newdata
is assumed
to be a replacement for the corresponding row of the original data.
The prediction for row i
of newdata
will be computed for the point process model that applies to row i
of the original data.
newdata
must include a column called id
specifying the row number, and therefore identifying which
of the point process models should apply.
The predictions for row i
of newdata
will be computed for the point process model that applies
to row k
of the original data, where k = newdata$id[i]
.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Baddeley, A., Bischof, L., Sintorn, I.-M., Haggarty, S., Bell, M. and Turner, R. Analysis of a designed experiment where the response is a spatial point pattern. In preparation.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
h <- hyperframe(Bugs=waterstriders) fit <- mppm(Bugs ~ x, data=h, interaction=Strauss(7)) # prediction on a grid p <- predict(fit) plot(p$trend) # prediction at specified locations loc <- with(h, runifpoint(20, Window(Bugs))) p2 <- predict(fit, locations=loc) plot(p2$trend)
h <- hyperframe(Bugs=waterstriders) fit <- mppm(Bugs ~ x, data=h, interaction=Strauss(7)) # prediction on a grid p <- predict(fit) plot(p$trend) # prediction at specified locations loc <- with(h, runifpoint(20, Window(Bugs))) p2 <- predict(fit, locations=loc) plot(p2$trend)
Given a fitted point process model obtained by ppm
,
evaluate the spatial trend or the conditional intensity of the model
at new locations.
## S3 method for class 'ppm' predict(object, window=NULL, ngrid=NULL, locations=NULL, covariates=NULL, type=c("trend", "cif", "intensity", "count"), se=FALSE, interval=c("none", "confidence", "prediction"), level = 0.95, X=data.ppm(object), correction, ignore.hardcore=FALSE, ..., dimyx=NULL, eps=NULL, rule.eps = c("adjust.eps", "grow.frame", "shrink.frame"), new.coef=NULL, check=TRUE, repair=TRUE)
## S3 method for class 'ppm' predict(object, window=NULL, ngrid=NULL, locations=NULL, covariates=NULL, type=c("trend", "cif", "intensity", "count"), se=FALSE, interval=c("none", "confidence", "prediction"), level = 0.95, X=data.ppm(object), correction, ignore.hardcore=FALSE, ..., dimyx=NULL, eps=NULL, rule.eps = c("adjust.eps", "grow.frame", "shrink.frame"), new.coef=NULL, check=TRUE, repair=TRUE)
object |
A fitted point process model, typically obtained from
the model-fitting algorithm |
window |
Optional. A window (object of class |
ngrid |
Optional. Dimensions of a rectangular grid of locations
inside |
locations |
Optional. Data giving the exact
|
covariates |
Values of external covariates required by the model. Either a data frame or a list of images. See Details. |
type |
Character string.
Indicates which property of the fitted model should be predicted.
Options are |
se |
Logical value indicating whether to calculate standard errors as well. |
interval |
String (partially matched) indicating whether to produce
estimates ( |
level |
Coverage probability for the confidence or prediction interval. |
X |
Optional. A point pattern (object of class |
correction |
Name of the edge correction to be used
in calculating the conditional intensity.
Options include |
ignore.hardcore |
Advanced use only. Logical value specifying whether to compute only the finite part of the interaction potential (effectively removing any hard core interaction terms). |
... |
Ignored. |
dimyx |
Equivalent to |
eps |
Width and height of pixels in the prediction grid. A numerical value, or numeric vector of length 2. |
rule.eps |
Argument passed to |
new.coef |
Numeric vector of parameter values to replace the
fitted model parameters |
check |
Logical value indicating whether to check the internal format
of |
repair |
Logical value indicating whether to repair the internal format
of |
This function computes properties of a fitted spatial point process
model (object of class "ppm"
). For a Poisson point process
it can compute the fitted intensity function, or the expected number of
points in a region. For a Gibbs point process it can compute the
spatial trend (first order potential), conditional intensity,
and approximate intensity of the process.
Point estimates, standard errors,
confidence intervals and prediction intervals are available.
Given a point pattern dataset, we may fit
a point process model to the data using the
model-fitting algorithm ppm
. This
returns an object of class "ppm"
representing
the fitted point process model (see ppm.object
).
The parameter estimates in this fitted model can be read off
simply by printing the ppm
object.
The spatial trend, conditional intensity and intensity of the
fitted model are evaluated using this function predict.ppm
.
The default action is to create a rectangular grid of points in the observation window of the data point pattern, and evaluate the spatial trend at these locations.
The argument type
specifies the values that are desired:
type="trend"
:the “spatial trend” of the fitted model is evaluated at each
required spatial location . See below.
type="cif"
:the conditional intensity of the
fitted model is evaluated at each required spatial location
,
with respect to the data point pattern
.
type="intensity"
:the intensity of the
fitted model is evaluated at each required spatial location
.
type="count"
:the expected total number of points (or the expected number
of points falling in window
) is evaluated.
If window
is a tessellation,
the expected number of points in each tile of the tessellation
is evaluated.
The spatial trend, conditional intensity, and intensity
are all equivalent if the fitted model is a Poisson point process.
However, if the model is not a Poisson process, then they are
all different. The “spatial trend” is the (exponentiated)
first order potential, and not the intensity of the process.
[For example if we fit the
stationary Strauss process with parameters
and
,
then the spatial trend is constant and equal to
,
while the intensity is a smaller value.]
The default is to compute an estimate of the desired quantity.
If interval="confidence"
or interval="prediction"
,
the estimate is replaced by a confidence interval or prediction interval.
If se=TRUE
, then a standard error is also calculated,
and is returned together with the (point or interval) estimate.
The spatial locations where predictions are required,
are determined by the (incompatible)
arguments ngrid
and locations
.
If the argument ngrid
is present, then
predictions are performed at a rectangular
grid of locations in the window window
.
The result of prediction will be a pixel image or images.
If locations
is present, then predictions
will be performed at the spatial locations given by
this dataset. These may be an arbitrary list of spatial locations,
or they may be a rectangular grid.
The result of prediction will be either a numeric vector
or a pixel image or images.
If neither ngrid
nor locations
is given, then
ngrid
is assumed. The value of ngrid
defaults to
spatstat.options("npixel")
, which is initialised to 128
when spatstat is loaded.
The argument locations
may be a point pattern,
a data frame or a list specifying arbitrary locations;
or it may be a binary image mask (an object of class "owin"
with type "mask"
) or a pixel image (object of class
"im"
) specifying (a subset of) a rectangular
grid of locations.
If locations
is a point pattern (object of class "ppp"
),
then prediction will be performed at the points of the point pattern.
The result of prediction will be a vector of predicted values,
one value for each point.
If the model is a marked point process, then
locations
should be a marked point pattern, with marks of the
same kind as the model; prediction will be performed at these
marked points.
The result of prediction will be a vector of predicted values,
one value for each (marked) point.
If locations
is a data frame or list, then it must contain
vectors locations$x
and locations$y
specifying the
coordinates of the prediction locations. Additionally, if
the model is a marked point process, then
locations
must also contain
a factor locations$marks
specifying the marks of the
prediction locations. These vectors must have equal length.
The result of prediction will be a vector of predicted values,
of the same length.
If locations
is a binary image mask, then prediction will be
performed at each pixel in this binary image where the pixel value
is TRUE
(in other words, at each pixel that is inside the
window). If the fitted model is an unmarked point process, then the
result of prediction will be an image. If the fitted model is a
marked point process, then prediction will
be performed for each possible value of the mark at each such
location, and the result of prediction will be a
list of images, one for each mark value.
If locations
is a pixel image (object of class "im"
),
then prediction will be performed at each pixel in this image where
the pixel value is defined (i.e.\ where the pixel value is not
NA
).
The argument covariates
gives the values of any spatial covariates
at the prediction locations.
If the trend formula in the fitted model
involves spatial covariates (other than
the Cartesian coordinates x
, y
)
then covariates
is required.
The format and use of covariates
are analogous to those of the
argument of the same name in ppm
.
It is either a data frame or a list of images.
If covariates
is a list of images, then
the names of the entries should correspond to
the names of covariates in the model formula trend
.
Each entry in the list must be an image object (of class "im"
,
see im.object
).
The software will look up
the pixel values of each image at the quadrature points.
If covariates
is a data frame, then the
i
th row of covariates
is assumed to contain covariate data for the i
th location.
When locations
is a data frame,
this just means that each row of covariates
contains the
covariate data for the location specified in the corresponding row of
locations
. When locations
is a binary image
mask, the row covariates[i,]
must correspond to the location
x[i],y[i]
where x = as.vector(raster.x(locations))
and y = as.vector(raster.y(locations))
.
Note that if you only want to use prediction in order to
generate a plot of the predicted values,
it may be easier to use plot.ppm
which calls
this function and plots the results.
If total
is given:
a numeric vector or matrix.
If locations
is given and is a data frame:
a vector of predicted values for the spatial locations
(and marks, if required) given in locations
.
If ngrid
is given, or if locations
is given
and is a binary image mask or a pixel image:
If object
is an unmarked point process,
the result is a pixel image object (of class "im"
, see
im.object
) containing the predictions.
If object
is a multitype point process,
the result is a list of pixel images, containing the predictions
for each type at the same grid of locations.
The “predicted values” are either values of the spatial trend
(if type="trend"
), values of the conditional intensity
(if type="cif"
or type="lambda"
),
values of the intensity (if type="intensity"
)
or numbers of points (if type="count"
).
If se=TRUE
, then the result is a list with two entries,
the first being the predicted values in the format described above,
and the second being the standard errors in the same format.
The current implementation invokes predict.glm
so that prediction is wrong if the trend formula in
object
involves terms in ns()
,
bs()
or poly()
.
This is a weakness of predict.glm
itself!
Error messages may be very opaque,
as they tend to come from deep in the workings of
predict.glm
.
If you are passing the covariates
argument
and the function crashes,
it is advisable to start by checking that all the conditions
listed above are satisfied.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31–38.
ppm
,
ppm.object
,
plot.ppm
,
print.ppm
,
fitted.ppm
,
spatstat.options
m <- ppm(cells ~ polynom(x,y,2), Strauss(0.05)) trend <- predict(m, type="trend") if(human <- interactive()) { image(trend) points(cells) } cif <- predict(m, type="cif") if(human) { persp(cif) } mj <- ppm(japanesepines ~ harmonic(x,y,2)) se <- predict(mj, se=TRUE) # image of standard error ci <- predict(mj, interval="c") # two images, confidence interval # prediction interval for total number of points predict(mj, type="count", interval="p") # prediction intervals for counts in tiles predict(mj, window=quadrats(japanesepines, 3), type="count", interval="p") # prediction at arbitrary locations predict(mj, locations=data.frame(x=0.3, y=0.4)) X <- runifpoint(5, Window(japanesepines)) predict(mj, locations=X, se=TRUE) # multitype rr <- matrix(0.06, 2, 2) ma <- ppm(amacrine ~ marks, MultiStrauss(rr)) Z <- predict(ma) Z <- predict(ma, type="cif") predict(ma, locations=data.frame(x=0.8, y=0.5,marks="on"), type="cif")
m <- ppm(cells ~ polynom(x,y,2), Strauss(0.05)) trend <- predict(m, type="trend") if(human <- interactive()) { image(trend) points(cells) } cif <- predict(m, type="cif") if(human) { persp(cif) } mj <- ppm(japanesepines ~ harmonic(x,y,2)) se <- predict(mj, se=TRUE) # image of standard error ci <- predict(mj, interval="c") # two images, confidence interval # prediction interval for total number of points predict(mj, type="count", interval="p") # prediction intervals for counts in tiles predict(mj, window=quadrats(japanesepines, 3), type="count", interval="p") # prediction at arbitrary locations predict(mj, locations=data.frame(x=0.3, y=0.4)) X <- runifpoint(5, Window(japanesepines)) predict(mj, locations=X, se=TRUE) # multitype rr <- matrix(0.06, 2, 2) ma <- ppm(amacrine ~ marks, MultiStrauss(rr)) Z <- predict(ma) Z <- predict(ma, type="cif") predict(ma, locations=data.frame(x=0.8, y=0.5,marks="on"), type="cif")
Given a model which has been fitted to point pattern data by recursive partitioning, compute the predicted intensity of the model.
## S3 method for class 'rppm' predict(object, ...) ## S3 method for class 'rppm' fitted(object, ...)
## S3 method for class 'rppm' predict(object, ...) ## S3 method for class 'rppm' fitted(object, ...)
object |
Fitted point process model of class |
... |
Optional arguments passed to |
These functions are methods for the generic functions
fitted
and predict
.
They compute the fitted intensity of a point process model.
The argument object
should be a fitted point process model
of class "rppm"
produced by the function rppm
.
The fitted
method computes the fitted intensity at the original data
points, yielding a numeric vector with one entry for each data point.
The predict
method computes the fitted intensity at
any locations. By default, predictions are
calculated at a regular grid of spatial locations, and the result
is a pixel image giving the predicted intensity values at these
locations.
Alternatively, predictions can be performed at other
locations, or a finer grid of locations, or only at certain specified
locations, using additional arguments ...
which will be interpreted by predict.ppm
.
Common arguments are ngrid
to increase the grid resolution,
window
to specify the prediction region, and locations
to specify the exact locations of predictions.
See predict.ppm
for details of these arguments.
Predictions are computed by evaluating the explanatory covariates at each desired location, and applying the recursive partitioning rule to each set of covariate values.
The result of fitted.rppm
is a numeric vector.
The result of predict.rppm
is a pixel image, a list of pixel images,
or a numeric vector.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- rppm(unmark(gorillas) ~ vegetation, data=gorillas.extra) plot(predict(fit)) lambdaX <- fitted(fit) lambdaX[1:5] # Mondriaan pictures plot(predict(rppm(redwoodfull ~ x + y))) points(redwoodfull)
fit <- rppm(unmark(gorillas) ~ vegetation, data=gorillas.extra) plot(predict(fit)) lambdaX <- fitted(fit) lambdaX[1:5] # Mondriaan pictures plot(predict(rppm(redwoodfull ~ x + y))) points(redwoodfull)
Given a fitted Spatial Logistic Regression model, this function computes the fitted probabilities for each pixel, or the fitted point process intensity, or the values of the linear predictor in each pixel.
## S3 method for class 'slrm' predict(object, ..., type = "intensity", newdata=NULL, window=NULL)
## S3 method for class 'slrm' predict(object, ..., type = "intensity", newdata=NULL, window=NULL)
object |
a fitted spatial logistic regression model.
An object of class |
... |
Optional arguments passed to |
type |
Character string (partially) matching one of
|
newdata |
Optional. List containing new covariate values for the prediction. See Details. |
window |
Optional. New window in which to predict.
An object of class |
This is a method for predict
for spatial logistic
regression models (objects of class "slrm"
, usually obtained
from the function slrm
).
The argument type
determines which quantity is computed.
If type="intensity"
), the value of the point process intensity
is computed at each pixel. If type="probabilities"
) the
probability of the presence of a random point in each pixel is
computed. If type="link"
, the value of the linear predictor is
computed at each pixel.
If newdata = NULL
(the default), the algorithm computes
fitted values of the model (based on the data that was originally used
to fit the model object
).
If newdata
is given, the algorithm computes predicted values
of the model, using the new values of the covariates provided by
newdata
. The argument newdata
should be a list;
names of entries in the list should correspond
to variables appearing in the model formula of the object
.
Each list entry may be a pixel image or a single numeric
value.
A pixel image (object of class "im"
) containing the predicted
values for each pixel.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
X <- rpoispp(42) fit <- slrm(X ~ x+y) plot(predict(fit)) X <- copper$SouthPoints Y <- copper$SouthLines Z <- distmap(Y) fitc <- slrm(X ~ Z) pc <- predict(fitc) Znew <- distmap(copper$Lines)[copper$SouthWindow] pcnew <- predict(fitc, newdata=list(Z=Znew))
X <- rpoispp(42) fit <- slrm(X ~ x+y) plot(predict(fit)) X <- copper$SouthPoints Y <- copper$SouthLines Z <- distmap(Y) fitc <- slrm(X ~ Z) pc <- predict(fitc) Znew <- distmap(copper$Lines)[copper$SouthWindow] pcnew <- predict(fitc, newdata=list(Z=Znew))
Default print
method for a fitted point process model.
## S3 method for class 'ppm' print(x,..., what=c("all", "model", "trend", "interaction", "se", "errors"))
## S3 method for class 'ppm' print(x,..., what=c("all", "model", "trend", "interaction", "se", "errors"))
x |
A fitted point process model, typically obtained from
the model-fittingg algorithm |
what |
Character vector (partially-matched) indicating what information should be printed. |
... |
Ignored. |
This is the print
method for the class "ppm"
.
It prints information about the fitted model in a sensible format.
The argument what
makes it possible to print only some
of the information.
If what
is missing, then by default,
standard errors for the estimated coefficients of the model
will be printed only if the model is a Poisson point process.
To print the standard errors for a non-Poisson model,
call print.ppm
with the argument what
given explicitly,
or reset the default rule by typing
spatstat.options(print.ppm.SE="always")
.
none.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
ppm.object
for details of the class "ppm"
.
ppm
for generating these objects.
m <- ppm(cells ~1, Strauss(0.05)) m
m <- ppm(cells ~1, Strauss(0.05)) m
Fits point process models by maximising the profile likelihood, profile pseudolikelihood, profile composite likelihood or AIC.
profilepl(s, f, ..., aic=FALSE, rbord=NULL, verbose = TRUE, fast=TRUE)
profilepl(s, f, ..., aic=FALSE, rbord=NULL, verbose = TRUE, fast=TRUE)
s |
Data frame containing values of the irregular parameters over which the criterion will be computed. |
f |
Function (such as |
... |
Data passed to |
aic |
Logical value indicating whether to find the parameter values
which minimise the AIC ( |
rbord |
Radius for border correction (same for all models). If omitted, this will be computed from the interactions. |
verbose |
Logical value indicating whether to print progress reports. |
fast |
Logical value indicating whether to use a faster, less accurate model-fitting technique when computing the profile pseudolikelihood. See Section on Speed and Accuracy. |
The model-fitting function ppm
fits point process
models to point pattern data. However,
only the ‘regular’ parameters of the model can be fitted by
ppm
. The model may also depend on ‘irregular’
parameters that must be fixed in any call to ppm
.
This function profilepl
is a wrapper which finds the values of the
irregular parameters that give the best fit.
If aic=FALSE
(the default),
the best fit is the model which maximises the
likelihood (if the models are Poisson processes) or maximises
the pseudolikelihood or logistic likelihood.
If aic=TRUE
then the best fit is the model which
minimises the Akaike Information Criterion AIC.ppm
.
The argument s
must be a data frame whose columns contain
values of the irregular parameters over which the maximisation is
to be performed.
An irregular parameter may affect either the interpoint interaction or the spatial trend.
in a call to ppm
, the argument interaction
determines the interaction between points. It is usually
a call to a function such as Strauss
. The
arguments of this call are irregular parameters.
For example, the interaction radius parameter of the Strauss
process, determined by the argument
r
to the function Strauss
, is an irregular parameter.
in a call to ppm
, the spatial trend may depend on
covariates, which are supplied by the argument covariates
.
These covariates may be functions written by the user,
of the form function(x,y,...)
, and the extra arguments
...
are irregular parameters.
The argument f
determines the interaction
for each model to be fitted. It would typically be one of the functions
Poisson
,
AreaInter
,
BadGey
,
DiggleGatesStibbard
,
DiggleGratton
,
Fiksel
,
Geyer
,
Hardcore
,
LennardJones
,
OrdThresh
,
Softcore
,
Strauss
or
StraussHard
.
Alternatively it could be a function written by the user.
Columns of s
which match the names of arguments of f
will be interpreted as interaction parameters. Other columns will be
interpreted as trend parameters.
The data frame s
must provide values for each argument of
f
, except for the optional arguments, which are those arguments of
f
that have the default value NA
.
To find the best fit,
each row of s
will be taken in turn. Interaction parameters in this
row will be passed to f
, resulting in an interaction object.
Then ppm
will be applied to the data ...
using this interaction. Any trend parameters will be passed to
ppm
through the argument covfunargs
.
This results in a fitted point process model.
The value of the log pseudolikelihood or AIC from this model is stored.
After all rows of s
have been processed in this way, the
row giving the maximum value of log pseudolikelihood will be found.
The object returned by profilepl
contains the profile
pseudolikelihood (or profile AIC) function,
the best fitting model, and other data.
It can be plotted (yielding a
plot of the log pseudolikelihood or AIC values against the irregular
parameters) or printed (yielding information about the best fitting
values of the irregular parameters).
In general, f
may be any function that will return
an interaction object (object of class "interact"
)
that can be used in a call to ppm
. Each argument of
f
must be a single value.
An object of class "profilepl"
. There are methods
for plot
,
print
,
summary
,
simulate
,
as.ppm
,
fitin
and
parameters
for objects of this class.
The components of the object include
fit |
Best-fitting model |
param |
The data frame |
iopt |
Row index of the best-fitting parameters in |
To extract the best fitting model you can also use as.ppm
.
Computation of the profile pseudolikelihood can be time-consuming.
We recommend starting with a small experiment in which
s
contains only a few rows of values. This will indicate
roughly the optimal values of the parameters.
Then a full calculation using more finely
spaced values can identify the exact optimal values.
It is normal that the procedure appears to slow down at the end. During the computation of the profile pseudolikelihood, the model-fitting procedure is accelerated by omitting some calculations that are not needed for computing the pseudolikelihood. When the optimal parameter values have been identified, they are used to fit the final model in its entirety. Fitting the final model can take longer than computing the profile pseudolikelihood.
If fast=TRUE
(the default), then additional shortcuts are taken
in order to accelerate the computation of the profile
log pseudolikelihood. These shortcuts mean that the values of the profile
log pseudolikelihood in the result ($prof
)
may not be equal to the values that would be obtained if the model was
fitted normally. Currently this happens only for the area interaction
AreaInter
. It may be wise to do a small experiment with
fast=TRUE
and then a definitive calculation with fast=FALSE
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
# one irregular parameter rr <- data.frame(r=seq(0.05,0.15, by=0.01)) ps <- profilepl(rr, Strauss, cells) ps plot(ps) # two irregular parameters rs <- expand.grid(r=seq(0.05,0.15, by=0.01),sat=1:3) pg <- profilepl(rs, Geyer, cells) pg as.ppm(pg) ## more information summary(pg) # multitype pattern with a common interaction radius RR <- data.frame(R=seq(0.03,0.05,by=0.01)) MS <- function(R) { MultiStrauss(radii=diag(c(R,R))) } pm <- profilepl(RR, MS, amacrine ~marks)
# one irregular parameter rr <- data.frame(r=seq(0.05,0.15, by=0.01)) ps <- profilepl(rr, Strauss, cells) ps plot(ps) # two irregular parameters rs <- expand.grid(r=seq(0.05,0.15, by=0.01),sat=1:3) pg <- profilepl(rs, Geyer, cells) pg as.ppm(pg) ## more information summary(pg) # multitype pattern with a common interaction radius RR <- data.frame(R=seq(0.03,0.05,by=0.01)) MS <- function(R) { MultiStrauss(radii=diag(c(R,R))) } pm <- profilepl(RR, MS, amacrine ~marks)
Given a model which has been fitted to point pattern data by recursive partitioning, apply pruning to reduce the complexity of the partition tree.
## S3 method for class 'rppm' prune(tree, ...)
## S3 method for class 'rppm' prune(tree, ...)
tree |
Fitted point process model of class |
... |
Arguments passed to |
This is a method for the generic function prune
for the class "rppm"
. An object of this class is a
point process model, fitted to point pattern data by
recursive partitioning, by the function rppm
.
The recursive partition tree will be pruned using
prune.rpart
. The result is another
object of class "rppm"
.
Object of class "rppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
rppm
,
plot.rppm
,
predict.rppm
.
# Murchison gold data mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) fit <- rppm(gold ~ dfault + greenstone, data=mur) fit prune(fit, cp=0.1)
# Murchison gold data mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) fit <- rppm(gold ~ dfault + greenstone, data=mur) fit prune(fit, cp=0.1)
Given a fitted point process model, calculate the pseudo-R-squared value, which measures the fraction of variation in the data that is explained by the model.
pseudoR2(object, ...) ## S3 method for class 'ppm' pseudoR2(object, ..., keepoffset=TRUE) ## S3 method for class 'slrm' pseudoR2(object, ..., keepoffset=TRUE)
pseudoR2(object, ...) ## S3 method for class 'ppm' pseudoR2(object, ..., keepoffset=TRUE) ## S3 method for class 'slrm' pseudoR2(object, ..., keepoffset=TRUE)
object |
Fitted point process model.
An object of class |
keepoffset |
Logical value indicating whether to retain offset terms in the model when computing the deviance difference. See Details. |
... |
Additional arguments passed to
|
The function pseudoR2
is generic, with methods
for fitted point process models
of class "ppm"
and "slrm"
.
This function computes McFadden's pseudo-Rsquared
where is the deviance of the fitted model
object
,
and is the deviance of the null model.
Deviance is defined as twice the negative log-likelihood
or log-pseudolikelihood.
The null model is usually obtained by re-fitting the model
using the trend formula ~1
.
However if the original model formula included offset
terms,
and if keepoffset=TRUE
(the default),
then the null model formula consists of these offset terms. This
ensures that the pseudoR2
value is non-negative.
A single numeric value.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- ppm(swedishpines ~ x+y) pseudoR2(fit) xcoord <- as.im(function(x,y) x, Window(swedishpines)) fut <- ppm(swedishpines ~ offset(xcoord/200) + y) pseudoR2(fut)
fit <- ppm(swedishpines ~ x+y) pseudoR2(fit) xcoord <- as.im(function(x,y) x, Window(swedishpines)) fut <- ppm(swedishpines ~ offset(xcoord/200) + y) pseudoR2(fut)
Computes the sibling probability of a cluster point process model.
psib(object) ## S3 method for class 'kppm' psib(object)
psib(object) ## S3 method for class 'kppm' psib(object)
object |
Fitted cluster point process model
(object of class |
In a Poisson cluster process, two points are called siblings
if they belong to the same cluster, that is, if they had the same
parent point. If two points of the process are
separated by a distance , the probability that
they are siblings is
where
is the
pair correlation function of the process.
The value is the probability that,
if two points of the process are situated very close to each other,
they came from the same cluster. This probability
is an index of the strength of clustering, with high values
suggesting strong clustering.
This concept was proposed in Baddeley, Rubak and Turner (2015, page 479) and Baddeley (2017). It was shown in Baddeley et al (2022) that the sibling probability is directly related to the strength of clustering.
A single number.
Adrian Baddeley [email protected].
Baddeley, A. (2017) Local composite likelihood for spatial point processes. Spatial Statistics 22, 261–295.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Baddeley, A., Davies, T.M., Hazelton, M.L., Rakshit, S. and Turner, R.
(2022)
Fundamental problems in fitting spatial cluster process models.
Spatial Statistics 52, 100709.
DOI: 10.1016/j.spasta.2022.100709
fit <- kppm(redwood ~1, "Thomas") psib(fit)
fit <- kppm(redwood ~1, "Thomas") psib(fit)
Given a point process model fitted to a point pattern dataset, and any choice of functional summary statistic, this function computes the pseudoscore test statistic of goodness-of-fit for the model.
psst(object, fun, r = NULL, breaks = NULL, ..., model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), truecoef=NULL, hi.res=NULL, funargs = list(correction="best"), verbose=TRUE)
psst(object, fun, r = NULL, breaks = NULL, ..., model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), truecoef=NULL, hi.res=NULL, funargs = list(correction="best"), verbose=TRUE)
object |
Object to be analysed.
Either a fitted point process model (object of class |
fun |
Summary function to be applied to each point pattern. |
r |
Optional.
Vector of values of the argument |
breaks |
Optional alternative to |
... |
Ignored. |
model |
Optional. A fitted point process model (object of
class |
trend , interaction , rbord
|
Optional. Arguments passed to |
truecoef |
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with |
hi.res |
Optional. List of parameters passed to |
funargs |
List of additional arguments to be passed to |
verbose |
Logical value determining whether to print progress reports during the computation. |
Let be a point pattern dataset consisting of points
in a window
.
Consider a point process model fitted to
, with
conditional intensity
at location
.
For the purpose of testing goodness-of-fit, we regard the fitted model
as the null hypothesis. Given a functional summary statistic
,
consider a family of alternative models obtained by exponential
tilting of the null model by
.
The pseudoscore for the null model is
where the operator is
the difference between the values of for the
point pattern with and without the point
.
According to the Georgii-Nguyen-Zessin formula, should have
mean zero if the model is correct (ignoring the fact that the
parameters of the model have been estimated). Hence
can be
used as a diagnostic for goodness-of-fit.
This algorithm computes by direct evaluation of the sum and
integral. It is computationally intensive, but it is available for
any summary statistic
.
The diagnostic is also called
the pseudoresidual of
. On the right
hand side of the equation for
given above,
the sum over points of
is called the
pseudosum and the integral is called the pseudocompensator.
A function value table (object of class "fv"
),
essentially a data frame of function values.
Columns in this data frame include dat
for the pseudosum,
com
for the compensator and res
for the
pseudoresidual.
There is a plot method for this class. See fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Alternative functions:
Kres
,
Gres
.
if(live <- interactive()) { fit0 <- ppm(cells ~ 1) } else { fit0 <- ppm(cells ~ 1, nd=8) } G0 <- psst(fit0, Gest) G0 if(live) plot(G0)
if(live <- interactive()) { fit0 <- ppm(cells ~ 1) } else { fit0 <- ppm(cells ~ 1, nd=8) } G0 <- psst(fit0, Gest) G0 if(live) plot(G0)
Given a point process model fitted to a point pattern dataset, this function computes the pseudoscore diagnostic of goodness-of-fit for the model, against moderately clustered or moderately inhibited alternatives of area-interaction type.
psstA(object, r = NULL, breaks = NULL, ..., model = NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), ppmcorrection = "border", correction = "all", truecoef = NULL, hi.res = NULL, nr=spatstat.options("psstA.nr"), ngrid=spatstat.options("psstA.ngrid"))
psstA(object, r = NULL, breaks = NULL, ..., model = NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), ppmcorrection = "border", correction = "all", truecoef = NULL, hi.res = NULL, nr=spatstat.options("psstA.nr"), ngrid=spatstat.options("psstA.ngrid"))
object |
Object to be analysed.
Either a fitted point process model (object of class |
r |
Optional.
Vector of values of the argument |
breaks |
This argument is for internal use only. |
... |
Extra arguments passed to |
model |
Optional. A fitted point process model (object of
class |
trend , interaction , rbord
|
Optional. Arguments passed to |
ppmcorrection |
Optional. Character string specifying the edge correction
for the pseudolikelihood to be used
in fitting the point process model. Passed to |
correction |
Optional. Character string specifying which diagnostic quantities
will be computed. Options are |
truecoef |
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with |
hi.res |
Optional. List of parameters passed to |
nr |
Optional. Number of |
ngrid |
Integer. Number of points in the square grid used to compute the approximate area. |
This function computes the pseudoscore test statistic which can be used as a diagnostic for goodness-of-fit of a fitted point process model.
Let be a point pattern dataset consisting of points
in a window
.
Consider a point process model fitted to
, with
conditional intensity
at location
.
For the purpose of testing goodness-of-fit, we regard the fitted model
as the null hypothesis.
The alternative hypothesis is a family of
hybrid models obtained by combining
the fitted model with the area-interaction process
(see
AreaInter
). The family of alternatives includes
models that are slightly more regular than the fitted model,
and others that are slightly more clustered than the fitted model.
The pseudoscore, evaluated at the null model, is
where
where is the area of the union of the discs of radius
centred at the points of
(i.e.
is the area
of the dilation of
by a distance
).
Thus
is the unclaimed area associated with
, that is, the area of that part of the disc
of radius
centred at the point
that is
not covered by any of the discs of radius
centred at
points of
.
According to the Georgii-Nguyen-Zessin formula, should have
mean zero if the model is correct (ignoring the fact that the
parameters of the model have been estimated). Hence
can be
used as a diagnostic for goodness-of-fit.
The diagnostic is also called
the pseudoresidual of
. On the right
hand side of the equation for
given above,
the sum over points of
is called the
pseudosum and the integral is called the pseudocompensator.
A function value table (object of class "fv"
),
essentially a data frame of function values.
Columns in this data frame include dat
for the pseudosum,
com
for the compensator and res
for the
pseudoresidual.
There is a plot method for this class. See fv.object
.
This computation can take a very long time.
To shorten the computation time, choose smaller values of the
arguments nr
and ngrid
, or reduce the values of their
defaults spatstat.options("psstA.nr")
and spatstat.options("psstA.ngrid")
.
Computation time is roughly proportional to
nr * npoints * ngrid^2
where npoints
is the number
of points in the point pattern.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Alternative functions:
psstG
,
psst
,
Gres
,
Kres
.
Point process models: ppm
.
Options: spatstat.options
if(live <- interactive()) { X <- rStrauss(200,0.1,0.05) } else { pso <- spatstat.options(psstA.ngrid=16,psstA.nr=10, ndummy.min=16,npixel=32) X <- cells } plot(psstA(X)) plot(psstA(X, interaction=Strauss(0.05))) if(!live) spatstat.options(pso)
if(live <- interactive()) { X <- rStrauss(200,0.1,0.05) } else { pso <- spatstat.options(psstA.ngrid=16,psstA.nr=10, ndummy.min=16,npixel=32) X <- cells } plot(psstA(X)) plot(psstA(X, interaction=Strauss(0.05))) if(!live) spatstat.options(pso)
Given a point process model fitted to a point pattern dataset, this function computes the pseudoscore diagnostic of goodness-of-fit for the model, against moderately clustered or moderately inhibited alternatives of saturation type.
psstG(object, r = NULL, breaks = NULL, ..., model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), truecoef = NULL, hi.res = NULL)
psstG(object, r = NULL, breaks = NULL, ..., model=NULL, trend = ~1, interaction = Poisson(), rbord = reach(interaction), truecoef = NULL, hi.res = NULL)
object |
Object to be analysed.
Either a fitted point process model (object of class |
r |
Optional.
Vector of values of the argument |
breaks |
Optional alternative to |
... |
Ignored. |
model |
Optional. A fitted point process model (object of
class |
trend , interaction , rbord
|
Optional. Arguments passed to |
truecoef |
Optional. Numeric vector. If present, this will be treated as
if it were the true coefficient vector of the point process model,
in calculating the diagnostic. Incompatible with |
hi.res |
Optional. List of parameters passed to |
This function computes the pseudoscore test statistic which can be used as a diagnostic for goodness-of-fit of a fitted point process model.
Consider a point process model fitted to , with
conditional intensity
at location
.
For the purpose of testing goodness-of-fit, we regard the fitted model
as the null hypothesis.
The alternative hypothesis is a family of
hybrid models obtained by combining
the fitted model with the Geyer saturation process
(see
Geyer
) with saturation parameter 1.
The family of alternatives includes
models that are more regular than the fitted model,
and others that are more clustered than the fitted model.
For any point pattern , and any
, let
be the number of points in
whose nearest
neighbour (the nearest other point in
)
is closer than
units. Then the pseudoscore for the null model is
where the operator is
the difference between the values of for the
point pattern with and without the point
.
According to the Georgii-Nguyen-Zessin formula, should have
mean zero if the model is correct (ignoring the fact that the
parameters of the model have been estimated). Hence
can be
used as a diagnostic for goodness-of-fit.
The diagnostic is also called
the pseudoresidual of
. On the right
hand side of the equation for
given above,
the sum over points of
is called the
pseudosum and the integral is called the pseudocompensator.
A function value table (object of class "fv"
),
essentially a data frame of function values.
Columns in this data frame include dat
for the pseudosum,
com
for the compensator and res
for the
pseudoresidual.
There is a plot method for this class. See
fv.object
.
Adrian Baddeley [email protected], Ege Rubak [email protected] and Jesper Moller.
Baddeley, A., Rubak, E. and Moller, J. (2011) Score, pseudo-score and residual diagnostics for spatial point process models. Statistical Science 26, 613–646.
Alternative functions:
psstA
,
psst
,
Kres
,
Gres
.
if(live <- interactive()) { X <- rStrauss(200,0.1,0.05) } else { pso <- spatstat.options(ndummy.min=16,npixel=32) X <- cells } plot(psstG(X)) plot(psstG(X, interaction=Strauss(0.05))) if(!live) spatstat.options(pso)
if(live <- interactive()) { X <- rStrauss(200,0.1,0.05) } else { pso <- spatstat.options(ndummy.min=16,npixel=32) X <- cells } plot(psstG(X)) plot(psstG(X, interaction=Strauss(0.05))) if(!live) spatstat.options(pso)
Given a point process model fitted to a point pattern, produce a Q-Q plot based on residuals from the model.
qqplot.ppm(fit, nsim=100, expr=NULL, ..., type="raw", style="mean", fast=TRUE, verbose=TRUE, plot.it=TRUE, dimyx=NULL, nrep=if(fast) 5e4 else 1e5, control=update(default.rmhcontrol(fit), nrep=nrep), saveall=FALSE, monochrome=FALSE, limcol=if(monochrome) "black" else "red", maxerr=max(100, ceiling(nsim/10)), check=TRUE, repair=TRUE, envir.expr)
qqplot.ppm(fit, nsim=100, expr=NULL, ..., type="raw", style="mean", fast=TRUE, verbose=TRUE, plot.it=TRUE, dimyx=NULL, nrep=if(fast) 5e4 else 1e5, control=update(default.rmhcontrol(fit), nrep=nrep), saveall=FALSE, monochrome=FALSE, limcol=if(monochrome) "black" else "red", maxerr=max(100, ceiling(nsim/10)), check=TRUE, repair=TRUE, envir.expr)
fit |
The fitted point process model, which is to be assessed
using the Q-Q plot. An object of class |
nsim |
The number of simulations from the “reference” point process model. |
expr |
Determines the simulation mechanism which provides the “theoretical” quantiles for the Q-Q plot. See Details. |
... |
Arguments passed to |
type |
String indicating the type of residuals or weights to be used.
Current options are |
style |
Character string controlling the type of Q-Q plot.
Options are |
fast |
Logical flag controlling the speed and accuracy of computation.
Use |
verbose |
Logical flag controlling whether the algorithm prints progress reports during long computations. |
plot.it |
Logical flag controlling whether the function produces a plot or simply returns a value (silently). |
dimyx |
Dimensions of the pixel grid on which the smoothed residual field will be calculated. A vector of two integers. |
nrep |
If |
control |
List of parameters controlling the Metropolis-Hastings algorithm
|
saveall |
Logical flag indicating whether to save all the intermediate calculations. |
monochrome |
Logical flag indicating whether the plot should be
in black and white ( |
limcol |
String. The colour to be used when plotting the 95% limit curves. |
maxerr |
Maximum number of failures tolerated while generating simulated realisations. See Details. |
check |
Logical value indicating whether to check the internal format
of |
repair |
Logical value indicating whether to repair the internal format
of |
envir.expr |
Optional. An environment in which the expression
|
This function generates a Q-Q plot of the residuals from a
fitted point process model. It is an addendum to the suite of
diagnostic plots produced by the function diagnose.ppm
,
kept separate because it is computationally intensive. The
quantiles of the theoretical distribution are estimated by simulation.
In classical statistics, a Q-Q plot of residuals is a useful
diagnostic for checking the distributional assumptions. Analogously,
in spatial statistics, a Q-Q plot of the (smoothed) residuals from a
fitted point process model is a useful way
to check the interpoint interaction part of the model
(Baddeley et al, 2005). The systematic part of the model
(spatial trend, covariate effects, etc) is assessed using
other plots made by diagnose.ppm
.
The argument fit
represents the fitted point process
model. It must be an object of class "ppm"
(typically produced
by the maximum pseudolikelihood fitting algorithm ppm
).
Residuals will be computed for this fitted model using
residuals.ppm
,
and the residuals will be kernel-smoothed to produce a “residual
field”. The values of this residual field will provide the
“data” quantiles for the Q-Q plot.
The argument expr
is not usually specified.
It provides a way to modify the “theoretical” or “reference”
quantiles for the Q-Q plot.
In normal usage we set expr=NULL
. The default
is to generate nsim
simulated realisations
of the fitted model fit
, re-fit this model to
each of the simulated patterns,
evaluate the residuals from
these fitted models, and use the kernel-smoothed residual field
from these fitted models as a sample from the reference distribution
for the Q-Q plot.
In advanced use, expr
may be an expression
.
It will be re-evaluated nsim
times, and should include
random computations so that the results are not identical
each time. The result of evaluating expr
should be either a point pattern (object of class
"ppp"
) or a fitted point process model (object of class
"ppm"
). If the value is a point pattern, then the
original fitted model fit
will be fitted to this new point
pattern using update.ppm
, to yield another fitted
model. Smoothed residuals obtained from these
nsim
fitted models will yield the “theoretical” quantiles for the
Q-Q plot.
Alternatively expr
can be a list of point patterns,
or an envelope
object that contains a list of point patterns
(typically generated by calling envelope
with
savepatterns=TRUE
). These point patterns will be used
as the simulated patterns.
Simulation is performed (if expr=NULL
)
using the Metropolis-Hastings algorithm rmh
.
Each simulated realisation is the result of
running the Metropolis-Hastings algorithm
from an independent random starting state each time.
The iterative and termination behaviour of the Metropolis-Hastings
algorithm are governed by the argument control
.
See rmhcontrol
for information about this argument.
As a shortcut, the argument nrep
determines
the number of Metropolis-Hastings iterations used to generate
each simulated realisation, if control
is absent.
By default, simulations are generated in an expanded
window. Use the argument control
to change this,
as explained in the section on Warning messages.
The argument type
selects the type of residual or weight
that will be computed. For options, see diagnose.ppm
.
The argument style
determines the type of Q-Q plot. It is
highly recommended to use the default, style="mean"
.
style="classical"
The quantiles of the residual field for the data (on the
axis) are plotted against the
quantiles of the pooled simulations (on the
axis).
This plot is biased, and therefore difficult to interpret,
because of strong autocorrelations in the residual field
and the large differences in sample size.
style="mean"
The order statistics of the residual field for the data are plotted
against the sample means, over the nsim
simulations,
of the corresponding order statistics of the residual field
for the simulated datasets.
Dotted lines show the 2.5 and 97.5 percentiles, over the
nsim
simulations, of each order statistic.
The argument fast
is a simple way to control
the accuracy and speed of computation.
If fast=FALSE
, the residual field is computed on
a fine grid of pixels (by default 100 by 100 pixels, see below)
and the Q-Q plot is based on the complete set of order statistics
(usually 10,000 quantiles).
If fast=TRUE
, the residual field is computed on a coarse
grid (at most 40 by 40 pixels) and the Q-Q plot is based on the
percentiles only. This is about 7 times faster.
It is recommended to use fast=TRUE
for interactive data
analysis and fast=FALSE
for definitive plots for
publication.
The argument dimyx
gives full control over the resolution of the
pixel grid used to calculate the smoothed residuals.
Its interpretation is the same as the argument dimyx
to the function as.mask
.
Note that dimyx[1]
is the number of
pixels in the direction, and
dimyx[2]
is the number
in the direction.
If
dimyx
is not present, then the default pixel grid dimensions
are controlled by spatstat.options("npixel")
.
Since the computation is so time-consuming, qqplot.ppm
returns
a list containing all the data necessary to re-display the Q-Q plot.
It is advisable to assign the result of qqplot.ppm
to something
(or use .Last.value
if you forgot to.)
The return value is an object of class "qqppm"
. There are methods for
plot.qqppm
and print.qqppm
. See the
Examples.
The argument saveall
is usually set to FALSE
.
If saveall=TRUE
, then the intermediate results of calculation for each
simulated realisation are saved and returned. The return value
includes a 3-dimensional array sim
containing the
smoothed residual field images for each of the nsim
realisations. When saveall=TRUE
, the return value is an object of very
large size, and should not be saved on disk.
Errors may occur during the simulation process, because random data are generated. For example:
one of the simulated patterns may be empty.
one of the simulated patterns may cause an error in the code that fits the point process model.
the user-supplied argument expr
may have a bug.
Empty point patterns do not cause a problem for the code,
but they are reported.
Other problems that would lead to a crash are trapped;
the offending simulated data are discarded, and the simulation is
retried. The argument maxerr
determines the maximum number of
times that such errors will be tolerated (mainly as a
safeguard against an infinite loop).
An object of class "qqppm"
containing the information
needed to reproduce the Q-Q plot.
Entries x
and y
are numeric vectors containing
quantiles of the simulations and of the data, respectively.
Produces a Q-Q plot if plot.it
is TRUE.
A warning message will be issued if any of the simulations trapped an error (a potential crash).
A warning message will be issued if all, or many, of the simulated point patterns are empty. This usually indicates a problem with the simulation procedure.
The default behaviour of qqplot.ppm
is to simulate patterns
on an expanded window (specified through the argument
control
) in order to avoid edge effects.
The model's trend is extrapolated over this expanded
window. If the trend is strongly inhomogeneous, the
extrapolated trend may have very large (or even infinite)
values. This can cause the simulation algorithm to
produce empty patterns.
The only way to suppress this problem entirely is to
prohibit the expansion of the window, by setting
the control
argument to something like
control=list(nrep=1e6, expand=1)
. Here expand=1
means there will be no expansion. See rmhcontrol
for more information about the argument control
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
diagnose.ppm
,
lurking
,
residuals.ppm
,
eem
,
ppm.object
,
ppm
,
rmh
,
rmhcontrol
fit <- ppm(cells ~1, Poisson()) diagnose.ppm(fit) # no suggestion of departure from stationarity if(interactive()) { qqplot.ppm(fit, 80) # strong evidence of non-Poisson interaction diagnose.ppm(fit, type="pearson") qqplot.ppm(fit, type="pearson") } # capture the plot coordinates mypreciousdata <- qqplot.ppm(fit, 4, type="pearson") ## or use the idiom .Last.value if you forgot to assign them qqplot.ppm(fit, 4, type="pearson") mypreciousdata <- .Last.value plot(mypreciousdata) ###################################################### # Q-Q plots based on fixed n # The above QQ plots used simulations from the (fitted) Poisson process. # But I want to simulate conditional on n, instead of Poisson # Do this by setting rmhcontrol(p=1) fixit <- list(p=1) if(interactive()) {qqplot.ppm(fit, 100, control=fixit)} ###################################################### # Inhomogeneous Poisson data X <- rpoispp(function(x,y){1000 * exp(-3*x)}, 1000) plot(X) # Inhomogeneous Poisson model fit <- ppm(X ~x, Poisson()) if(interactive()) {qqplot.ppm(fit, 100)} # conclusion: fitted inhomogeneous Poisson model looks OK ###################################################### # Advanced use of 'expr' argument # # set the initial conditions in Metropolis-Hastings algorithm # expr <- expression(rmh(fit, start=list(n.start=42), verbose=FALSE)) if(interactive()) qqplot.ppm(fit, 100, expr)
fit <- ppm(cells ~1, Poisson()) diagnose.ppm(fit) # no suggestion of departure from stationarity if(interactive()) { qqplot.ppm(fit, 80) # strong evidence of non-Poisson interaction diagnose.ppm(fit, type="pearson") qqplot.ppm(fit, type="pearson") } # capture the plot coordinates mypreciousdata <- qqplot.ppm(fit, 4, type="pearson") ## or use the idiom .Last.value if you forgot to assign them qqplot.ppm(fit, 4, type="pearson") mypreciousdata <- .Last.value plot(mypreciousdata) ###################################################### # Q-Q plots based on fixed n # The above QQ plots used simulations from the (fitted) Poisson process. # But I want to simulate conditional on n, instead of Poisson # Do this by setting rmhcontrol(p=1) fixit <- list(p=1) if(interactive()) {qqplot.ppm(fit, 100, control=fixit)} ###################################################### # Inhomogeneous Poisson data X <- rpoispp(function(x,y){1000 * exp(-3*x)}, 1000) plot(X) # Inhomogeneous Poisson model fit <- ppm(X ~x, Poisson()) if(interactive()) {qqplot.ppm(fit, 100)} # conclusion: fitted inhomogeneous Poisson model looks OK ###################################################### # Advanced use of 'expr' argument # # set the initial conditions in Metropolis-Hastings algorithm # expr <- expression(rmh(fit, start=list(n.start=42), verbose=FALSE)) if(interactive()) qqplot.ppm(fit, 100, expr)
Given a fitted point process model, this function extracts the quadrature scheme used to fit the model.
quad.ppm(object, drop=FALSE, clip=FALSE)
quad.ppm(object, drop=FALSE, clip=FALSE)
object |
fitted point process model (an object of class |
drop |
Logical value determining whether to delete quadrature points that were not used to fit the model. |
clip |
Logical value determining whether to erode the window,
if |
An object of class "ppm"
represents a point process model
that has been fitted to data. It is typically produced by
the model-fitting algorithm ppm
.
The maximum pseudolikelihood algorithm in ppm
approximates the pseudolikelihood
integral by a sum over a finite set of quadrature points,
which is constructed by augmenting
the original data point pattern by a set of “dummy” points.
The fitted model object returned by ppm
contains complete information about this quadrature scheme.
See ppm
or ppm.object
for further
information.
This function quad.ppm
extracts the quadrature scheme.
A typical use of this function would be to inspect the quadrature scheme
(points and weights) to gauge the accuracy of the approximation to the
exact pseudolikelihood.
Some quadrature points may not have been used in
fitting the model. This happens if the border correction is used,
and in other cases (e.g. when the value of a covariate is NA
at these points). The argument drop
specifies whether these
unused quadrature points shall be deleted (drop=TRUE
) or
retained (drop=FALSE
) in the return value.
The quadrature scheme has a window, which by default is set to
equal the window of the original data. However this window may be
larger than the actual domain of integration of the pseudolikelihood
or composite likelihood that was used to fit the model.
If clip=TRUE
then the window of the quadrature scheme is
set to the actual domain of integration. This option only has an effect
when the model was fitted using the border correction; then
the window is obtained by eroding the original data window
by the border correction distance.
See ppm.object
for a list of all operations that can be
performed on objects of class "ppm"
.
See quad.object
for a list of all operations that can be
performed on objects of class "quad"
.
This function can also be applied to objects of class "kppm"
and "lppm"
.
A quadrature scheme (object of class "quad"
).
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
fit <- ppm(cells ~1, Strauss(r=0.1)) Q <- quad.ppm(fit) plot(Q) npoints(Q$data) npoints(Q$dummy)
fit <- ppm(cells ~1, Strauss(r=0.1)) Q <- quad.ppm(fit) plot(Q) npoints(Q$data) npoints(Q$dummy)
Performs a chi-squared goodness-of-fit test of a Poisson point process model fitted to multiple point patterns.
## S3 method for class 'mppm' quadrat.test(X, ...)
## S3 method for class 'mppm' quadrat.test(X, ...)
X |
An object of class |
... |
Arguments passed to |
This function performs a test of goodness-of-fit
for a Poisson point process model, based on quadrat counts.
It can also be used to perform a test of Complete Spatial Randomness
for a list of point patterns.
The function quadrat.test
is generic, with methods for
point patterns (class "ppp"
), point process models
(class "ppm"
) and
multiple point process models (class
"mppm"
).
For this function, the argument X
should be a
multiple point process model (object of class "mppm"
)
obtained by fitting a point process model to a list of
point patterns using the function mppm
.
To perform the test, the data point patterns are extracted from X
.
For each point pattern
the window of observation is divided
into rectangular tiles, and the number of data points in each tile is
counted, as described in quadratcount
.
The expected number of points in each quadrat is calculated, as determined by the fitted model.
Then we perform a single test of goodness-of-fit
based on these observed and expected counts.
An object of class "htest"
.
Printing the object gives comprehensible output
about the outcome of the test.
The -value of the test is stored in the
component
p.value
.
The return value also belongs to
the special class "quadrat.test"
. Plotting the object
will display, for each window, the position of the quadrats,
annotated by their observed and expected
counts and the Pearson residuals. See the examples.
The return value also has an attribute "components"
which is a list containing the results of
tests of goodness-of-fit
for each individual point pattern.
If the intention is to test Complete Spatial Randomness (CSR) there are two options:
CSR with the same intensity of points in each point pattern;
CSR with a different, unrelated intensity of points in each point pattern.
In the first case,
suppose P
is a list of point patterns we want to test.
Then fit the multiple model fit1 <- mppm(P ~1)
which signifies a
Poisson point process model with a constant intensity. Then
apply quadrat.test(fit1)
.
In the second case, fit the model fit2 <- mppm(P ~id)
which signifies a Poisson point process with a different constant
intensity for each point pattern. Then apply quadrat.test(fit2)
.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
H <- hyperframe(X=waterstriders) # Poisson with constant intensity for all patterns fit1 <- mppm(X~1, H) quadrat.test(fit1, nx=2) # uniform Poisson with different intensity for each pattern fit2 <- mppm(X ~ id, H) quadrat.test(fit2, nx=2)
H <- hyperframe(X=waterstriders) # Poisson with constant intensity for all patterns fit1 <- mppm(X~1, H) quadrat.test(fit1, nx=2) # uniform Poisson with different intensity for each pattern fit2 <- mppm(X ~ id, H) quadrat.test(fit2, nx=2)
Performs a test of Complete Spatial Randomness for a given point pattern, based on quadrat counts. Alternatively performs a goodness-of-fit test of a fitted inhomogeneous Poisson model. By default performs chi-squared tests; can also perform Monte Carlo based tests.
## S3 method for class 'ppm' quadrat.test(X, nx=5, ny=nx, alternative=c("two.sided", "regular", "clustered"), method=c("Chisq", "MonteCarlo"), conditional=TRUE, CR=1, df.est=NULL, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL, nsim=1999) ## S3 method for class 'slrm' quadrat.test(X, nx=5, ny=nx, alternative=c("two.sided", "regular", "clustered"), method=c("Chisq", "MonteCarlo"), conditional=TRUE, CR=1, df.est=NULL, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL, nsim=1999)
## S3 method for class 'ppm' quadrat.test(X, nx=5, ny=nx, alternative=c("two.sided", "regular", "clustered"), method=c("Chisq", "MonteCarlo"), conditional=TRUE, CR=1, df.est=NULL, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL, nsim=1999) ## S3 method for class 'slrm' quadrat.test(X, nx=5, ny=nx, alternative=c("two.sided", "regular", "clustered"), method=c("Chisq", "MonteCarlo"), conditional=TRUE, CR=1, df.est=NULL, ..., xbreaks=NULL, ybreaks=NULL, tess=NULL, nsim=1999)
X |
A point pattern (object of class |
nx , ny
|
Numbers of quadrats in the |
alternative |
Character string (partially matched) specifying the alternative hypothesis. |
method |
Character string (partially matched) specifying the test to use:
either |
conditional |
Logical. Should the Monte Carlo test be conducted
conditionally upon the observed number of points of the pattern?
Ignored if |
CR |
Optional. Numerical value. The exponent for the Cressie-Read test statistic. See Details. |
df.est |
Optional. Advanced use only. The number of fitted parameters, or the number of degrees of freedom lost by estimation of parameters. |
... |
Ignored. |
xbreaks |
Optional. Numeric vector giving the |
ybreaks |
Optional. Numeric vector giving the |
tess |
Tessellation (object of class |
nsim |
The number of simulated samples to generate when
|
These functions perform tests or Monte Carlo tests
of goodness-of-fit for a point process model, based on quadrat counts.
The function quadrat.test
is generic, with methods for
point patterns (class "ppp"
), split point patterns
(class "splitppp"
), point process models
(class "ppm"
or "slrm"
)
and quadrat count tables (class "quadratcount"
).
if X
is a point pattern, we test the null hypothesis
that the data pattern is a realisation of Complete Spatial
Randomness (the uniform Poisson point process). Marks in the point
pattern are ignored. (If lambda
is given then the null
hypothesis is the Poisson process with intensity lambda
.)
if X
is a split point pattern, then for each of the
component point patterns (taken separately) we test
the null hypotheses of Complete Spatial Randomness.
See quadrat.test.splitppp
for documentation.
If X
is a fitted point process model, then it should be
a Poisson point process model. The
data to which this model was fitted are extracted from the model
object, and are treated as the data point pattern for the test.
We test the null hypothesis
that the data pattern is a realisation of the (inhomogeneous) Poisson point
process specified by X
.
In all cases, the window of observation is divided
into tiles, and the number of data points in each tile is
counted, as described in quadratcount
.
The quadrats are rectangular by default, or may be regions of arbitrary shape
specified by the argument tess
.
The expected number of points in each quadrat is also calculated,
as determined by CSR (in the first case) or by the fitted model
(in the second case).
Then the Pearson statistic
is computed.
If method="Chisq"
then a test of
goodness-of-fit is performed by comparing the test statistic
to the
distribution
with
degrees of freedom, where
m
is the number of
quadrats and is the number of fitted parameters
(equal to 1 for
quadrat.test.ppp
). The default is to
compute the two-sided -value, so that the test will
be declared significant if
is either very large or very
small. One-sided
-values can be obtained by specifying the
alternative
. An important requirement of the
test is that the expected counts in each quadrat
be greater than 5.
If method="MonteCarlo"
then a Monte Carlo test is performed,
obviating the need for all expected counts to be at least 5. In the
Monte Carlo test, nsim
random point patterns are generated
from the null hypothesis (either CSR or the fitted point process
model). The Pearson statistic is computed as above.
The
-value is determined by comparing the
statistic for the observed point pattern, with the values obtained
from the simulations. Again the default is to
compute the two-sided
-value.
If conditional
is TRUE
then the simulated samples are
generated from the multinomial distribution with the number of “trials”
equal to the number of observed points and the vector of probabilities
equal to the expected counts divided by the sum of the expected counts.
Otherwise the simulated samples are independent Poisson counts, with
means equal to the expected counts.
If the argument CR
is given, then instead of the
Pearson statistic, the Cressie-Read (1984) power divergence
test statistic
is computed, where is the
th observed count
and
is the corresponding expected count.
The value
CR=1
gives the Pearson statistic;
CR=0
gives the likelihood ratio test statistic ;
CR=-1/2
gives the Freeman-Tukey statistic ;
CR=-1
gives the modified likelihood ratio test statistic ;
and
CR=-2
gives Neyman's modified statistic .
In all cases the asymptotic distribution of this test statistic is
the same
distribution as above.
The return value is an object of class "htest"
.
Printing the object gives comprehensible output
about the outcome of the test.
The return value also belongs to
the special class "quadrat.test"
. Plotting the object
will display the quadrats, annotated by their observed and expected
counts and the Pearson residuals. See the examples.
An object of class "htest"
. See chisq.test
for explanation.
The return value is also an object of the special class
"quadrattest"
, and there is a plot method for this class.
See the examples.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Cressie, N. and Read, T.R.C. (1984) Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B 46, 440–464.
quadrat.test.splitppp
,
quadratcount
,
quadrats
,
quadratresample
,
chisq.test
,
cdf.test
.
To test a Poisson point process model against a specific alternative,
use anova.ppm
.
# fitted model: inhomogeneous Poisson fitx <- ppm(simdat ~ x) quadrat.test(fitx) # an equivalent test (results differ due to discretisation effects): quadrat.test(simdat, lambda=predict(fitx), df.est=length(coef(fitx)))
# fitted model: inhomogeneous Poisson fitx <- ppm(simdat ~ x) quadrat.test(fitx) # an equivalent test (results differ due to discretisation effects): quadrat.test(simdat, lambda=predict(fitx), df.est=length(coef(fitx)))
Given a point process model fitted to a list of point patterns,
extract the fixed effects of the model.
A method for ranef
.
## S3 method for class 'mppm' ranef(object, ...)
## S3 method for class 'mppm' ranef(object, ...)
object |
A fitted point process model (an object of class |
... |
Ignored. |
This is a method for the generic function ranef
.
The argument object
must be a fitted point process model
(object of class "mppm"
) produced by the
fitting algorithm mppm
). This represents a
point process model that has been fitted
to a list of several point pattern datasets. See mppm
for information.
This function extracts the coefficients of the random effects of the model.
A data frame, or list of data frames, as described in the help for
ranef.lme
.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
H <- hyperframe(Y = waterstriders) # Tweak data to exaggerate differences H$Y[[1]] <- rthin(H$Y[[1]], 0.3) m1 <- mppm(Y ~ id, data=H, Strauss(7)) ranef(m1) m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7)) ranef(m2)
H <- hyperframe(Y = waterstriders) # Tweak data to exaggerate differences H$Y[[1]] <- rthin(H$Y[[1]], 0.3) m1 <- mppm(Y ~ id, data=H, Strauss(7)) ranef(m1) m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7)) ranef(m2)
Generates simulated realisations from a determinantal point process.
rdpp(eig, index, basis = "fourierbasis", window = boxx(rep(list(0:1), ncol(index))), reject_max = 10000, progress = 0, debug = FALSE, ...)
rdpp(eig, index, basis = "fourierbasis", window = boxx(rep(list(0:1), ncol(index))), reject_max = 10000, progress = 0, debug = FALSE, ...)
eig |
vector of values between 0 and 1 specifying the non-zero eigenvalues for the process. |
index |
|
basis |
character string giving the name of the basis. |
window |
window (of class |
reject_max |
integer giving the maximal number of trials for rejection sampling. |
progress |
integer giving the interval for making a progress report. The value zero turns reporting off. |
debug |
logical value indicating whether debug informationb should be outputted. |
... |
Ignored. |
A point pattern (object of class "ppp"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
index <- expand.grid(-2:2,-2:2) eig <- exp(-rowSums(index^2)) X <- rdpp(eig, index) X ## To simulate a det. projection p. p. with the given indices set eig=1: XX <- rdpp(1, index) XX
index <- expand.grid(-2:2,-2:2) eig <- exp(-rowSums(index^2)) X <- rdpp(eig, index) X ## To simulate a det. projection p. p. with the given indices set eig=1: XX <- rdpp(1, index) XX
Computes the interaction distance of a point process model.
## S3 method for class 'ppm' reach(x, ..., epsilon=0) ## S3 method for class 'interact' reach(x, ...) ## S3 method for class 'fii' reach(x, ..., epsilon)
## S3 method for class 'ppm' reach(x, ..., epsilon=0) ## S3 method for class 'interact' reach(x, ...) ## S3 method for class 'fii' reach(x, ..., epsilon)
x |
Either a fitted point process model (object of class
|
epsilon |
Numerical threshold below which interaction is treated as zero. See details. |
... |
Other arguments are ignored. |
The function reach
computes the
‘interaction distance’ or ‘interaction range’ of a point process
model.
The definition of the interaction distance depends on the
type of point process model. This help page explains the
interaction distance for a Gibbs point process. For other kinds of
models, see reach.kppm
and
reach.dppm
.
For a Gibbs point process model, the interaction distance
is the shortest distance such that any two points in the
process which are separated by a distance greater than
do not
interact with each other.
For example, the interaction range of a Strauss process
(see Strauss
or rStrauss
)
with parameters is equal to
, unless
in which case the model is
Poisson and the interaction
range is
.
The interaction range of a Poisson process is zero.
The interaction range of the Ord threshold process
(see
OrdThresh
) is infinite, since two points may
interact at any distance apart.
The function reach
is generic, with methods
for the case where x
is
a fitted point process model
(object of class "ppm"
, usually obtained from the model-fitting
function ppm
);
an interpoint interaction structure (object of class
"interact"
)
created by one of the functions
Poisson
,
Strauss
,
StraussHard
,
MultiStrauss
,
MultiStraussHard
,
Softcore
,
DiggleGratton
,
Pairwise
,
PairPiece
,
Geyer
,
LennardJones
,
Saturated
,
OrdThresh
or
Ord
;
a fitted interpoint interaction (object of class
"fii"
)
extracted from a fitted point process model
by the command fitin
;
a point process model for simulation (object of class
"rmhmodel"
), usually obtained from rmhmodel
.
When x
is an "interact"
object,
reach(x)
returns the maximum possible interaction range
for any point process model with interaction structure given by x
.
For example, reach(Strauss(0.2))
returns 0.2
.
When x
is a "ppm"
object,
reach(x)
returns the interaction range
for the point process model represented by x
.
For example, a fitted Strauss process model
with parameters beta,gamma,r
will return
either 0
or r
, depending on whether the fitted
interaction parameter gamma
is equal or not equal to 1.
For some point process models, such as the soft core process
(see Softcore
), the interaction distance is
infinite, because the interaction terms are positive for all
pairs of points. A practical solution is to compute
the distance at which the interaction contribution
from a pair of points falls below a threshold epsilon
,
on the scale of the log conditional intensity. This is done
by setting the argument epsilon
to a positive value.
The interaction distance, or NA
if this cannot be
computed from the information given.
Methods for reach
are also defined for
point process models of class "kppm"
and "dppm"
.
Their technical definition is different from this one.
See reach.kppm
and reach.dppm
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
ppm
,
Poisson
,
Strauss
,
StraussHard
,
MultiStrauss
,
MultiStraussHard
,
Softcore
,
DiggleGratton
,
Pairwise
,
PairPiece
,
Geyer
,
LennardJones
,
Saturated
,
OrdThresh
,
Ord
.
See reach.kppm
and reach.dppm
for other types of point process models.
reach(Poisson()) # returns 0 reach(Strauss(r=7)) # returns 7 fit <- ppm(swedishpines ~ 1, Strauss(r=7)) reach(fit) # returns 7 reach(OrdThresh(42)) # returns Inf reach(MultiStrauss(matrix(c(1,3,3,1),2,2))) # returns 3
reach(Poisson()) # returns 0 reach(Strauss(r=7)) # returns 7 fit <- ppm(swedishpines ~ 1, Strauss(r=7)) reach(fit) # returns 7 reach(OrdThresh(42)) # returns Inf reach(MultiStrauss(matrix(c(1,3,3,1),2,2))) # returns 3
Returns the range of interaction for a determinantal point process model.
## S3 method for class 'dppm' reach(x, ...) ## S3 method for class 'detpointprocfamily' reach(x, ...)
## S3 method for class 'dppm' reach(x, ...) ## S3 method for class 'detpointprocfamily' reach(x, ...)
x |
Model of class |
... |
Additional arguments passed to the range function of the given model. |
The range of interaction for a determinantal point
process model may defined as the smallest number such that
for all
, where
is the pair
correlation function. For many models the range is infinite, but one
may instead use a value where the pair correlation function is
sufficiently close to 1. For example in the Matern model this defaults
to finding
such that
.
Numeric
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
reach(dppMatern(lambda=100, alpha=.01, nu=1, d=2))
reach(dppMatern(lambda=100, alpha=.01, nu=1, d=2))
Returns the range of interaction for a Cox or cluster point process model.
## S3 method for class 'kppm' reach(x, ..., epsilon)
## S3 method for class 'kppm' reach(x, ..., epsilon)
x |
Fitted point process model of class |
epsilon |
Optional numerical value. Differences smaller than
|
... |
Additional arguments passed to the range function of the given model. |
The range of interaction for a fitted point process model
of class "kppm"
may defined as the smallest number such that
for all
, where
is the pair
correlation function.
For many models the range is infinite, but one
may instead use a value where the pair correlation function is
sufficiently close to 1. The argument epsilon
specifies the
tolerance; there is a sensible default.
Numeric
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- kppm(redwood ~ 1) reach(fit)
fit <- kppm(redwood ~ 1) reach(fit)
Given a point process model fitted to a multitype point pattern, this function computes the fitted spatially-varying probability of each type of point, or the ratios of such probabilities, according to the fitted model. Optionally the standard errors of the estimates are also computed.
## S3 method for class 'ppm' relrisk(X, ..., at = c("pixels", "points"), relative = FALSE, se = FALSE, casecontrol = TRUE, control = 1, case, ngrid = NULL, window = NULL)
## S3 method for class 'ppm' relrisk(X, ..., at = c("pixels", "points"), relative = FALSE, se = FALSE, casecontrol = TRUE, control = 1, case, ngrid = NULL, window = NULL)
X |
A fitted point process model (object of class |
... |
Ignored. |
at |
String specifying whether to compute the probability values
at a grid of pixel locations ( |
relative |
Logical.
If |
se |
Logical value indicating whether to compute standard errors as well. |
casecontrol |
Logical. Whether to treat a bivariate point pattern as consisting of cases and controls, and return only the probability or relative risk of a case. Ignored if there are more than 2 types of points. See Details. |
control |
Integer, or character string, identifying which mark value corresponds to a control. |
case |
Integer, or character string, identifying which mark value
corresponds to a case (rather than a control)
in a bivariate point pattern.
This is an alternative to the argument |
ngrid |
Optional. Dimensions of a rectangular grid of locations
inside |
window |
Optional. A window (object of class |
The command relrisk
is generic and can be used to
estimate relative risk in different ways.
This function relrisk.ppm
is the method for fitted point
process models (class "ppm"
). It computes parametric
estimates of relative risk, using the fitted model.
If X
is a bivariate point pattern
(a multitype point pattern consisting of two types of points)
then by default,
the points of the first type (the first level of marks(X)
)
are treated as controls or non-events, and points of the second type
are treated as cases or events. Then by default this command computes
the spatially-varying probability of a case,
i.e. the probability
that a point at spatial location
will be a case. If
relative=TRUE
, it computes the
spatially-varying relative risk of a case relative to a
control, .
If X
is a multitype point pattern with types,
or if
X
is a bivariate point pattern
and casecontrol=FALSE
,
then by default this command computes, for each type ,
a nonparametric estimate of
the spatially-varying probability of an event of type
.
This is the probability
that a point at spatial location
will belong to type
.
If
relative=TRUE
, the command computes the
relative risk of an event of type
relative to a control,
,
where events of type
are treated as controls.
The argument
control
determines which type
is treated as a control.
If at = "pixels"
the calculation is performed for
every spatial location on a fine pixel grid, and the result
is a pixel image representing the function
or a list of pixel images representing the functions
or
for
.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as
NA
.
If at = "points"
the calculation is performed
only at the data points . By default
the result is a vector of values
giving the estimated probability of a case
at each data point, or a matrix of values
giving the estimated probability of
each possible type
at each data point.
If
relative=TRUE
then the relative risks
or
are
returned.
An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as
Inf
.
Probabilities and risks are computed from the fitted intensity of the model,
using predict.ppm
.
If se=TRUE
then standard errors will also be computed,
based on asymptotic theory, using vcov.ppm
.
If se=FALSE
(the default), the format is described below.
If se=TRUE
, the result is a list of two entries,
estimate
and SE
, each having the format described below.
If X
consists of only two types of points,
and if casecontrol=TRUE
,
the result is a pixel image (if at="pixels"
)
or a vector (if at="points"
).
The pixel values or vector values
are the probabilities of a case if relative=FALSE
,
or the relative risk of a case (probability of a case divided by the
probability of a control) if relative=TRUE
.
If X
consists of more than two types of points,
or if casecontrol=FALSE
, the result is:
(if at="pixels"
)
a list of pixel images, with one image for each possible type of point.
The result also belongs to the class "solist"
so that it can
be printed and plotted.
(if at="points"
)
a matrix of probabilities, with rows corresponding to
data points , and columns corresponding
to types
.
The pixel values or matrix entries
are the probabilities of each type of point if relative=FALSE
,
or the relative risk of each type (probability of each type divided by the
probability of a control) if relative=TRUE
.
If relative=FALSE
, the resulting values always lie between 0
and 1. If relative=TRUE
, the results are either non-negative
numbers, or the values Inf
or NA
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
There is another method relrisk.ppp
for point pattern datasets
which computes nonparametric estimates of relative risk
by kernel smoothing.
See also
relrisk
,
relrisk.ppp
,
ppm
fit <- ppm(chorley ~ marks * (x+y)) rr <- relrisk(fit, relative=TRUE, control="lung", se=TRUE) plot(rr$estimate) plot(rr$SE) rrX <- relrisk(fit, at="points", relative=TRUE, control="lung")
fit <- ppm(chorley ~ marks * (x+y)) rr <- relrisk(fit, relative=TRUE, control="lung", se=TRUE) plot(rr$estimate) plot(rr$SE) rrX <- relrisk(fit, at="points", relative=TRUE, control="lung")
Computes a measure of the degree of repulsion between points in a determinantal point process model.
repul(model, ...) ## S3 method for class 'dppm' repul(model, ...)
repul(model, ...) ## S3 method for class 'dppm' repul(model, ...)
model |
A fitted point process model
of determinantal type (object of class |
... |
Ignored. |
The repulsiveness index of a determinantal
point process model was defined by
Lavancier, Moller and Rubak (2015) as
where is the intensity of the model and
is the pair correlation function, and
the integral is taken over all two-dimensional vectors
.
Values of are dimensionless.
Larger values of
indicate stronger repulsion
between points.
If the model is stationary, the result is a single number.
If the model is not stationary, the result is a pixel image (obtained by multiplying the spatially-varying intensity by the integral defined above).
A numeric value or a pixel image.
Adrian Baddeley [email protected].
Lavancier, F., Moller, J. and Rubak, E. (2015), Determinantal point process models and statistical inference. Journal of Royal Statistical Society: Series B (Statistical Methodology), 77, 853–877.
jpines <- residualspaper$Fig1 fit <- dppm(jpines ~ 1, dppGauss) repul(fit)
jpines <- residualspaper$Fig1 fit <- dppm(jpines ~ 1, dppGauss) repul(fit)
Given a point pattern and an estimate of its intensity function obtained in any fashion, compute the residual measure.
residualMeasure(Q, lambda, type = c("raw", "inverse", "Pearson", "pearson"), ...)
residualMeasure(Q, lambda, type = c("raw", "inverse", "Pearson", "pearson"), ...)
Q |
A point pattern (object of class |
lambda |
Predicted intensity. An image (object of class |
type |
Character string (partially matched) specifying the type of residuals. |
... |
Arguments passed to |
This command constructs the residual measure
for the model in which Q
is the observed point pattern
or quadrature scheme, and lambda
is the estimated intensity
function obtained in some fashion.
A measure (object of class "msr"
).
Adrian Baddeley [email protected].
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
## nonparametric regression estimate of intensity ## as a function of terrain elevation f <- rhohat(bei, bei.extra$elev) ## predicted intensity as a function of location lam <- predict(f) ## residuals res <- residualMeasure(bei, lam) res plot(res)
## nonparametric regression estimate of intensity ## as a function of terrain elevation f <- rhohat(bei, bei.extra$elev) ## predicted intensity as a function of location lam <- predict(f) ## residuals res <- residualMeasure(bei, lam) res plot(res)
Given a determinantal point process model fitted to a point pattern, compute residuals.
## S3 method for class 'dppm' residuals(object, ...)
## S3 method for class 'dppm' residuals(object, ...)
object |
The fitted determinatal point process model (an object of class |
... |
Arguments passed to |
This function extracts the intensity component of the model using
as.ppm
and then applies residuals.ppm
to compute the residuals.
Use plot.msr
to plot the residuals directly.
An object of class "msr"
representing a signed measure or vector-valued measure
(see msr
). This object can be plotted.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- dppm(swedishpines ~ x, dppGauss, method="c") rr <- residuals(fit)
fit <- dppm(swedishpines ~ x, dppGauss, method="c") rr <- residuals(fit)
Given a Cox or cluster point process model fitted to a point pattern, compute residuals.
## S3 method for class 'kppm' residuals(object, ...)
## S3 method for class 'kppm' residuals(object, ...)
object |
The fitted point process model (an object of class |
... |
Arguments passed to |
This function extracts the intensity component of the model using
as.ppm
and then applies residuals.ppm
to compute the residuals.
Use plot.msr
to plot the residuals directly.
An object of class "msr"
representing a signed measure or vector-valued measure
(see msr
). This object can be plotted.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
fit <- kppm(redwood ~ x, "Thomas") rr <- residuals(fit)
fit <- kppm(redwood ~ x, "Thomas") rr <- residuals(fit)
Given a point process model fitted to multiple point patterns, compute residuals for each pattern.
## S3 method for class 'mppm' residuals(object, type = "raw", ..., fittedvalues = fitted.mppm(object))
## S3 method for class 'mppm' residuals(object, type = "raw", ..., fittedvalues = fitted.mppm(object))
object |
Fitted point process model (object of class |
... |
Ignored. |
type |
Type of residuals: either |
fittedvalues |
Advanced use only. Fitted values of the model to be used in the calculation. |
Baddeley et al (2005) defined residuals for the fit of
a point process model to spatial point pattern data.
For an explanation of these residuals, see the help file for
residuals.ppm
.
This function computes the residuals
for a point process model fitted to multiple point patterns.
The object
should be an object of class "mppm"
obtained
from mppm
.
The return value is a list.
The number of entries in the list equals the
number of point patterns in the original data. Each entry in the list
has the same format as the output of
residuals.ppm
.
That is, each entry in the list is a signed measure (object of
class "msr"
) giving the residual measure for the corresponding
point pattern.
A list of signed measures (objects of class "msr"
)
giving the residual measure for each of the original point patterns.
See Details.
Adrian Baddeley [email protected], Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders)) r <- residuals(fit) # compute total residual for each point pattern rtot <- sapply(r, integral.msr) # standardise the total residuals areas <- sapply(windows.mppm(fit), area.owin) rtot/sqrt(areas)
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders)) r <- residuals(fit) # compute total residual for each point pattern rtot <- sapply(r, integral.msr) # standardise the total residuals areas <- sapply(windows.mppm(fit), area.owin) rtot/sqrt(areas)
Given a point process model fitted to a point pattern, compute residuals.
## S3 method for class 'ppm' residuals(object, type="raw", ..., check=TRUE, drop=FALSE, fittedvalues=NULL, new.coef=NULL, dropcoef=FALSE, quad=NULL)
## S3 method for class 'ppm' residuals(object, type="raw", ..., check=TRUE, drop=FALSE, fittedvalues=NULL, new.coef=NULL, dropcoef=FALSE, quad=NULL)
object |
The fitted point process model (an object of class |
type |
String indicating the type of residuals to be calculated.
Current options are
|
... |
Ignored. |
check |
Logical value indicating whether to check the internal format
of |
drop |
Logical value determining whether to delete quadrature points
that were not used to fit the model. See |
fittedvalues |
Vector of fitted values for the conditional intensity at the quadrature points, from which the residuals will be computed. For expert use only. |
new.coef |
Optional. Numeric vector of coefficients for the model,
replacing |
dropcoef |
Internal use only. |
quad |
Optional. Data specifying how to re-fit the model.
A list of arguments passed to |
This function computes several kinds of residuals for the fit of
a point process model to a spatial point pattern dataset
(Baddeley et al, 2005).
Use plot.msr
to plot the residuals directly,
or diagnose.ppm
to produce diagnostic plots based on these residuals.
The argument object
must be a fitted point process model
(object of class "ppm"
). Such objects are produced by the maximum
pseudolikelihood fitting algorithm ppm
.
This fitted model object contains complete
information about the original data pattern.
Residuals are attached both to the data points and to some
other points in the window of observation (namely, to the dummy
points of the quadrature scheme used to fit the model).
If the fitted model is correct, then the sum of the
residuals over all (data and dummy) points in a spatial region
has mean zero. For further explanation, see Baddeley et al (2005).
The type of residual
is chosen by the argument type
. Current options are
"raw"
:the raw residuals
at the quadrature points ,
where
is the indicator equal to 1 if
is a data point and 0 if
is a dummy point;
is the quadrature weight attached to
; and
is the conditional intensity of the fitted model at .
These are the spatial analogue of the martingale residuals
of a one-dimensional counting process.
"inverse"
:the ‘inverse-lambda’ residuals (Baddeley et al, 2005)
obtained by dividing the raw residuals by
the fitted conditional intensity. These are
a counterpart of the exponential energy marks (see eem
).
"pearson"
:the Pearson residuals (Baddeley et al, 2005)
obtained by dividing the raw residuals by the
square root of the fitted conditional intensity.
The Pearson residuals are standardised, in the sense
that if the model (true and fitted) is Poisson,
then the sum of the Pearson residuals in a spatial region
has variance equal to the area of
.
"score"
:the score residuals (Baddeley et al, 2005)
obtained by multiplying the raw residuals
by the covariates
for quadrature point
.
The score residuals always sum to zero.
The result of residuals.ppm
is a measure
(object of class "msr"
).
Use plot.msr
to plot the residuals directly,
or diagnose.ppm
to produce diagnostic plots
based on these residuals.
Use integral.msr
to compute the total residual.
By default,
the window of the measure is the same as the original window
of the data. If drop=TRUE
then the window is the
domain of integration of the pseudolikelihood or composite likelihood.
This only matters when the model object
was fitted using
the border correction: in that case, if drop=TRUE
the
window of the residuals is the erosion of the original data window
by the border correction distance rbord
.
An object of class "msr"
representing a signed measure or vector-valued measure
(see msr
). This object can be plotted.
Sometimes we want to modify the calculation of residuals by using
different values for the model parameters. This capability is
provided by the arguments new.coef
and quad
.
If new.coef
is given, then the residuals will be computed
by taking the model parameters to be new.coef
.
This should be a numeric vector
of the same length as the vector of fitted model parameters
coef(object)
.
If new.coef
is missing and quad
is given,
then the model parameters will
be determined by re-fitting the model using a new
quadrature scheme specified by quad
.
Residuals will be computed for the
original model object
using these new parameter values.
The argument quad
should normally be
a list of arguments in name=value
format that will be
passed to quadscheme
(together with
the original data points) to determine the new quadrature scheme.
It may also be a quadrature scheme (object of class
"quad"
) to which the model should be fitted, or a
point pattern (object of class "ppp"
) specifying the
dummy points in a new quadrature scheme.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
msr
,
diagnose.ppm
,
ppm.object
,
ppm
fit <- ppm(cells, ~x, Strauss(r=0.15)) # Pearson residuals rp <- residuals(fit, type="pe") rp # simulated data X <- rStrauss(100,0.7,0.05) # fit Strauss model fit <- ppm(X, ~1, Strauss(0.05)) res.fit <- residuals(fit) # check that total residual is 0 integral.msr(residuals(fit, drop=TRUE)) # true model parameters truecoef <- c(log(100), log(0.7)) res.true <- residuals(fit, new.coef=truecoef)
fit <- ppm(cells, ~x, Strauss(r=0.15)) # Pearson residuals rp <- residuals(fit, type="pe") rp # simulated data X <- rStrauss(100,0.7,0.05) # fit Strauss model fit <- ppm(X, ~1, Strauss(0.05)) res.fit <- residuals(fit) # check that total residual is 0 integral.msr(residuals(fit, drop=TRUE)) # true model parameters truecoef <- c(log(100), log(0.7)) res.true <- residuals(fit, new.coef=truecoef)
Given a point process model that was fitted to a point pattern by recursive partitioning (regression tree) methods, compute the residual measure.
## S3 method for class 'rppm' residuals(object, type=c("raw", "inverse", "Pearson"), ...)
## S3 method for class 'rppm' residuals(object, type=c("raw", "inverse", "Pearson"), ...)
object |
The fitted point process model (an object of class |
type |
String (partially matched) indicating the type of residuals to be calculated. |
... |
Ignored. |
This function computes the residual measure for a point process model that was fitted to point pattern data by recursive partitioning of the covariates.
The argument object
must be a fitted model object
of class "rppm"
). Such objects are
created by the fitting algorithm rppm
.
The type of residual is chosen by the argument type
.
A measure (object of class "msr"
).
Adrian Baddeley [email protected].
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
fit <- rppm(bei ~ elev + grad, data=bei.extra) res <- residuals(fit) plot(res)
fit <- rppm(bei ~ elev + grad, data=bei.extra) res <- residuals(fit) plot(res)
Given a spatial logistic regression model fitted to a point pattern, compute the residuals for each pixel.
## S3 method for class 'slrm' residuals(object, type=c("raw", "deviance", "pearson", "working", "response", "partial", "score"), ...)
## S3 method for class 'slrm' residuals(object, type=c("raw", "deviance", "pearson", "working", "response", "partial", "score"), ...)
object |
The fitted point process model (an object of class |
type |
String (partially matched) indicating the type of residuals to be calculated. |
... |
Ignored. |
This function computes several kinds of residuals for the fit of a spatial logistic regression model to a spatial point pattern dataset.
The argument object
must be a fitted spatial logistic
regression model (object of class "slrm"
). Such objects are
created by the fitting algorithm slrm
.
The residuals are computed for each pixel that was used to fit the original model. The residuals are returned as a pixel image (if the residual values are scalar), or a list of pixel images (if the residual values are vectors).
The type of residual is chosen by the argument type
.
For a given pixel, suppose is the fitted probability of
presence of a point, and
is the presence indicator
(equal to 1 if the pixel contains any data points, and equal to 0
otherwise). Then
type="raw"
or type="response"
specifies
the response residual
type="pearson"
is the Pearson residual
type="deviance"
is the deviance residual
type="score"
specifies the score residuals
where x
is the vector of canonical covariate values
for the pixel
type="working"
specifies the working residuals
as defined in residuals.glm
type="partial"
specifies the partial residuals
as defined in residuals.glm
A pixel image (if the residual values are scalar), or a list of pixel images (if the residual values are vectors).
Adrian Baddeley [email protected]
d <- if(interactive()) 128 else 32 H <- unmark(humberside) fit <- slrm(H ~ x + y, dimyx=d) plot(residuals(fit)) plot(residuals(fit, type="score"))
d <- if(interactive()) 128 else 32 H <- unmark(humberside) fit <- slrm(H ~ x + y, dimyx=d) plot(residuals(fit)) plot(residuals(fit, type="score"))
Given a fitted model (of any kind) extract the values of the response variable. For a point process model, the observed point pattern is extracted.
response(object) ## S3 method for class 'lm' response(object) ## S3 method for class 'glm' response(object) ## S3 method for class 'ppm' response(object) ## S3 method for class 'kppm' response(object) ## S3 method for class 'dppm' response(object) ## S3 method for class 'slrm' response(object) ## S3 method for class 'rppm' response(object) ## S3 method for class 'mppm' response(object)
response(object) ## S3 method for class 'lm' response(object) ## S3 method for class 'glm' response(object) ## S3 method for class 'ppm' response(object) ## S3 method for class 'kppm' response(object) ## S3 method for class 'dppm' response(object) ## S3 method for class 'slrm' response(object) ## S3 method for class 'rppm' response(object) ## S3 method for class 'mppm' response(object)
object |
A fitted model (object of class
|
For fitted linear models of class "lm"
and fitted generalized linear models of class "glm"
,
the numerical values of the response variable are extracted
if they are available, and otherwise NULL
is returned.
For fitted point process models of class
"ppm"
,
"kppm"
,
"dppm"
,
"slrm"
, "lppm"
or "rppm"
,
the original data point pattern is extracted.
For a fitted point process model of class "mppm"
,
the list of original data point patterns is extracted.
For response.lm
and response.glm
,
a numeric vector, or NULL
.
For
response.ppm
,
response.kppm
,
response.dppm
,
response.slrm
and response.rppm
,
a two-dimensional spatial point pattern (class "ppp"
).
For response.mppm
, a list of
two-dimensional spatial point patterns (objects of class
"ppp"
).
The list also belongs to classes "solist"
and "ppplist"
.
Adrian Baddeley [email protected].
fit <- ppm(cells ~ x) response(fit)
fit <- ppm(cells ~ x) response(fit)
Performs Richardson Extrapolation on a sequence of approximate values.
rex(x, r = 2, k = 1, recursive = FALSE)
rex(x, r = 2, k = 1, recursive = FALSE)
x |
A numeric vector or matrix, whose columns are successive estimates or approximations to a vector of parameters. |
r |
A number greater than 1. The ratio of successive step sizes. See Details. |
k |
Integer. The order of convergence assumed. See Details. |
recursive |
Logical value indicating whether to perform one step of
Richardson extrapolation ( |
Richardson extrapolation is a general technique for improving numerical approximations, often used in numerical integration (Brezinski and Zaglia, 1991). It can also be used to improve parameter estimates in statistical models (Baddeley and Turner, 2014).
The successive columns of x
are assumed to
have been obtained using approximations with step sizes
where
is the initial step size (which does not need to be
specified).
Estimates based on a step size are assumed to have an error
of order
.
Thus, the default values r=2
and k=1
imply that the errors in
the second column of x
should be roughly as large
as the errors in the first column, and so on.
A matrix whose columns contain a sequence of improved estimates.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Baddeley, A. and Turner, R. (2014) Bias correction for parameter estimates of spatial point process models. Journal of Statistical Computation and Simulation 84, 1621–1643. DOI: 10.1080/00949655.2012.755976
Brezinski, C. and Zaglia, M.R. (1991) Extrapolation Methods. Theory and Practice. North-Holland.
# integrals of sin(x) and cos(x) from 0 to pi # correct answers: 2, 0 est <- function(nsteps) { xx <- seq(0, pi, length=nsteps) ans <- pi * c(mean(sin(xx)), mean(cos(xx))) names(ans) <- c("sin", "cos") ans } X <- cbind(est(10), est(20), est(40)) X rex(X) rex(X, recursive=TRUE) # fitted Gibbs point process model fit0 <- ppm(cells ~ 1, Strauss(0.07), nd=16) fit1 <- update(fit0, nd=32) fit2 <- update(fit0, nd=64) co <- cbind(coef(fit0), coef(fit1), coef(fit2)) co rex(co, k=2, recursive=TRUE)
# integrals of sin(x) and cos(x) from 0 to pi # correct answers: 2, 0 est <- function(nsteps) { xx <- seq(0, pi, length=nsteps) ans <- pi * c(mean(sin(xx)), mean(cos(xx))) names(ans) <- c("sin", "cos") ans } X <- cbind(est(10), est(20), est(40)) X rex(X) rex(X, recursive=TRUE) # fitted Gibbs point process model fit0 <- ppm(cells ~ 1, Strauss(0.07), nd=16) fit1 <- update(fit0, nd=32) fit2 <- update(fit0, nd=64) co <- cbind(coef(fit0), coef(fit1), coef(fit2)) co rex(co, k=2, recursive=TRUE)
Computes a nonparametric estimate of the intensity of a point process, as a function of a (continuous) spatial covariate.
## S3 method for class 'ppm' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "mountain", "valley", "piecewise"), subset=NULL, do.CI=TRUE, jitter=TRUE, jitterfactor=1, interpolate=TRUE, dimyx=NULL, eps=NULL, rule.eps = c("adjust.eps", "grow.frame", "shrink.frame"), n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL) ## S3 method for class 'slrm' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "mountain", "valley", "piecewise"), subset=NULL, do.CI=TRUE, jitter=TRUE, jitterfactor=1, interpolate=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL)
## S3 method for class 'ppm' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "mountain", "valley", "piecewise"), subset=NULL, do.CI=TRUE, jitter=TRUE, jitterfactor=1, interpolate=TRUE, dimyx=NULL, eps=NULL, rule.eps = c("adjust.eps", "grow.frame", "shrink.frame"), n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL) ## S3 method for class 'slrm' rhohat(object, covariate, ..., weights=NULL, method=c("ratio", "reweight", "transform"), horvitz=FALSE, smoother=c("kernel", "local", "decreasing", "increasing", "mountain", "valley", "piecewise"), subset=NULL, do.CI=TRUE, jitter=TRUE, jitterfactor=1, interpolate=TRUE, n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL, bwref=bw, covname, confidence=0.95, positiveCI, breaks=NULL)
object |
A point pattern (object of class |
covariate |
Either a |
weights |
Optional weights attached to the data points.
Either a numeric vector of weights for each data point,
or a pixel image (object of class |
method |
Character string determining the estimation method. See Details. |
horvitz |
Logical value indicating whether to use Horvitz-Thompson weights. See Details. |
smoother |
Character string determining the smoothing algorithm and the type of curve that will be estimated. See Details. |
subset |
Optional. A spatial window (object of class |
do.CI |
Logical value specifying whether to calculate standard errors and confidence bands. |
jitter |
Logical value. If |
jitterfactor |
Numeric value controlling the scale of noise added to the
covariate values at the data points when |
interpolate |
Logical value specifying whether to use spatial interpolation
to obtain the values of the covariate at the data points,
when the covariate is a pixel image
(object of class |
dimyx , eps , rule.eps
|
Arguments controlling the pixel resolution at which the covariate will be evaluated. See Details. |
bw |
Smoothing bandwidth or bandwidth rule
(passed to |
adjust |
Smoothing bandwidth adjustment factor
(passed to |
n , from , to
|
Arguments passed to |
bwref |
Optional. An alternative value of |
... |
Additional arguments passed to |
covname |
Optional. Character string to use as the name of the covariate. |
confidence |
Confidence level for confidence intervals. A number between 0 and 1. |
positiveCI |
Logical value.
If |
breaks |
Breakpoints for the piecewise-constant function
computed when |
This command estimates the relationship between point process intensity and a given spatial covariate. Such a relationship is sometimes called a resource selection function (if the points are organisms and the covariate is a descriptor of habitat) or a prospectivity index (if the points are mineral deposits and the covariate is a geological variable). This command uses nonparametric methods which do not assume a particular form for the relationship.
If object
is a point pattern, and baseline
is missing or
null, this command assumes that object
is a realisation of a
point process with intensity function
of the form
where is the spatial
covariate function given by
covariate
, and
is the resource selection function
or prospectivity index.
A nonparametric estimator of the function
is computed.
If object
is a point pattern, and baseline
is given,
then the intensity function is assumed to be
where is the baseline intensity at location
.
A nonparametric estimator of the relative intensity
is computed.
If object
is a fitted point process model, suppose X
is
the original data point pattern to which the model was fitted. Then
this command assumes X
is a realisation of a Poisson point
process with intensity function of the form
where is the intensity of the fitted model
object
. A nonparametric estimator of
the relative intensity is computed.
The nonparametric estimation procedure is controlled by the
arguments smoother
, method
and horvitz
.
The argument smoother
selects the type of estimation technique.
If smoother="kernel"
(the default),
the nonparametric estimator is a kernel smoothing estimator
of (Guan, 2008; Baddeley et al, 2012).
The estimated function
will be
a smooth function of
which takes nonnegative values.
If
do.CI=TRUE
(the default),
confidence bands are also computed, assuming a Poisson point process.
See the section on Smooth estimates.
If smoother="local"
,
the nonparametric estimator is a local regression estimator
of (Baddeley et al, 2012) obtained using
local likelihood.
The estimated function
will be
a smooth function of
.
If
do.CI=TRUE
(the default),
confidence bands are also computed, assuming a Poisson point process.
See the section on Smooth estimates.
If smoother="increasing"
, we assume that
is an increasing function of
,
and use the nonparametric maximum likelihood estimator
of
described by Sager (1982).
The estimated function will be a step function, that is increasing
as a function of
. Confidence bands are not computed.
See the section on Monotone estimates.
If smoother="decreasing"
, we assume that
is a decreasing function of
,
and use the nonparametric maximum likelihood estimator
of
described by Sager (1982).
The estimated function will be a step function, that is decreasing
as a function of
. Confidence bands are not computed.
See the section on Monotone estimates.
If smoother="mountain"
, we assume that
is a function with an inverted U shape,
with a single peak at a value
, so that
is an increasing function of
for
and a decreasing function of
for
.
We compute the nonparametric maximum likelihood estimator.
The estimated function will be a step function, which is
increasing and then decreasing as a function of
.
Confidence bands are not computed.
See the section on Unimodal estimates.
If smoother="valley"
, we assume that
is a function with a U shape,
with a single minimum at a value
, so that
is a decreasing function of
for
and an increasing function of
for
.
We compute the nonparametric maximum likelihood estimator.
The estimated function will be a step function, which is
decreasing and then increasing as a function of
.
Confidence bands are not computed.
See the section on Unimodal estimates.
If smoother="piecewise"
, the estimate of
is piecewise constant.
The range of covariate values is divided into several intervals
(ranges or bands). The endpoints of these intervals are the
breakpoints, which may be specified by the argument
breaks
;
there is a sensible default. The estimate of
takes a constant value on each interval.
The estimate of
in each interval of covariate
values is simply the average intensity
(number of points per unit area) in the relevant sub-region.
If
do.CI=TRUE
(the default),
confidence bands are computed assuming a Poisson process.
See Baddeley (2018) for a comparison of these estimation techniques
(except for "mountain"
and "valley"
).
If the argument weights
is present, then the contribution
from each data point X[i]
to the estimate of is
multiplied by
weights[i]
.
If the argument subset
is present, then the calculations are
performed using only the data inside this spatial region.
This technique assumes that covariate
has continuous values.
It is not applicable to covariates with categorical (factor) values
or discrete values such as small integers.
For a categorical covariate, use
intensity.quadratcount
applied to the result of
quadratcount(X, tess=covariate)
.
The argument covariate
should be a pixel image, or a function,
or one of the strings "x"
or "y"
signifying the
cartesian coordinates. It will be evaluated on a fine grid of locations,
with spatial resolution controlled by the arguments
dimyx,eps,rule.eps
which are passed to as.mask
.
A function value table (object of class "fv"
)
containing the estimated values of
(and confidence limits) for a sequence of values of
.
Also belongs to the class
"rhohat"
which has special methods for print
, plot
and predict
.
Smooth estimators of
were proposed by Baddeley and Turner (2005) and Baddeley et al (2012).
Similar estimators were proposed by Guan (2008) and in the literature
on relative distributions (Handcock and Morris, 1999).
The estimated function will be a smooth function
of
.
The smooth estimation procedure involves computing several density estimates
and combining them. The algorithm used to compute density estimates is
determined by smoother
:
If smoother="kernel"
,
the smoothing procedure is based on
fixed-bandwidth kernel density estimation,
performed by density.default
.
If smoother="local"
, the smoothing procedure
is based on local likelihood density estimation, performed by
locfit
.
The argument method
determines how the density estimates will be
combined to obtain an estimate of :
If method="ratio"
, then is
estimated by the ratio of two density estimates,
The numerator is a (rescaled) density estimate obtained by
smoothing the values
of the covariate
observed at the data points
. The denominator
is a density estimate of the reference distribution of
.
See Baddeley et al (2012), equation (8). This is similar but not
identical to an estimator proposed by Guan (2008).
If method="reweight"
, then is
estimated by applying density estimation to the
values
of the covariate
observed at the data points
,
with weights inversely proportional to the reference density of
.
See Baddeley et al (2012), equation (9).
If method="transform"
,
the smoothing method is variable-bandwidth kernel
smoothing, implemented by applying the Probability Integral Transform
to the covariate values, yielding values in the range 0 to 1,
then applying edge-corrected density estimation on the interval
, and back-transforming.
See Baddeley et al (2012), equation (10).
If horvitz=TRUE
, then the calculations described above
are modified by using Horvitz-Thompson weighting.
The contribution to the numerator from
each data point is weighted by the reciprocal of the
baseline value or fitted intensity value at that data point;
and a corresponding adjustment is made to the denominator.
Pointwise confidence intervals for the true value of
are also calculated for each
,
and will be plotted as grey shading.
The confidence intervals are derived using the central limit theorem,
based on variance calculations which assume a Poisson point process.
If
positiveCI=FALSE
, the lower limit of the confidence
interval may sometimes be negative, because the confidence intervals
are based on a normal approximation to the estimate of .
If
positiveCI=TRUE
, the confidence limits are always
positive, because the confidence interval is based on a normal
approximation to the estimate of .
For consistency with earlier versions, the default is
positiveCI=FALSE
for smoother="kernel"
and positiveCI=TRUE
for smoother="local"
.
The nonparametric maximum likelihood estimator
of a monotone function was described by Sager (1982).
This method assumes that
is either an increasing
function of
, or a decreasing function of
.
The estimated function will be a step function,
increasing or decreasing as a function of
.
This estimator is chosen by specifying
smoother="increasing"
or smoother="decreasing"
.
The argument method
is ignored this case.
To compute the estimate of , the algorithm first
computes several primitive step-function estimates, and then takes
the maximum of these primitive functions.
If smoother="decreasing"
, each primitive step function
takes the form when
,
and
when
, where
and
is a primitive estimate of intensity
based on the data for
. The jump location
will be the value of the covariate
at one of the
data points. The primitive estimate
is the average intensity (number of points divided by area)
for the region of space where the covariate value is less than
or equal to
.
If horvitz=TRUE
, then the calculations described above
are modified by using Horvitz-Thompson weighting.
The contribution to the numerator from
each data point is weighted by the reciprocal of the
baseline value or fitted intensity value at that data point;
and a corresponding adjustment is made to the denominator.
Confidence intervals are not available for the monotone estimators.
If smoother="valley"
then we estimate a U-shaped function.
A function is U-shaped if it is
decreasing when
and
increasing when
, where
is
called the critical value. The nonparametric maximum likelihood
estimate of such a function can be computed by profiling over
.
The algorithm considers all possible candidate values of the critical value
, and estimates the function
separately on the left and right of
using the monotone
estimators described above. These function estimates are combined into
a single function, and the Poisson point process likelihood is
computed. The optimal value of
is the one which maximises the Poisson point process likelihood.
If smoother="mountain"
then we estimate a function which has
an inverted U shape. A function is
inverted-U-shaped if it is
increasing when
and
decreasing when
. The nonparametric maximum likelihood
estimate of such a function can be computed by profiling over
using the same technique mutatis mutandis.
Confidence intervals are not available for the unimodal estimators.
By default, rhohat
adds a small amount of random noise to the
data. This is designed to suppress the effects of
discretisation in pixel images.
This strategy means that rhohat
does not produce exactly the same result when the computation is
repeated. If you need the results to be exactly reproducible, set
jitter=FALSE
.
By default, the values of the covariate at the data points
will be randomly perturbed by adding a small amount
of noise using the function jitter
. To reduce this
effect, set jitterfactor
to a number smaller than 1. To
suppress this effect entirely, set jitter=FALSE
.
Smoothing algorithm by Adrian Baddeley [email protected], Ya-Mei Chang, Yong Song, and Rolf Turner [email protected].
Nonparametric maximum likelihood algorithm by Adrian Baddeley [email protected].
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221–236.
Baddeley, A. and Turner, R. (2005) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling, Lecture Notes in Statistics number 185. Pages 23–74. Springer-Verlag, New York, 2006. ISBN: 0-387-28311-0.
Baddeley, A. (2018) A statistical commentary on mineral prospectivity analysis. Chapter 2, pages 25–65 in Handbook of Mathematical Geosciences: Fifty Years of IAMG, edited by B.S. Daya Sagar, Q. Cheng and F.P. Agterberg. Springer, Berlin.
Guan, Y. (2008) On consistent nonparametric intensity estimation for inhomogeneous spatial point processes. Journal of the American Statistical Association 103, 1238–1247.
Handcock, M.S. and Morris, M. (1999) Relative Distribution Methods in the Social Sciences. Springer, New York.
Sager, T.W. (1982) Nonparametric maximum likelihood estimation of spatial patterns. Annals of Statistics 10, 1125–1136.
rho2hat
,
methods.rhohat
,
parres
.
See ppm
for a parametric method for the same problem.
X <- rpoispp(function(x,y){exp(3+3*x)}) fit <- ppm(X ~x) rr <- rhohat(fit, "y")
X <- rpoispp(function(x,y){exp(3+3*x)}) fit <- ppm(X ~x) rr <- rhohat(fit, "y")
Given a point process model fitted to data, generate a random simulation of the model, using the Metropolis-Hastings algorithm.
## S3 method for class 'ppm' rmh(model, start=NULL, control=default.rmhcontrol(model, w=w), ..., w = NULL, project=TRUE, nsim=1, drop=TRUE, saveinfo=TRUE, verbose=TRUE, new.coef=NULL)
## S3 method for class 'ppm' rmh(model, start=NULL, control=default.rmhcontrol(model, w=w), ..., w = NULL, project=TRUE, nsim=1, drop=TRUE, saveinfo=TRUE, verbose=TRUE, new.coef=NULL)
model |
A fitted point process model (object of class
|
start |
Data determining the initial state
of the Metropolis-Hastings algorithm. See
|
control |
Data controlling the iterative behaviour of
the Metropolis-Hastings algorithm. See |
... |
Further arguments passed to |
w |
Optional. Window in which the simulations should be generated. Default is the window of the original data. |
project |
Logical flag indicating what to do if the fitted model is
invalid (in the sense that the values of the fitted coefficients do not
specify a valid point process).
If |
nsim |
Number of simulated point patterns that should be generated. |
drop |
Logical. If |
saveinfo |
Logical value indicating whether to save auxiliary information. |
verbose |
Logical flag indicating whether to print progress reports. |
new.coef |
New values for the canonical parameters of the model.
A numeric vector of the same length as |
This function generates simulated realisations from a point
process model that has been fitted to point pattern data. It is
a method for the generic function rmh
for the
class "ppm"
of fitted point process models. To simulate
other kinds of point process models, see rmh
or rmh.default
.
The argument model
describes the fitted model. It must be
an object of class "ppm"
(see ppm.object
),
and will typically be the result of a call to the point process
model fitting function ppm
.
The current implementation enables simulation from any fitted model
involving the interactions
AreaInter
,
DiggleGratton
,
DiggleGatesStibbard
,
Geyer
,
Hardcore
,
MultiStrauss
,
MultiStraussHard
,
PairPiece
,
Poisson
,
Strauss
,
StraussHard
and Softcore
,
including nonstationary models. See the examples.
It is also possible to simulate hybrids of several such models.
See Hybrid
and the examples.
It is possible that the fitted coefficients of a point process model
may be “illegal”, i.e. that there may not exist a
mathematically well-defined point process with the given parameter
values. For example, a Strauss process with interaction
parameter does not exist,
but the model-fitting procedure used in
ppm
will sometimes
produce values of greater than 1.
In such cases, if
project=FALSE
then an error will occur,
while if project=TRUE
then rmh.ppm
will find
the nearest legal model and simulate
this model instead. (The nearest legal model is obtained by
projecting the vector of coefficients onto the set of
valid coefficient vectors. The result is usually the Poisson process
with the same fitted intensity.)
The arguments start
and control
are lists of
parameters determining the initial state and the iterative
behaviour, respectively, of the Metropolis-Hastings algorithm.
The argument start
is passed directly to rmhstart
.
See rmhstart
for details of the parameters of the
initial state, and their default values.
The argument control
is first passed to
rmhcontrol
. Then if any additional arguments ...
are given, update.rmhcontrol
is called to update the
parameter values. See rmhcontrol
for details of
the iterative behaviour parameters, and default.rmhcontrol
for their default values.
Note that if you specify expansion of the simulation window
using the parameter expand
(so that the
model will be simulated on a window larger than the original data
window) then the model must be capable of extrapolation to this
larger window. This is usually not possible for models which
depend on external covariates, because the domain of a covariate image
is usually the same as the domain of the fitted model.
After extracting the relevant information from the fitted model
object model
, rmh.ppm
invokes the default
rmh
algorithm rmh.default
, unless the model
is Poisson. If the model is Poisson then the Metropolis-Hastings
algorithm is not needed, and the model is simulated directly, using
one of rpoispp
, rmpoispp
,
rpoint
or rmpoint
.
See rmh.default
for further information about the
implementation, or about the Metropolis-Hastings algorithm.
A point pattern (an object of class "ppp"
; see
ppp.object
) or a list of point patterns.
See Warnings in rmh.default
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
simulate.ppm
,
rmh
,
rmhmodel
,
rmhcontrol
,
default.rmhcontrol
,
update.rmhcontrol
,
rmhstart
,
rmh.default
,
ppp.object
,
ppm
,
Interactions:
AreaInter
,
DiggleGratton
,
DiggleGatesStibbard
,
Geyer
,
Hardcore
,
Hybrid
,
MultiStrauss
,
MultiStraussHard
,
PairPiece
,
Poisson
,
Strauss
,
StraussHard
,
Softcore
live <- interactive() op <- spatstat.options() spatstat.options(rmh.nrep=1e5) Nrep <- 1e5 X <- swedishpines if(live) plot(X, main="Swedish Pines data") # Poisson process fit <- ppm(X, ~1, Poisson()) Xsim <- rmh(fit) if(live) plot(Xsim, main="simulation from fitted Poisson model") # Strauss process fit <- ppm(X, ~1, Strauss(r=7)) Xsim <- rmh(fit) if(live) plot(Xsim, main="simulation from fitted Strauss model") if(live) { # Strauss process simulated on a larger window # then clipped to original window Xsim <- rmh(fit, control=list(nrep=Nrep, expand=1.1, periodic=TRUE)) Xsim <- rmh(fit, nrep=Nrep, expand=2, periodic=TRUE) } if(live) { X <- rSSI(0.05, 100) # piecewise-constant pairwise interaction function fit <- ppm(X, ~1, PairPiece(seq(0.02, 0.1, by=0.01))) Xsim <- rmh(fit) } # marked point pattern Y <- amacrine if(live) { # marked Poisson models fit <- ppm(Y) fit <- ppm(Y,~marks) fit <- ppm(Y,~polynom(x,2)) fit <- ppm(Y,~marks+polynom(x,2)) fit <- ppm(Y,~marks*polynom(x,y,2)) Ysim <- rmh(fit) } # multitype Strauss models MS <- MultiStrauss(radii=matrix(0.07, ncol=2, nrow=2), types = levels(Y$marks)) if(live) { fit <- ppm(Y ~marks, MS) Ysim <- rmh(fit) } fit <- ppm(Y ~ marks*polynom(x,y,2), MS) Ysim <- rmh(fit) if(live) plot(Ysim, main="simulation from fitted inhomogeneous Multitype Strauss") spatstat.options(op) if(live) { # Hybrid model fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2))) Y <- rmh(fit) }
live <- interactive() op <- spatstat.options() spatstat.options(rmh.nrep=1e5) Nrep <- 1e5 X <- swedishpines if(live) plot(X, main="Swedish Pines data") # Poisson process fit <- ppm(X, ~1, Poisson()) Xsim <- rmh(fit) if(live) plot(Xsim, main="simulation from fitted Poisson model") # Strauss process fit <- ppm(X, ~1, Strauss(r=7)) Xsim <- rmh(fit) if(live) plot(Xsim, main="simulation from fitted Strauss model") if(live) { # Strauss process simulated on a larger window # then clipped to original window Xsim <- rmh(fit, control=list(nrep=Nrep, expand=1.1, periodic=TRUE)) Xsim <- rmh(fit, nrep=Nrep, expand=2, periodic=TRUE) } if(live) { X <- rSSI(0.05, 100) # piecewise-constant pairwise interaction function fit <- ppm(X, ~1, PairPiece(seq(0.02, 0.1, by=0.01))) Xsim <- rmh(fit) } # marked point pattern Y <- amacrine if(live) { # marked Poisson models fit <- ppm(Y) fit <- ppm(Y,~marks) fit <- ppm(Y,~polynom(x,2)) fit <- ppm(Y,~marks+polynom(x,2)) fit <- ppm(Y,~marks*polynom(x,y,2)) Ysim <- rmh(fit) } # multitype Strauss models MS <- MultiStrauss(radii=matrix(0.07, ncol=2, nrow=2), types = levels(Y$marks)) if(live) { fit <- ppm(Y ~marks, MS) Ysim <- rmh(fit) } fit <- ppm(Y ~ marks*polynom(x,y,2), MS) Ysim <- rmh(fit) if(live) plot(Ysim, main="simulation from fitted inhomogeneous Multitype Strauss") spatstat.options(op) if(live) { # Hybrid model fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2))) Y <- rmh(fit) }
Converts a fitted point process model into a format that can be used to simulate the model by the Metropolis-Hastings algorithm.
## S3 method for class 'ppm' rmhmodel(model, w, ..., verbose=TRUE, project=TRUE, control=rmhcontrol(), new.coef=NULL)
## S3 method for class 'ppm' rmhmodel(model, w, ..., verbose=TRUE, project=TRUE, control=rmhcontrol(), new.coef=NULL)
model |
Fitted point process model (object of class |
w |
Optional. Window in which the simulations should be generated. |
... |
Ignored. |
verbose |
Logical flag indicating whether to print progress reports while the model is being converted. |
project |
Logical flag indicating what to do if the fitted model does not correspond to a valid point process. See Details. |
control |
Parameters determining the iterative behaviour of the simulation
algorithm. Passed to |
new.coef |
New values for the canonical parameters of the model.
A numeric vector of the same length as |
The generic function rmhmodel
takes a
description of a point process model in some format, and
converts it into an object of class "rmhmodel"
so that simulations of the model can be generated using
the Metropolis-Hastings algorithm rmh
.
This function rmhmodel.ppm
is the method for
the class "ppm"
of fitted point process models.
The argument model
should be a fitted point process model
(object of class "ppm"
) typically obtained from the
model-fitting function ppm
.
This will be converted into an object of class "rmhmodel"
.
The optional argument w
specifies the window
in which the pattern is to be generated. If specified, it must be in
a form which can be coerced to an object of class owin
by as.owin
.
Not all fitted point process models
obtained from ppm
can be simulated.
We have not yet implemented simulation code for
the LennardJones
and OrdThresh
models.
It is also possible that a fitted point process model
obtained from ppm
may not correspond to a valid
point process. For example a fitted model with the Strauss
interpoint interaction may have any value of the interaction parameter
; however the Strauss
process is not well-defined for
(Kelly and Ripley, 1976).
The argument project
determines what to do in such cases.
If project=FALSE
, a fatal error will occur.
If project=TRUE
, the fitted model parameters will be
adjusted to the nearest values which do correspond to a valid
point process. For example a Strauss process with will be projected to a Strauss process with
, equivalent to a Poisson process.
An object of class "rmhmodel"
, which is essentially
a list of parameter values for the model.
There is a print
method for this class, which prints
a sensible description of the model chosen.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79–140.
Kelly, F.P. and Ripley, B.D. (1976) On Strauss's model for clustering. Biometrika 63, 357–360.
rmhmodel
,
rmhmodel.list
,
rmhmodel.default
,
rmh
,
rmhcontrol
,
rmhstart
,
ppm
,
AreaInter
, BadGey
, DiggleGatesStibbard
, DiggleGratton
, Fiksel
, Geyer
, Hardcore
, Hybrid
, LennardJones
, MultiStrauss
, MultiStraussHard
, PairPiece
, Penttinen
, Poisson
, Softcore
, Strauss
, StraussHard
and Triplets
.
fit1 <- ppm(cells ~1, Strauss(0.07)) mod1 <- rmhmodel(fit1) fit2 <- ppm(cells ~x, Geyer(0.07, 2)) mod2 <- rmhmodel(fit2) fit3 <- ppm(cells ~x, Hardcore(0.07)) mod3 <- rmhmodel(fit3) # Then rmh(mod1), etc
fit1 <- ppm(cells ~1, Strauss(0.07)) mod1 <- rmhmodel(fit1) fit2 <- ppm(cells ~x, Geyer(0.07, 2)) mod2 <- rmhmodel(fit2) fit3 <- ppm(cells ~x, Hardcore(0.07)) mod3 <- rmhmodel(fit3) # Then rmh(mod1), etc
Computes the Receiver Operating Characteristic curve for a point pattern or a fitted point process model.
## S3 method for class 'ppm' roc(X, ...) ## S3 method for class 'kppm' roc(X, ...) ## S3 method for class 'slrm' roc(X, ...)
## S3 method for class 'ppm' roc(X, ...) ## S3 method for class 'kppm' roc(X, ...) ## S3 method for class 'slrm' roc(X, ...)
X |
Point pattern (object of class |
... |
Arguments passed to |
This command computes Receiver Operating
Characteristic curve. The area under the ROC is computed by auc
.
For a point pattern X
and a covariate Z
, the
ROC is a plot showing the ability of the
covariate to separate the spatial domain
into areas of high and low density of points.
For each possible threshold , the algorithm calculates
the fraction
of area in the study region where the
covariate takes a value greater than
, and the
fraction
of data points for which the covariate value
is greater than
. The ROC is a plot of
against
for all thresholds
.
For a fitted point process model, the ROC shows the ability of the fitted model intensity to separate the spatial domain into areas of high and low density of points. The ROC is not a diagnostic for the goodness-of-fit of the model (Lobo et al, 2007).
(For spatial logistic regression models (class "slrm"
)
replace “intensity” by “probability of presence”
in the text above.)
Function value table (object of class "fv"
)
which can be plotted to show the ROC curve.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Lobo, J.M., Jimenez-Valverde, A. and Real, R. (2007) AUC: a misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography 17(2) 145–151.
Nam, B.-H. and D'Agostino, R. (2002) Discrimination index, the area under the ROC curve. Pages 267–279 in Huber-Carol, C., Balakrishnan, N., Nikulin, M.S. and Mesbah, M., Goodness-of-fit tests and model validity, Birkhauser, Basel.
fit <- ppm(swedishpines ~ x+y) plot(roc(fit))
fit <- ppm(swedishpines ~ x+y) plot(roc(fit))
Fits a recursive partition model to point pattern data.
rppm(..., rpargs=list())
rppm(..., rpargs=list())
... |
Arguments passed to |
rpargs |
Optional list of arguments passed to |
This function attempts to find a simple rule for predicting low and high intensity regions of points in a point pattern, using explanatory covariates.
The arguments ...
specify the point pattern data
and explanatory covariates in the same way as they would be
in the function ppm
.
The recursive partitioning algorithm rpart
is then used to find a partitioning rule.
An object of class "rppm"
. There are methods
for print
, plot
, fitted
, predict
and
prune
for this class.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984) Classification and Regression Trees. Wadsworth.
plot.rppm
,
predict.rppm
,
update.rppm
,
prune.rppm
.
# New Zealand trees data: trees planted along border # Use covariates 'x', 'y' nzfit <- rppm(nztrees ~ x + y) nzfit prune(nzfit, cp=0.035) # Murchison gold data: numeric and logical covariates mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) # mfit <- rppm(gold ~ dfault + greenstone, data=mur) mfit # Gorillas data: factor covariates # (symbol '.' indicates 'all variables') gfit <- rppm(unmark(gorillas) ~ . , data=gorillas.extra) gfit
# New Zealand trees data: trees planted along border # Use covariates 'x', 'y' nzfit <- rppm(nztrees ~ x + y) nzfit prune(nzfit, cp=0.035) # Murchison gold data: numeric and logical covariates mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) # mfit <- rppm(gold ~ dfault + greenstone, data=mur) mfit # Gorillas data: factor covariates # (symbol '.' indicates 'all variables') gfit <- rppm(unmark(gorillas) ~ . , data=gorillas.extra) gfit
Creates an instance of a saturated pairwise interaction point process model with piecewise constant potential function. The model can then be fitted to point pattern data.
SatPiece(r, sat)
SatPiece(r, sat)
r |
vector of jump points for the potential function |
sat |
vector of saturation values, or a single saturation value |
This is a generalisation of the Geyer saturation point process model,
described in Geyer
, to the case of multiple interaction
distances. It can also be described as the saturated analogue of a
pairwise interaction process with piecewise-constant pair potential,
described in PairPiece
.
The saturated point process with interaction radii
,
saturation thresholds
,
intensity parameter
and
interaction parameters
,
is the point process
in which each point
in the pattern
contributes a factor
to the probability density of the point pattern, where
where denotes the
number of points in the pattern
which lie
at a distance between
and
from the point
. We take
so that
is the number of points of
that lie within a distance
of the point
.
SatPiece
is used to fit this model to data.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the piecewise constant Saturated pairwise
interaction is yielded by the function SatPiece()
.
See the examples below.
Simulation of this point process model is not yet implemented. This model is not locally stable (the conditional intensity is unbounded).
The argument r
specifies the vector of interaction distances.
The entries of r
must be strictly increasing, positive numbers.
The argument sat
specifies the vector of saturation parameters.
It should be a vector of the same length as r
, and its entries
should be nonnegative numbers. Thus sat[1]
corresponds to the
distance range from 0
to r[1]
, and sat[2]
to the
distance range from r[1]
to r[2]
, etc.
Alternatively sat
may be a single number, and this saturation
value will be applied to every distance range.
Infinite values of the
saturation parameters are also permitted; in this case
and there is effectively no ‘saturation’ for the distance range in
question. If all the saturation parameters are set to
Inf
then
the model is effectively a pairwise interaction process, equivalent to
PairPiece
(however the interaction parameters
obtained from
SatPiece
are the
square roots of the parameters
obtained from
PairPiece
).
If r
is a single number, this model is virtually equivalent to the
Geyer process, see Geyer
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected] and Rolf Turner [email protected] in collaboration with Hao Wang and Jeff Picka
ppm
,
pairsat.family
,
Geyer
,
PairPiece
,
BadGey
.
SatPiece(c(0.1,0.2), c(1,1)) # prints a sensible description of itself SatPiece(c(0.1,0.2), 1) ppm(cells ~1, SatPiece(c(0.07, 0.1, 0.13), 2)) # fit a stationary piecewise constant Saturated pairwise interaction process ppm(cells ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2)) # nonstationary process with log-cubic polynomial trend
SatPiece(c(0.1,0.2), c(1,1)) # prints a sensible description of itself SatPiece(c(0.1,0.2), 1) ppm(cells ~1, SatPiece(c(0.07, 0.1, 0.13), 2)) # fit a stationary piecewise constant Saturated pairwise interaction process ppm(cells ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2)) # nonstationary process with log-cubic polynomial trend
Experimental.
Saturated(pot, name)
Saturated(pot, name)
pot |
An S language function giving the user-supplied pairwise interaction potential. |
name |
Character string. |
This is experimental. It constructs a member of the
“saturated pairwise” family pairsat.family
.
An object of class "interact"
describing the interpoint interaction
structure of a point process.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
ppm
,
pairsat.family
,
Geyer
,
SatPiece
,
ppm.object
Generates simulated realisations from a determinantal point process model.
## S3 method for class 'dppm' simulate(object, nsim = 1, seed = NULL, ..., W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object)) ## S3 method for class 'detpointprocfamily' simulate(object, nsim = 1, seed = NULL, ..., W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object))
## S3 method for class 'dppm' simulate(object, nsim = 1, seed = NULL, ..., W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object)) ## S3 method for class 'detpointprocfamily' simulate(object, nsim = 1, seed = NULL, ..., W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object))
object |
Determinantal point process model. An object of class
|
nsim |
Number of simulated realisations. |
seed |
an object specifying whether and how to initialise the random
number generator. Either |
... |
Arguments passed on to |
W |
Object specifying the window of simulation (defaults to a unit
box if nothing else is sensible – see Details). Can be any single
argument acceptable to |
trunc |
Numeric value specifying how the model truncation is preformed. See Details. |
correction |
Character string specifying the type of correction to use.
The options are |
rbord |
Numeric value specifying the extent of the border correction if this correction is used. See Details. |
These functions are methods for the generic function
simulate
for the classes "detpointprocfamily"
and
"dppm"
of determinantal point process models.
The return value is a list of nsim
point patterns.
It also carries an attribute "seed"
that
captures the initial state of the random number generator.
This follows the convention used in
simulate.lm
(see simulate
).
It can be used to force a sequence of simulations to be
repeated exactly, as shown in the examples for
simulate
.
The exact simulation of a determinantal point process model involves
an infinite series, which typically has no analytical solution. In the
implementation a truncation is performed. The truncation
trunc
can be specified either directly as a positive integer
or as a fraction between 0 and 1. In the latter case the truncation is chosen
such that the expected number of points in a simulation is
trunc
times the theoretical expected number of points in the
model. The default is 0.99.
The window of the returned point pattern(s) can be specified via the
argument W
. For a fitted model (of class "dppm"
) it
defaults to the observation window of the data used to fit the
model. For inhomogeneous models it defaults to the window of the
intensity image. Otherwise it defaults to a unit box. For
non-rectangular windows simulation is done in the containing rectangle
and then restricted to the window. For inhomogeneous models a
stationary model is first simulated using the maximum intensity and
then the result is obtained by thinning.
The default is to use periodic edge correction for simulation such
that opposite edges are glued together. If border correction is used
then the simulation is done in an extended window. Edge effects are
theoretically completely removed by doubling the size of the window in
each spatial dimension, but for practical purposes much less extension
may be sufficient. The numeric rbord
determines the extent of
the extra space added to the window.
A list of length nsim
containing simulated point patterns.
If the patterns are two-dimensional, then they are
objects of class "ppp"
, and the list has class "solist"
.
Otherwise, the patterns are objects of class "ppx"
and the
list has class "anylist"
.
The return value also carries an attribute "seed"
that
captures the initial state of the random number generator.
See Details.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Lavancier, F. Moller, J. and Rubak, E. (2015) Determinantal point process models and statistical inference Journal of the Royal Statistical Society, Series B 77, 853–977.
if(interactive()) { nsim <- 2 lam <- 100 } else { nsim <- 1 lam <- 30 } model <- dppGauss(lambda=lam, alpha=.05, d=2) simulate(model, nsim)
if(interactive()) { nsim <- 2 lam <- 100 } else { nsim <- 1 lam <- 30 } model <- dppGauss(lambda=lam, alpha=.05, d=2) simulate(model, nsim)
Generates simulated realisations from a fitted cluster point process model.
## S3 method for class 'kppm' simulate(object, nsim = 1, seed=NULL, ..., window=NULL, covariates=NULL, n.cond = NULL, w.cond = NULL, verbose=TRUE, retry=10, drop=FALSE)
## S3 method for class 'kppm' simulate(object, nsim = 1, seed=NULL, ..., window=NULL, covariates=NULL, n.cond = NULL, w.cond = NULL, verbose=TRUE, retry=10, drop=FALSE)
object |
Fitted cluster point process model. An object of class |
nsim |
Number of simulated realisations. |
seed |
an object specifying whether and how to initialise
the random number generator. Either |
... |
Additional arguments passed to the relevant random generator. See Details. |
window |
Optional. Window (object of class |
covariates |
Optional. A named list containing new values for the covariates in the model. |
n.cond |
Optional. Integer specifying a fixed number of points. See the section on Conditional Simulation. |
w.cond |
Optional. Conditioning region. A window (object of class |
verbose |
Logical. Whether to print progress reports (when |
retry |
Number of times to repeat the simulation if it fails (e.g. because of insufficient memory). |
drop |
Logical. If |
This function is a method for the generic function
simulate
for the class "kppm"
of fitted
cluster point process models.
Simulations are performed by
rThomas
,
rMatClust
,
rCauchy
,
rVarGamma
or rLGCP
depending on the model.
Additional arguments ...
are passed to the relevant function
performing the simulation.
For example the argument saveLambda
is recognised by all of the
simulation functions.
The return value is a list of point patterns.
It also carries an attribute "seed"
that
captures the initial state of the random number generator.
This follows the convention used in
simulate.lm
(see simulate
).
It can be used to force a sequence of simulations to be
repeated exactly, as shown in the examples for simulate
.
A list of length nsim
containing simulated point patterns
(objects of class "ppp"
). (For conditional simulation,
the length of the result may be shorter than nsim
).
The return value also carries an attribute "seed"
that
captures the initial state of the random number generator.
See Details.
If n.cond
is specified, it should be a single integer.
Simulation will be conditional on the event
that the pattern contains exactly n.cond
points
(or contains exactly n.cond
points inside
the region w.cond
if it is given).
Conditional simulation uses the rejection algorithm described
in Section 6.2 of Moller, Syversveen and Waagepetersen (1998).
There is a maximum number of proposals which will be attempted.
Consequently the return value may contain fewer
than nsim
point patterns.
The simulation algorithm for log-Gaussian Cox processes
has been completely re-written
in spatstat.random version 3.2-0
to avoid depending on
the package RandomFields which is now defunct (and is sadly missed).
It is no longer possible to replicate results
of simulate.kppm
for log-Gaussian Cox processes
that were obtained using previous versions of spatstat.random.
The current code for simulating log-Gaussian Cox processes is a new implementation and should be considered vulnerable to new bugs.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Moller, J., Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
kppm
,
rThomas
,
rMatClust
,
rCauchy
,
rVarGamma
,
rLGCP
,
simulate.ppm
,
simulate
if(offline <- !interactive()) { spatstat.options(npixel=32, ndummy.min=16) } fit <- kppm(redwood ~x, "Thomas") simulate(fit, 2) simulate(fit, n.cond=60) if(offline) reset.spatstat.options()
if(offline <- !interactive()) { spatstat.options(npixel=32, ndummy.min=16) } fit <- kppm(redwood ~x, "Thomas") simulate(fit, 2) simulate(fit, n.cond=60) if(offline) reset.spatstat.options()
Generates simulated realisations from a point process model that was fitted to several point patterns.
## S3 method for class 'mppm' simulate(object, nsim=1, ..., verbose=TRUE)
## S3 method for class 'mppm' simulate(object, nsim=1, ..., verbose=TRUE)
object |
Point process model fitted to several point patterns.
An object of class |
nsim |
Number of simulated realisations (of each original pattern). |
... |
Further arguments passed to |
verbose |
Logical value indicating whether to print progress reports. |
This function is a method for the generic function
simulate
for the class "mppm"
of fitted
point process models for replicated point pattern data.
The result is a hyperframe with n
rows and nsim
columns,
where n
is the number of original point pattern
datasets to which the model was fitted. Each column of the hyperframe
contains a simulated version of the original data.
For each of the original point pattern datasets, the
fitted model for this dataset is extracted using
subfits
, then nsim
simulated realisations
of this model are generated using simulate.ppm
,
and these are stored in the corresponding row of the output.
A hyperframe.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
H <- hyperframe(Bugs=waterstriders) fit <- mppm(Bugs ~ id, H) y <- simulate(fit, nsim=2) y plot(y[1,,drop=TRUE], main="Simulations for Waterstriders pattern 1") plot(y[,1,drop=TRUE], main="Simulation 1 for each Waterstriders pattern")
H <- hyperframe(Bugs=waterstriders) fit <- mppm(Bugs ~ id, H) y <- simulate(fit, nsim=2) y plot(y[1,,drop=TRUE], main="Simulations for Waterstriders pattern 1") plot(y[,1,drop=TRUE], main="Simulation 1 for each Waterstriders pattern")
Generates simulated realisations from a fitted Gibbs or Poisson point process model.
## S3 method for class 'ppm' simulate(object, nsim=1, ..., singlerun = FALSE, start = NULL, control = default.rmhcontrol(object, w=w), w = window, window = NULL, project=TRUE, new.coef=NULL, verbose=FALSE, progress=(nsim > 1), drop=FALSE)
## S3 method for class 'ppm' simulate(object, nsim=1, ..., singlerun = FALSE, start = NULL, control = default.rmhcontrol(object, w=w), w = window, window = NULL, project=TRUE, new.coef=NULL, verbose=FALSE, progress=(nsim > 1), drop=FALSE)
object |
Fitted point process model.
An object of class |
nsim |
Number of simulated realisations. |
singlerun |
Logical. Whether to generate the simulated realisations
from a single long run of the Metropolis-Hastings algorithm
( |
start |
Data determining the initial state
of the Metropolis-Hastings algorithm. See
|
control |
Data controlling the running of
the Metropolis-Hastings algorithm. See |
w , window
|
Optional. The window in which the model is defined.
An object of class |
... |
Further arguments passed to |
project |
Logical flag indicating what to do if the fitted model is
invalid (in the sense that the values of the fitted coefficients do not
specify a valid point process).
If |
verbose |
Logical flag indicating whether to print progress reports
from |
progress |
Logical flag indicating whether to print progress reports for the sequence of simulations. |
new.coef |
New values for the canonical parameters of the model.
A numeric vector of the same length as |
drop |
Logical. If |
This function is a method for the generic function
simulate
for the class "ppm"
of fitted
point process models.
Simulations are performed by rmh.ppm
.
If singlerun=FALSE
(the default), the simulated patterns are
the results of independent runs of the Metropolis-Hastings
algorithm. If singlerun=TRUE
, a single long run of the
algorithm is performed, and the state of the simulation is saved
every nsave
iterations to yield the simulated patterns.
In the case of a single run, the behaviour is controlled
by the parameters nsave,nburn,nrep
. These
are described in rmhcontrol
. They may be passed
in the ...
arguments or included in control
.
It is sufficient to specify two
of the three parameters nsave,nburn,nrep
.
A list of length nsim
containing simulated point patterns
(objects of class "ppp"
).
It also belongs to the class "solist"
, so that it can be
plotted, and the class "timed"
, so that the total computation
time is recorded.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
fit <- ppm(japanesepines, ~1, Strauss(0.1)) simulate(fit, 2) simulate(fit, 2, singlerun=TRUE, nsave=1e4, nburn=1e4)
fit <- ppm(japanesepines, ~1, Strauss(0.1)) simulate(fit, 2) simulate(fit, 2, singlerun=TRUE, nsave=1e4, nburn=1e4)
Generates simulated realisations from a fitted spatial logistic regresson model
## S3 method for class 'slrm' simulate(object, nsim = 1, seed=NULL, ..., window=NULL, covariates=NULL, verbose=TRUE, drop=FALSE)
## S3 method for class 'slrm' simulate(object, nsim = 1, seed=NULL, ..., window=NULL, covariates=NULL, verbose=TRUE, drop=FALSE)
object |
Fitted spatial logistic regression model. An object of class |
nsim |
Number of simulated realisations. |
seed |
an object specifying whether and how to initialise
the random number generator. Either |
... |
Ignored. |
window |
Optional. Window (object of class |
covariates |
Optional. A named list containing new values for the covariates in the model. |
verbose |
Logical. Whether to print progress reports (when |
drop |
Logical. If |
This function is a method for the generic function
simulate
for the class "slrm"
of fitted
spatial logistic regression models.
Simulations are performed by rpoispp
after the intensity has been computed by predict.slrm
.
The return value is a list of point patterns.
It also carries an attribute "seed"
that
captures the initial state of the random number generator.
This follows the convention used in
simulate.lm
(see simulate
).
It can be used to force a sequence of simulations to be
repeated exactly, as shown in the examples for simulate
.
A list of length nsim
containing simulated point patterns
(objects of class "ppp"
).
The return value also carries an attribute "seed"
that
captures the initial state of the random number generator.
See Details.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
slrm
,
rpoispp
,
simulate.ppm
,
simulate.kppm
,
simulate
X <- copper$SouthPoints fit <- slrm(X ~ 1) simulate(fit, 2) fitxy <- slrm(X ~ x+y) simulate(fitxy, 2, window=square(2))
X <- copper$SouthPoints fit <- slrm(X ~ 1) simulate(fit, 2) fitxy <- slrm(X ~ x+y) simulate(fitxy, 2, window=square(2))
Fits a spatial logistic regression model to a spatial point pattern.
slrm(formula, ..., data = NULL, offset = TRUE, link = "logit", dataAtPoints=NULL, splitby=NULL)
slrm(formula, ..., data = NULL, offset = TRUE, link = "logit", dataAtPoints=NULL, splitby=NULL)
formula |
The model formula. See Details. |
... |
Optional arguments passed to |
data |
Optional. A list containing data required in the formula. The names of entries in the list should correspond to variable names in the formula. The entries should be point patterns, pixel images or windows. |
offset |
Logical flag indicating whether the model formula should be augmented by an offset equal to the logarithm of the pixel area. |
link |
The link function for the regression model. A character string, specifying a link function for binary regression. |
dataAtPoints |
Optional.
Exact values of the covariates at the data points.
A data frame, with column names corresponding to
variables in the |
splitby |
Optional. Character string identifying a window. The window will be used to split pixels into sub-pixels. |
This function fits a Spatial Logistic Regression model (Tukey, 1972; Agterberg, 1974) to a spatial point pattern dataset. The logistic function may be replaced by another link function.
The formula
specifies the form of the model to be fitted,
and the data to which it should be fitted. The formula
must be an R formula with a left and right hand
side.
The left hand side of the formula
is the name of the
point pattern dataset, an object of class "ppp"
.
The right hand side of the formula
is an expression,
in the usual R formula syntax, representing the functional form of
the linear predictor for the model.
Each variable name that appears in the formula may be
one of the reserved names x
and y
,
referring to the Cartesian coordinates;
the name of an entry in the list data
, if this argument is given;
the name of an object in the
parent environment, that is, in the environment where the call
to slrm
was issued.
Each object appearing on the right hand side of the formula may be
a pixel image (object of class "im"
)
containing the values of a covariate;
a window (object of class "owin"
), which will be
interpreted as a logical covariate which is TRUE
inside the
window and FALSE
outside it;
a function
in the R language, with arguments
x,y
, which can be evaluated at any location to
obtain the values of a covariate.
See the Examples below.
The fitting algorithm discretises the point pattern onto a pixel grid. The value in each pixel is 1 if there are any points of the point pattern in the pixel, and 0 if there are no points in the pixel. The dimensions of the pixel grid will be determined as follows:
The pixel grid will be determined by the extra
arguments ...
if they are specified (for example the argument
dimyx
can be used to specify the number of pixels).
Otherwise, if the right hand side of the formula
includes
the names of any pixel images containing covariate values,
these images will determine the pixel grid for the discretisation.
The covariate image with the finest grid (the smallest pixels) will
be used.
Otherwise, the default pixel grid size is given by
spatstat.options("npixel")
.
The covariates are evaluated at the centre of each pixel.
If dataAtPoints
is given, then
the covariate values at the corresponding pixels
are overwritten by the entries of dataAtPoints
(and the spatial coordinates are overwritten by the exact spatial
coordinates of the data points).
If link="logit"
(the default), the algorithm fits a Spatial Logistic
Regression model. This model states that the probability
that a given pixel contains a data point, is related to the
covariates through
where is the linear predictor of the model
(a linear combination of the covariates,
whose form is specified by the
formula
).
If link="cloglog"
then the algorithm fits a model stating that
.
If offset=TRUE
(the default), the model formula will be
augmented by adding an offset term equal to the logarithm of the pixel
area. This ensures that the fitted parameters are
approximately independent of pixel size.
If offset=FALSE
, the offset is not included, and the
traditional form of Spatial Logistic Regression is fitted.
An object of class "slrm"
representing the fitted model.
There are many methods for this class, including methods for
print
, fitted
, predict
,
anova
, coef
, logLik
, terms
,
update
, formula
and vcov
.
Automated stepwise model selection is possible using
step
. Confidence intervals for the parameters can be
computed using confint
.
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Agterberg, F.P. (1974) Automatic contouring of geological maps to detect target areas for mineral exploration. Journal of the International Association for Mathematical Geology 6, 373–395.
Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K.,
Schuhmacher, D., Shah, R. and Turner, R. (2010)
Spatial logistic regression and change-of-support
for spatial Poisson point processes.
Electronic Journal of Statistics
4, 1151–1201.
DOI: 10.1214/10-EJS581
Tukey, J.W. (1972) Discussion of paper by F.P. Agterberg and S.C. Robinson. Bulletin of the International Statistical Institute 44 (1) p. 596. Proceedings, 38th Congress, International Statistical Institute.
anova.slrm
,
coef.slrm
,
fitted.slrm
,
logLik.slrm
,
plot.slrm
,
predict.slrm
,
vcov.slrm
if(offline <- !interactive()) op <- spatstat.options(npixel=32) X <- copper$SouthPoints slrm(X ~ 1) slrm(X ~ x+y) slrm(X ~ x+y, link="cloglog") # specify a grid of 2-km-square pixels slrm(X ~ 1, eps=2) Y <- copper$SouthLines Z <- distmap(Y) slrm(X ~ Z) slrm(X ~ Z, dataAtPoints=list(Z=nncross(X,Y,what="dist"))) mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) slrm(gold ~ dfault, data=mur) slrm(gold ~ dfault + greenstone, data=mur) slrm(gold ~ dfault, data=mur, splitby="greenstone") if(offline) spatstat.options(op)
if(offline <- !interactive()) op <- spatstat.options(npixel=32) X <- copper$SouthPoints slrm(X ~ 1) slrm(X ~ x+y) slrm(X ~ x+y, link="cloglog") # specify a grid of 2-km-square pixels slrm(X ~ 1, eps=2) Y <- copper$SouthLines Z <- distmap(Y) slrm(X ~ Z) slrm(X ~ Z, dataAtPoints=list(Z=nncross(X,Y,what="dist"))) mur <- solapply(murchison, rescale, s=1000, unitname="km") mur$dfault <- distfun(mur$faults) slrm(gold ~ dfault, data=mur) slrm(gold ~ dfault + greenstone, data=mur) slrm(gold ~ dfault, data=mur, splitby="greenstone") if(offline) spatstat.options(op)
Apply kernel smoothing to a signed measure or vector-valued measure.
## S3 method for class 'msr' Smooth(X, ..., drop=TRUE)
## S3 method for class 'msr' Smooth(X, ..., drop=TRUE)
X |
Object of class |
... |
Arguments passed to |
drop |
Logical. If |
This function applies kernel smoothing to a signed measure or
vector-valued measure X
. The Gaussian kernel is used.
The object X
would typically have been created by
residuals.ppm
or msr
.
A pixel image or a list of pixel images.
For scalar-valued measures, a pixel image (object of class
"im"
) provided drop=TRUE
.
For vector-valued measures (or if drop=FALSE
),
a list of pixel images; the list also
belongs to the class "solist"
so that it can be printed and plotted.
Adrian Baddeley [email protected]
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Moller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes. Annals of the Institute of Statistical Mathematics 60, 627–649.
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") plot(Smooth(rp)) plot(Smooth(rs))
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") rs <- residuals(fit, type="score") plot(Smooth(rp)) plot(Smooth(rs))
Creates an instance of the Soft Core point process model which can then be fitted to point pattern data.
Softcore(kappa, sigma0=NA)
Softcore(kappa, sigma0=NA)
kappa |
The exponent |
sigma0 |
Optional. Initial estimate of the parameter |
The (stationary)
Soft Core point process with parameters and
and exponent
is the pairwise interaction point process in which
each point contributes a factor
to the
probability density of the point pattern, and each pair of points
contributes a factor
to the density, where is the distance between the two points.
See the Examples for a plot of this interaction curve.
Thus the process has probability density
where represent the
points of the pattern,
is the number of points in the
pattern,
is the normalising constant,
and the sum on the right hand side is
over all unordered pairs of points of the pattern.
This model describes an “ordered” or “inhibitive” process,
with the strength of inhibition decreasing smoothly with distance.
The interaction is controlled by the parameters
and
.
The spatial scale of interaction is controlled by the
parameter , which is a positive real number
interpreted as a distance, expressed in the same units of distance as
the spatial data. The parameter
is the distance at which the
pair potential reaches the threshold value 0.37.
The shape of the interaction function
is controlled by the exponent
which is a dimensionless number
in the range
, with larger values corresponding to
a flatter shape (or a more gradual decay rate).
The process is well-defined only for
in
.
The limit of the model as
is the
hard core process with hard core distance
.
The “strength” of the interaction is determined by both of the
parameters and
.
The larger the value of
, the wider the range of
distances over which the interaction has an effect.
If
is very small, the interaction is very weak
for all practical purposes (theoretically if
the model reduces to the Poisson point process).
The nonstationary Soft Core process is similar except that
the contribution of each individual point
is a function
of location, rather than a constant beta.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Soft Core process pairwise interaction is
yielded by the function Softcore()
. See the examples below.
The main argument is the exponent kappa
.
When kappa
is fixed, the model becomes an exponential family
with canonical parameters
and
The canonical parameters are estimated by ppm()
, not fixed in
Softcore()
.
The optional argument sigma0
can be used to improve
numerical stability. If sigma0
is given, it should be a positive
number, and it should be a rough estimate of the
parameter .
An object of class "interact"
describing the interpoint interaction
structure of the Soft Core process with exponent .
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Ogata, Y, and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure. Annals of the Institute of Statistical Mathematics, B 33, 315–338.
Ogata, Y, and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. Journal of the Royal Statistical Society, series B 46, 496–518.
ppm
,
pairwise.family
,
ppm.object
# fit the stationary Soft Core process to `cells' fit5 <- ppm(cells ~1, Softcore(kappa=0.5), correction="isotropic") # study shape of interaction and explore effect of parameters fit2 <- update(fit5, Softcore(kappa=0.2)) fit8 <- update(fit5, Softcore(kappa=0.8)) plot(fitin(fit2), xlim=c(0, 0.4), main="Pair potential (sigma = 0.1)", xlab=expression(d), ylab=expression(h(d)), legend=FALSE) plot(fitin(fit5), add=TRUE, col=4) plot(fitin(fit8), add=TRUE, col=3) legend("bottomright", col=c(1,4,3), lty=1, legend=expression(kappa==0.2, kappa==0.5, kappa==0.8))
# fit the stationary Soft Core process to `cells' fit5 <- ppm(cells ~1, Softcore(kappa=0.5), correction="isotropic") # study shape of interaction and explore effect of parameters fit2 <- update(fit5, Softcore(kappa=0.2)) fit8 <- update(fit5, Softcore(kappa=0.8)) plot(fitin(fit2), xlim=c(0, 0.4), main="Pair potential (sigma = 0.1)", xlab=expression(d), ylab=expression(h(d)), legend=FALSE) plot(fitin(fit5), add=TRUE, col=4) plot(fitin(fit8), add=TRUE, col=3) legend("bottomright", col=c(1,4,3), lty=1, legend=expression(kappa==0.2, kappa==0.5, kappa==0.8))
Decomposes a measure into components, each component being a measure.
## S3 method for class 'msr' split(x, f, drop = FALSE, ...)
## S3 method for class 'msr' split(x, f, drop = FALSE, ...)
x |
Measure (object of class |
f |
Factor or tessellation determining the decomposition.
Argument passed to |
drop |
Logical value indicating whether empty components should be retained
in the list ( |
... |
Ignored. |
An object of class "msr"
represents a signed (i.e. real-valued)
or vector-valued measure in the spatstat package.
See msr
for explanation.
This function is a method for the generic
split
. It divides the measure x
into
components, each of which is a measure.
A measure x
is represented in spatstat by
a finite set of sample points with values attached to them.
The function split.msr
divides this pattern of sample points
into several sub-patterns of points using split.ppp
.
For each sub-pattern, the values attached to these points are
extracted from x
, and these values and sample points
determine a measure, which is a
component or piece of the original x
.
The argument f
can be missing, if the sample points of x
are multitype points. In this case, x
represents a measure
associated with marked spatial locations, and the command split(x)
separates x
into a list of component measures, one for each
possible mark.
Otherwise the argument f
is passed to split.ppp
.
It should be either a factor (of length equal to the number of
sample points of x
) or a tessellation (object of class
"tess"
representing a division of space into tiles)
as documented under split.ppp
.
A list, each of whose entries is a measure (object of class
"msr"
).
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
## split by tessellation a <- residuals(ppm(cells ~ x)) aa <- split(a, dirichlet(runifpoint(4))) aa sapply(aa, integral) ## split by type of point b <- residuals(ppm(amacrine ~ marks + x)) bb <- split(b) bb
## split by tessellation a <- residuals(ppm(cells ~ x)) aa <- split(a, dirichlet(runifpoint(4))) aa sapply(aa, integral) ## split by type of point b <- residuals(ppm(amacrine ~ marks + x)) bb <- split(b) bb
Creates an instance of the Strauss point process model which can then be fitted to point pattern data.
Strauss(r)
Strauss(r)
r |
The interaction radius of the Strauss process |
The (stationary) Strauss process with interaction radius and
parameters
and
is the pairwise interaction point process
in which each point contributes a factor
to the
probability density of the point pattern, and each pair of points
closer than
units apart contributes a factor
to the density.
Thus the probability density is
where represent the
points of the pattern,
is the number of points in the
pattern,
is the number of distinct unordered pairs of
points that are closer than
units apart,
and
is the normalising constant.
The interaction parameter must be less than
or equal to
so that this model describes an “ordered” or “inhibitive” pattern.
The nonstationary Strauss process is similar except that
the contribution of each individual point
is a function
of location, rather than a constant beta.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Strauss process pairwise interaction is
yielded by the function Strauss()
. See the examples below.
Note the only argument is the interaction radius r
.
When r
is fixed, the model becomes an exponential family.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
Strauss()
.
An object of class "interact"
describing the interpoint interaction
structure of the Strauss process with interaction radius .
Adrian Baddeley [email protected] and Rolf Turner [email protected].
Kelly, F.P. and Ripley, B.D. (1976) On Strauss's model for clustering. Biometrika 63, 357–360.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
ppm
,
pairwise.family
,
ppm.object
Strauss(r=0.1) # prints a sensible description of itself ppm(cells ~1, Strauss(r=0.07)) # fit the stationary Strauss process to `cells' ppm(cells ~polynom(x,y,3), Strauss(r=0.07)) # fit a nonstationary Strauss process with log-cubic polynomial trend
Strauss(r=0.1) # prints a sensible description of itself ppm(cells ~1, Strauss(r=0.07)) # fit the stationary Strauss process to `cells' ppm(cells ~polynom(x,y,3), Strauss(r=0.07)) # fit a nonstationary Strauss process with log-cubic polynomial trend
Creates an instance of the “Strauss/ hard core” point process model which can then be fitted to point pattern data.
StraussHard(r, hc=NA)
StraussHard(r, hc=NA)
r |
The interaction radius of the Strauss interaction |
hc |
The hard core distance. Optional. |
A Strauss/hard core process with interaction radius ,
hard core distance
, and
parameters
and
,
is a pairwise interaction point process
in which
distinct points are not allowed to come closer
than a distance apart
each pair of points closer than units apart
contributes a factor
to the probability density.
This is a hybrid of the Strauss process and the hard core process.
The probability density is zero if any pair of points
is closer than units apart, and otherwise equals
where represent the
points of the pattern,
is the number of points in the
pattern,
is the number of distinct unordered pairs of
points that are closer than
units apart,
and
is the normalising constant.
The interaction parameter may take any
positive value (unlike the case for the Strauss process).
If
,
the model describes an “ordered” or “inhibitive” pattern.
If
,
the model is “ordered” or “inhibitive” up to the distance
, but has an “attraction” between points lying at
distances in the range between
and
.
If , the process reduces to a classical
hard core process with hard core distance
.
If
, the process reduces to a classical
hard core process with hard core distance
.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Strauss/hard core process
pairwise interaction is
yielded by the function StraussHard()
. See the examples below.
The canonical parameter
is estimated by
ppm()
, not fixed in
StraussHard()
.
If the hard core distance argument hc
is missing or NA
,
it will be estimated from the data when ppm
is called.
The estimated value of hc
is the minimum nearest neighbour distance
multiplied by , where
is the
number of data points.
An object of class "interact"
describing the interpoint interaction
structure of the “Strauss/hard core”
process with Strauss interaction radius
and hard core distance
hc
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
ppm
,
pairwise.family
,
ppm.object
StraussHard(r=1,hc=0.02) # prints a sensible description of itself # ppm(cells ~1, StraussHard(r=0.1, hc=0.05)) # fit the stationary Strauss/hard core process to `cells' ppm(cells ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05)) # fit a nonstationary Strauss/hard core process # with log-cubic polynomial trend
StraussHard(r=1,hc=0.02) # prints a sensible description of itself # ppm(cells ~1, StraussHard(r=0.1, hc=0.05)) # fit the stationary Strauss/hard core process to `cells' ppm(cells ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05)) # fit a nonstationary Strauss/hard core process # with log-cubic polynomial trend
Takes a Gibbs point process model that has been fitted to several point patterns simultaneously, and produces a list of fitted point process models for the individual point patterns.
subfits(object, what="models", verbose=FALSE, new.coef=NULL) subfits.old(object, what="models", verbose=FALSE, new.coef=NULL) subfits.new(object, what="models", verbose=FALSE)
subfits(object, what="models", verbose=FALSE, new.coef=NULL) subfits.old(object, what="models", verbose=FALSE, new.coef=NULL) subfits.new(object, what="models", verbose=FALSE)
object |
An object of class |
what |
What should be returned.
Either |
verbose |
Logical flag indicating whether to print progress reports. |
new.coef |
Advanced use only. Numeric vector or matrix of coefficients
to replaced the fitted coefficients |
object
is assumed to have been generated by
mppm
. It represents a point process model that has been
fitted to a list of several point patterns, with covariate data.
For each of the individual point pattern
datasets, this function derives the corresponding fitted model
for that dataset only (i.e. a point process model for the th
point pattern, that is consistent with
object
).
If what="models"
,
the result is a list of point process models (a list of objects of class
"ppm"
), one model for each point pattern dataset in the
original fit.
If what="interactions"
,
the result is a list of fitted interpoint interactions (a list of
objects of class
"fii"
).
Two different algorithms are provided, as
subfits.old
and subfits.new
.
Currently subfits
is the same as the old algorithm
subfits.old
because the newer algorithm is too memory-hungry.
A list of point process models (a list of objects of class
"ppm"
) or a list of fitted interpoint interactions (a list of
objects of class "fii"
).
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
H <- hyperframe(Wat=waterstriders) fit <- mppm(Wat~x, data=H) subfits(fit) H$Wat[[3]] <- rthin(H$Wat[[3]], 0.1) fit2 <- mppm(Wat~x, data=H, random=~1|id) subfits(fit2)
H <- hyperframe(Wat=waterstriders) fit <- mppm(Wat~x, data=H) subfits(fit) H$Wat[[3]] <- rthin(H$Wat[[3]], 0.1) fit2 <- mppm(Wat~x, data=H, random=~1|id) subfits(fit2)
The canonical sufficient statistic of a point process model is evaluated for a given point pattern.
suffstat(model, X=data.ppm(model))
suffstat(model, X=data.ppm(model))
model |
A fitted point process model (object of class
|
X |
A point pattern (object of class |
The canonical sufficient statistic
of model
is evaluated for the point pattern X
.
This computation is useful for various Monte Carlo methods.
Here model
should be a point process model (object of class
"ppm"
, see ppm.object
), typically obtained
from the model-fitting function ppm
. The argument
X
should be a point pattern (object of class "ppp"
).
Every point process model fitted by ppm
has
a probability density of the form
where denotes a typical realisation (i.e. a point pattern),
is the vector of model coefficients,
is a normalising constant,
and
is a function of the realisation
, called the
“canonical sufficient statistic” of the model.
For example, the stationary Poisson process has canonical sufficient
statistic , the number of points in
.
The stationary Strauss process with interaction range
(and fitted with no edge correction) has canonical sufficient statistic
where
is the number of pairs
of points in
which are closer than a distance
to each other.
suffstat(model, X)
returns the value of , where
is
the canonical sufficient statistic associated with
model
,
evaluated when is the given point pattern
X
.
The result is a numeric vector, with entries which correspond to the
entries of the coefficient vector coef(model)
.
The sufficient statistic
does not depend on the fitted coefficients
of the model. However it does depend on the irregular parameters
which are fixed in the original call to
ppm
, for
example, the interaction range r
of the Strauss process.
The sufficient statistic also depends on the edge correction that was used to fit the model. For example in a Strauss process,
If the model is fitted with correction="none"
, the sufficient
statistic is where
is the
number of points and
is the number of pairs of points
which are closer than
units apart.
If the model is fitted with correction="periodic"
, the sufficient
statistic is the same as above, except that distances are measured
in the periodic sense.
If the model is fitted with
correction="translate"
, then is unchanged
but
is replaced by a weighted sum (the sum of the translation
correction weights for all pairs of points which are closer than
units apart).
If the model is fitted with
correction="border"
(the default), then points lying less than
units from the boundary of the observation window are
treated as fixed. Thus
is
replaced by the number
of points lying at least
units from
the boundary of the observation window, and
is replaced by
the number
of pairs of points, which are closer
than
units apart, and at least one of which lies
more than
units from the boundary of the observation window.
Non-finite values of the sufficient statistic (NA
or
-Inf
) may be returned if the point pattern X
is
not a possible realisation of the model (i.e. if X
has zero
probability of occurring under model
for all values of
the canonical coefficients ).
A numeric vector of sufficient statistics. The entries
correspond to the model coefficients coef(model)
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fitS <- ppm(swedishpines~1, Strauss(7)) suffstat(fitS) X <- rpoispp(intensity(swedishpines), win=Window(swedishpines)) suffstat(fitS, X)
fitS <- ppm(swedishpines~1, Strauss(7)) suffstat(fitS) X <- rpoispp(intensity(swedishpines), win=Window(swedishpines)) suffstat(fitS, X)
summary
method for class "dppm"
.
## S3 method for class 'dppm' summary(object, ..., quick=FALSE) ## S3 method for class 'summary.dppm' print(x, ...)
## S3 method for class 'dppm' summary(object, ..., quick=FALSE) ## S3 method for class 'summary.dppm' print(x, ...)
object |
A fitted determinantal point process model (object of
class |
quick |
Logical value controlling the scope of the summary. |
... |
Arguments passed to |
x |
Object of class |
This is a method for the generic summary
for the class "dppm"
. An object of class "dppm"
describes a fitted determinantal point process model.
See dppm
.
summary.dppm
extracts information about the
type of model that has been fitted, the data to which the model was
fitted, and the values of the fitted coefficients.
print.summary.dppm
prints this information in a
comprehensible format.
In normal usage, print.summary.dppm
is invoked implicitly
when the user calls summary.dppm
without assigning its value
to anything. See the examples.
summary.dppm
returns an object of class "summary.dppm"
,
while print.summary.dppm
returns NULL
.
The result of summary.dppm
includes at least the
following components:
Xname |
character string name of the original point pattern data |
stationary |
logical value indicating whether the model is stationary |
trend |
Object of class |
repul |
Repulsiveness index |
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
jpines <- residualspaper$Fig1 fit <- dppm(jpines ~ 1, dppGauss) summary(fit)
jpines <- residualspaper$Fig1 fit <- dppm(jpines ~ 1, dppGauss) summary(fit)
summary
method for class "kppm"
.
## S3 method for class 'kppm' summary(object, ..., quick=FALSE) ## S3 method for class 'summary.kppm' print(x, ...)
## S3 method for class 'kppm' summary(object, ..., quick=FALSE) ## S3 method for class 'summary.kppm' print(x, ...)
object |
A fitted Cox or cluster point process model (object of
class |
quick |
Logical value controlling the scope of the summary. |
... |
Arguments passed to |
x |
Object of class |
This is a method for the generic summary
for the class "kppm"
. An object of class "kppm"
describes a fitted Cox or cluster point process model.
See kppm
.
summary.kppm
extracts information about the
type of model that has been fitted, the data to which the model was
fitted, and the values of the fitted coefficients.
print.summary.kppm
prints this information in a
comprehensible format.
In normal usage, print.summary.kppm
is invoked implicitly
when the user calls summary.kppm
without assigning its value
to anything. See the examples.
You can also type coef(summary(object))
to extract a table
of the fitted coefficients of the point process model object
together with standard errors and confidence limits.
summary.kppm
returns an object of class "summary.kppm"
,
while print.summary.kppm
returns NULL
.
The result of summary.kppm
includes at least the
following components:
Xname |
character string name of the original point pattern data |
stationary |
logical value indicating whether the model is stationary |
clusters |
the |
modelname |
character string describing the model |
isPCP |
|
lambda |
Estimated intensity: numeric value, or pixel image |
mu |
Mean cluster size: numeric value, pixel image, or
|
clustpar |
list of fitted parameters for the cluster model |
clustargs |
list of fixed parameters for the cluster model, if any |
callstring |
character string representing the original call to
|
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
Baddeley, A., Davies, T.M., Hazelton, M.L., Rakshit, S. and Turner,
R. (2022) Fundamental problems in fitting spatial cluster
process models. Spatial Statistics 52, 100709.
DOI: 10.1016/j.spasta.2022.100709
fit <- kppm(redwood ~ 1, "Thomas") summary(fit) coef(summary(fit))
fit <- kppm(redwood ~ 1, "Thomas") summary(fit) coef(summary(fit))
summary
method for class "ppm"
.
## S3 method for class 'ppm' summary(object, ..., quick=FALSE, fine=FALSE) ## S3 method for class 'summary.ppm' print(x, ...)
## S3 method for class 'ppm' summary(object, ..., quick=FALSE, fine=FALSE) ## S3 method for class 'summary.ppm' print(x, ...)
object |
A fitted point process model. |
... |
Ignored. |
quick |
Logical flag controlling the scope of the summary. |
fine |
Logical value passed to |
x |
Object of class |
This is a method for the generic summary
for the class "ppm"
. An object of class "ppm"
describes a fitted point process model. See ppm.object
)
for details of this class.
summary.ppm
extracts information about the
type of model that has been fitted, the data to which the model was
fitted, and the values of the fitted coefficients.
(If quick=TRUE
then only the information about the type
of model is extracted.)
print.summary.ppm
prints this information in a
comprehensible format.
In normal usage, print.summary.ppm
is invoked implicitly
when the user calls summary.ppm
without assigning its value
to anything. See the examples.
You can also type coef(summary(object))
to extract a table
of the fitted coefficients of the point process model object
together with standard errors and confidence limits.
summary.ppm
returns an object of class "summary.ppm"
,
while print.summary.ppm
returns NULL
.
Adrian Baddeley [email protected] and Rolf Turner [email protected]
# invent some data X <- rpoispp(42) # fit a model to it fit <- ppm(X ~ x, Strauss(r=0.1)) # summarize the fitted model summary(fit) # `quick' option summary(fit, quick=TRUE) # coefficients with standard errors and CI coef(summary(fit)) coef(summary(fit, fine=TRUE)) # save the full summary s <- summary(fit) # print it print(s) s # extract stuff names(s) coef(s) s$args$correction s$name s$trend$value # multitype pattern fit <- ppm(demopat ~marks, Poisson()) summary(fit) # model with external covariates fitX <- ppm(X, ~Z, covariates=list(Z=function(x,y){x+y})) summary(fitX)
# invent some data X <- rpoispp(42) # fit a model to it fit <- ppm(X ~ x, Strauss(r=0.1)) # summarize the fitted model summary(fit) # `quick' option summary(fit, quick=TRUE) # coefficients with standard errors and CI coef(summary(fit)) coef(summary(fit, fine=TRUE)) # save the full summary s <- summary(fit) # print it print(s) s # extract stuff names(s) coef(s) s$args$correction s$name s$trend$value # multitype pattern fit <- ppm(demopat ~marks, Poisson()) summary(fit) # model with external covariates fitX <- ppm(X, ~Z, covariates=list(Z=function(x,y){x+y})) summary(fitX)
Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using the K function.
thomas.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
thomas.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the Thomas model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the Thomas process. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
This algorithm fits the Thomas point process model to a point pattern dataset
by the Method of Minimum Contrast, using the function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The function of the point pattern will be computed
using
Kest
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits the Thomas point process to X
,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical function of the Thomas process
and the observed
function.
For a more detailed explanation of the Method of Minimum Contrast,
see
mincontrast
.
The Thomas point process is described in
Moller and Waagepetersen (2003, pp. 61–62). It is a cluster
process formed by taking a pattern of parent points, generated
according to a Poisson process with intensity , and
around each parent point, generating a random number of offspring
points, such that the number of offspring of each parent is a Poisson
random variable with mean
, and the locations of the
offspring points of one parent are independent and isotropically
Normally distributed around the parent point with standard deviation
which is equal to the parameter
scale
. The
named vector of stating values can use either sigma2
() or
scale
as the name of the second
component, but the latter is recommended for consistency with other
cluster models.
The theoretical -function of the Thomas process is
The theoretical intensity
of the Thomas process is .
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The Thomas process can be simulated, using rThomas
.
Homogeneous or inhomogeneous Thomas process models can also
be fitted using the function kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Rasmus Plenge Waagepetersen [email protected]. Adapted for spatstat by Adrian Baddeley [email protected].
Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 659–667.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
lgcp.estK
,
matclust.estK
,
mincontrast
,
Kest
,
rThomas
to simulate the fitted model.
u <- thomas.estK(redwood, c(kappa=10, scale=0.1)) u plot(u)
u <- thomas.estK(redwood, c(kappa=10, scale=0.1)) u plot(u)
Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using the pair correlation function.
thomas.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
thomas.estpcf(X, startpar=c(kappa=1,scale=1), lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
X |
Data to which the Thomas model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the Thomas process. |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
This algorithm fits the Thomas point process model to a point pattern dataset
by the Method of Minimum Contrast, using the pair correlation function
pcf
.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf
or one of its relatives.
The algorithm fits the Thomas point process to X
,
by finding the parameters of the Thomas model
which give the closest match between the
theoretical pair correlation function of the Thomas process
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The Thomas point process is described in
Moller and Waagepetersen (2003, pp. 61–62). It is a cluster
process formed by taking a pattern of parent points, generated
according to a Poisson process with intensity , and
around each parent point, generating a random number of offspring
points, such that the number of offspring of each parent is a Poisson
random variable with mean
, and the locations of the
offspring points of one parent are independent and isotropically
Normally distributed around the parent point with standard deviation
which is equal to the parameter
scale
. The
named vector of stating values can use either sigma2
() or
scale
as the name of the second
component, but the latter is recommended for consistency with other
cluster models.
The theoretical pair correlation function of the Thomas process is
The theoretical intensity
of the Thomas process is .
In this algorithm, the Method of Minimum Contrast is first used to find
optimal values of the parameters
and
. Then the remaining parameter
is inferred from the estimated intensity
.
If the argument lambda
is provided, then this is used
as the value of . Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The Thomas process can be simulated, using rThomas
.
Homogeneous or inhomogeneous Thomas process models can also
be fitted using the function kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Adrian Baddeley [email protected]
Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by means of distance methods. Biometrics 32 659–667.
Moller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
Thomas, M. (1949) A generalisation of Poisson's binomial limit for use in ecology. Biometrika 36, 18–25.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
thomas.estK
mincontrast
,
pcf
,
rThomas
to simulate the fitted model.
u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1)) u plot(u, legendpos="topright") u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1), pcfargs=list(stoyan=0.12))
u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1)) u plot(u, legendpos="topright") u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1), pcfargs=list(stoyan=0.12))
Extract the history of evaluations of the objective function performed when a cluster process model was fitted.
traj(object)
traj(object)
object |
Fitted cluster point process model (object of class |
Under appropriate circumstances, the fitted model object
contains the history of evaluations of the objective function
that were performed by the optimisation algorithm. This history is
extracted by traj
.
The result is a data frame containing the input parameter values for
the objective function, and the corresponding value of the objective
function, that were considered by the optimisation algorithm. This
data frame also belongs to the class "traj"
which has
methods for plot
, print
and other purposes.
Either a data frame (belonging to class "traj"
)
or NULL
.
Adrian Baddeley [email protected].
fit <- kppm(redwood, trajectory=TRUE) h <- traj(fit)
fit <- kppm(redwood, trajectory=TRUE) h <- traj(fit)
An object describing the family of all Gibbs point processes with interaction order equal to 3.
Advanced Use Only!
This structure would not normally be touched by the user. It describes the interaction structure of Gibbs point processes which have infinite order of interaction, such as the triplet interaction process Triplets.
Anyway, triplet.family
is an object of class "isf"
containing a function triplet.family$eval
for
evaluating the sufficient statistics of a Gibbs
point process model taking an exponential family form.
Object of class "isf"
, see isf.object
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42, 283–322.
Triplets
to create the triplet interaction process
structure.
Other families:
pairwise.family
,
pairsat.family
,
inforder.family
,
ord.family
.
Creates an instance of Geyer's triplet interaction point process model which can then be fitted to point pattern data.
Triplets(r)
Triplets(r)
r |
The interaction radius of the Triplets process |
The (stationary) Geyer triplet process (Geyer, 1999)
with interaction radius and
parameters
and
is the point process
in which each point contributes a factor
to the
probability density of the point pattern, and each triplet of close points
contributes a factor
to the density.
A triplet of close points is a group of 3 points,
each pair of which is closer than
units
apart.
Thus the probability density is
where represent the
points of the pattern,
is the number of points in the
pattern,
is the number of unordered triples of
points that are closer than
units apart,
and
is the normalising constant.
The interaction parameter must be less than
or equal to
so that this model describes an “ordered” or “inhibitive” pattern.
The nonstationary Triplets process is similar except that
the contribution of each individual point
is a function
of location, rather than a constant beta.
The function ppm()
, which fits point process models to
point pattern data, requires an argument
of class "interact"
describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Triplets process pairwise interaction is
yielded by the function Triplets()
. See the examples below.
Note the only argument is the interaction radius r
.
When r
is fixed, the model becomes an exponential family.
The canonical parameters
and
are estimated by
ppm()
, not fixed in
Triplets()
.
An object of class "interact"
describing the interpoint interaction
structure of the Triplets process with interaction radius .
Adrian Baddeley [email protected] and Rolf Turner [email protected]
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80. Pages 79–140.
ppm
,
triplet.family
,
ppm.object
Triplets(r=0.1) # prints a sensible description of itself ppm(cells ~1, Triplets(r=0.2)) # fit the stationary Triplets process to `cells' ppm(cells ~polynom(x,y,3), Triplets(r=0.2)) # fit a nonstationary Triplets process with log-cubic polynomial trend
Triplets(r=0.1) # prints a sensible description of itself ppm(cells ~1, Triplets(r=0.2)) # fit the stationary Triplets process to `cells' ppm(cells ~polynom(x,y,3), Triplets(r=0.2)) # fit a nonstationary Triplets process with log-cubic polynomial trend
Inspect or change the name of the unit of length in a spatial dataset.
## S3 method for class 'dppm' unitname(x) ## S3 method for class 'kppm' unitname(x) ## S3 method for class 'minconfit' unitname(x) ## S3 method for class 'ppm' unitname(x) ## S3 method for class 'slrm' unitname(x) ## S3 replacement method for class 'dppm' unitname(x) <- value ## S3 replacement method for class 'kppm' unitname(x) <- value ## S3 replacement method for class 'minconfit' unitname(x) <- value ## S3 replacement method for class 'ppm' unitname(x) <- value ## S3 replacement method for class 'slrm' unitname(x) <- value
## S3 method for class 'dppm' unitname(x) ## S3 method for class 'kppm' unitname(x) ## S3 method for class 'minconfit' unitname(x) ## S3 method for class 'ppm' unitname(x) ## S3 method for class 'slrm' unitname(x) ## S3 replacement method for class 'dppm' unitname(x) <- value ## S3 replacement method for class 'kppm' unitname(x) <- value ## S3 replacement method for class 'minconfit' unitname(x) <- value ## S3 replacement method for class 'ppm' unitname(x) <- value ## S3 replacement method for class 'slrm' unitname(x) <- value
x |
A spatial dataset.
Either a point pattern (object of class |
value |
Name of the unit of length. See Details. |
Spatial datasets in the spatstat package may include the name of the unit of length. This name is used when printing or plotting the dataset, and in some other applications.
unitname(x)
extracts this name,
and unitname(x) <- value
sets the name to value
.
A valid name is either
a single character string
a vector of two character strings giving the singular and plural forms of the unit name
a list of length 3, containing two character strings giving the singular and plural forms of the basic unit, and a number specifying the multiple of this unit.
Note that re-setting the name of the unit of length does not
affect the numerical values in x
. It changes only the string
containing the name of the unit of length. To rescale the numerical
values, use rescale
.
The return value of unitname
is an object of class "unitname"
containing the name of the unit of length in x
. There are
methods for print
, summary
, as.character
,
rescale
and compatible
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
X <- runifrect(20) # if the unit of length is 1 metre: unitname(X) <- c("metre", "metres") # if the unit of length is 6 inches: unitname(X) <- list("inch", "inches", 6)
X <- runifrect(20) # if the unit of length is 1 metre: unitname(X) <- c("metre", "metres") # if the unit of length is 6 inches: unitname(X) <- list("inch", "inches", 6)
Converts a vector-valued measure into a list of scalar-valued measures.
## S3 method for class 'msr' unstack(x, ...)
## S3 method for class 'msr' unstack(x, ...)
x |
A measure (object of class |
... |
Ignored. |
This is a method for the generic
unstack
for the class "msr"
of measures.
If x
is a vector-valued measure, then
y <- unstack(x)
is a list of scalar-valued measures
defined by the components of x
.
The j
th entry of the list, y[[j]]
, is equivalent to
the j
th component of the vector measure x
.
If x
is a scalar-valued measure, then
the result is a list consisting of one entry, which is x
.
A list of measures, of class "solist"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit <- ppm(cells ~ x) m <- residuals(fit, type="score") m unstack(m)
fit <- ppm(cells ~ x) m <- residuals(fit, type="score") m unstack(m)
Set parameter values in a determinantal point process model object.
## S3 method for class 'detpointprocfamily' update(object, ...)
## S3 method for class 'detpointprocfamily' update(object, ...)
object |
object of class |
... |
arguments of the form |
Another object of class "detpointprocfamily"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
update
method for class "dppm"
.
## S3 method for class 'dppm' update(object, ..., evaluate=TRUE, envir=environment(terms(object)))
## S3 method for class 'dppm' update(object, ..., evaluate=TRUE, envir=environment(terms(object)))
object |
Fitted determinantal point process model.
An object of class |
... |
Arguments passed to |
evaluate |
Logical value indicating whether to return the updated fitted model
( |
envir |
Environment in which to re-evaluate the call to |
object
should be a fitted determinantal point process model,
obtained from the model-fitting function dppm
.
The model will be updated according to the new arguments provided.
If the argument trend
is provided, it determines the
intensity in the updated model. It should be an R formula
(with or without a left hand side). It may include the symbols
+
or -
to specify addition or deletion of terms
in the current model formula, as shown in the Examples below.
The symbol .
refers to the current contents of the
formula.
The intensity in the updated model is determined by the
argument trend
if it is provided, or otherwise by any unnamed
argument that is a formula, or otherwise by the formula of the
original model, formula(object)
.
The spatial point pattern data to which the new model is fitted
is determined by the left hand side of the updated model formula,
if this is present. Otherwise it is determined by the argument
X
if it is provided, or otherwise by any unnamed argument
that is a point pattern or a quadrature scheme.
The model is refitted using dppm
.
Another fitted cluster point process model (object of
class "dppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
dppm
, plot.dppm
,
predict.dppm
, simulate.dppm
,
methods.dppm
.
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") fitx <- update(fit, ~x) fit2 <- update(fit, flipxy(swedishpines))
fit <- dppm(swedishpines ~ x + y, dppGauss, method="c") fitx <- update(fit, ~x) fit2 <- update(fit, flipxy(swedishpines))
This command updates the object
using the
arguments given.
## S3 method for class 'interact' update(object, ...)
## S3 method for class 'interact' update(object, ...)
object |
Interpoint interaction (object of class |
... |
Additional or replacement values of parameters of |
This is a method for the generic function update
for the class "interact"
of interpoint interactions.
It updates the object
using the parameters given in the
extra arguments ...
.
The extra arguments must be given in
the form name=value
and must be recognisable to
the interaction object. They override any parameters
of the same name in object
.
Another object of class "interact"
, equivalent to object
except for changes in parameter values.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Str <- Strauss(r=1) Str update(Str, r=2) M <- MultiStrauss(radii=matrix(1,2,2)) update(M, types=c("on", "off"))
Str <- Strauss(r=1) Str update(Str, r=2) M <- MultiStrauss(radii=matrix(1,2,2)) update(M, types=c("on", "off"))
update
method for class "kppm"
.
## S3 method for class 'kppm' update(object, ..., evaluate=TRUE, envir=environment(terms(object)))
## S3 method for class 'kppm' update(object, ..., evaluate=TRUE, envir=environment(terms(object)))
object |
Fitted cluster point process model.
An object of class |
... |
Arguments passed to |
evaluate |
Logical value indicating whether to return the updated fitted model
( |
envir |
Environment in which to re-evaluate the call to |
object
should be a fitted cluster point process model,
obtained from the model-fitting function kppm
.
The model will be updated according to the new arguments provided.
If the argument trend
is provided, it determines the
intensity in the updated model. It should be an R formula
(with or without a left hand side). It may include the symbols
+
or -
to specify addition or deletion of terms
in the current model formula, as shown in the Examples below.
The symbol .
refers to the current contents of the
formula.
The intensity in the updated model is determined by the
argument trend
if it is provided, or otherwise by any unnamed
argument that is a formula, or otherwise by the formula of the
original model, formula(object)
.
The spatial point pattern data to which the new model is fitted
is determined by the left hand side of the updated model formula,
if this is present. Otherwise it is determined by the argument
X
if it is provided, or otherwise by any unnamed argument
that is a point pattern or a quadrature scheme.
The model is refitted using kppm
.
Another fitted cluster point process model (object of
class "kppm"
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
kppm
, plot.kppm
,
predict.kppm
, simulate.kppm
,
methods.kppm
,
vcov.kppm
fit <- kppm(redwood ~1, "Thomas") fitx <- update(fit, ~ . + x) fitM <- update(fit, clusters="MatClust") fitC <- update(fit, cells) fitCx <- update(fit, cells ~ x)
fit <- kppm(redwood ~1, "Thomas") fitx <- update(fit, ~ . + x) fitM <- update(fit, clusters="MatClust") fitC <- update(fit, cells) fitCx <- update(fit, cells ~ x)
update
method for class "ppm"
.
## S3 method for class 'ppm' update(object, ..., fixdummy=TRUE, use.internal=NULL, envir=environment(terms(object)))
## S3 method for class 'ppm' update(object, ..., fixdummy=TRUE, use.internal=NULL, envir=environment(terms(object)))
object |
An existing fitted point process model,
typically produced by |
... |
Arguments to be updated in the new call to |
fixdummy |
Logical flag indicating whether the quadrature scheme for the
call to |
use.internal |
Optional. Logical flag indicating whether the model should be
refitted using the internally saved data ( |
envir |
Environment in which to re-evaluate the call to |
This is a method for the generic function update
for the class "ppm"
. An object of class "ppm"
describes a fitted point process model. See ppm.object
)
for details of this class.
update.ppm
will modify the point process model
specified by object
according to the new arguments given,
then re-fit it.
The actual re-fitting is performed by the model-fitting
function ppm
.
If you are comparing several model fits to the same data,
or fits of the same model to different data, it is
strongly advisable to use update.ppm
rather than trying to fit them by hand.
This is because update.ppm
re-fits the model
in a way which is comparable to the original fit.
The arguments ...
are matched to the formal arguments
of ppm
as follows.
First, all the named arguments in ...
are matched
with the formal arguments of ppm
.
Use name=NULL
to remove the argument name
from the
call.
Second, any unnamed arguments in ...
are
matched with formal arguments of ppm
if the matching
is obvious from the class of the object. Thus ...
may contain
exactly one argument of class "ppp"
or "quad"
,
which will be interpreted as the named argument Q
;
exactly one argument of class "formula"
, which will be
interpreted as the named argument trend
(or as specifying
a change to the trend formula);
exactly one argument of class "interact"
, which will be
interpreted as the named argument interaction
;
exactly one argument of class "data.frame"
, which will be
interpreted as the named argument covariates
.
The trend
argument can be a formula that specifies a
change to the current trend formula. For example, the
formula ~ . + Z
specifies that the additional covariate
Z
will be added to the right hand side of the trend
formula in the existing object
.
The argument fixdummy=TRUE
ensures comparability of the
objects before and after updating.
When fixdummy=FALSE
, calling update.ppm
is exactly the same as calling ppm
with the updated
arguments. However, the original and updated models
are not strictly comparable (for example, their pseudolikelihoods
are not strictly comparable) unless they used the same set of dummy
points for the quadrature scheme. Setting fixdummy=TRUE
ensures that the re-fitting will be performed using the same set
of dummy points. This is highly recommended.
The value of use.internal
determines where to find data
to re-evaluate the model (data for the arguments mentioned in
the original call to ppm
that are not overwritten by
arguments to update.ppm
).
If use.internal=FALSE
, then arguments
to ppm
are re-evaluated in the frame where you
call update.ppm
. This is like the behaviour of the
other methods for update
. This means that if you have changed
any of the objects referred to in the call, these changes will be
taken into account. Also if the original call to ppm
included
any calls to random number generators, these calls will be recomputed,
so that you will get a different outcome of the random numbers.
If use.internal=TRUE
, then arguments to ppm
are extracted
from internal data stored inside the current fitted
model object
. This is useful if you don't want to
re-evaluate anything. It is also necessary if
if object
has been restored from a dump file
using load
or source
. In such cases,
we have lost the environment in which object
was fitted,
and data cannot be re-evaluated.
By default, if use.internal
is missing, update.ppm
will
re-evaluate the arguments if this is possible, and use internal data
if not.
Another fitted point process model (object of class "ppm"
).
Adrian Baddeley [email protected] and Rolf Turner [email protected]
data(cells) # fit the stationary Poisson process fit <- ppm(nztrees, ~ 1) # fit a nonstationary Poisson process fitP <- update(fit, trend=~x) fitP <- update(fit, ~x) # change the trend formula: add another term to the trend fitPxy <- update(fitP, ~ . + y) # change the trend formula: remove the x variable fitPy <- update(fitPxy, ~ . - x) # fit a stationary Strauss process fitS <- update(fit, interaction=Strauss(13)) fitS <- update(fit, Strauss(13)) # refit using a different edge correction fitS <- update(fitS, correction="isotropic") # re-fit the model to a subset # of the original point pattern nzw <- owin(c(0,148),c(0,95)) nzsub <- nztrees[,nzw] fut <- update(fitS, Q=nzsub) fut <- update(fitS, nzsub) # WARNING: the point pattern argument is called 'Q' ranfit <- ppm(rpoispp(42), ~1, Poisson()) ranfit # different random data! update(ranfit) # the original data update(ranfit, use.internal=TRUE)
data(cells) # fit the stationary Poisson process fit <- ppm(nztrees, ~ 1) # fit a nonstationary Poisson process fitP <- update(fit, trend=~x) fitP <- update(fit, ~x) # change the trend formula: add another term to the trend fitPxy <- update(fitP, ~ . + y) # change the trend formula: remove the x variable fitPy <- update(fitPxy, ~ . - x) # fit a stationary Strauss process fitS <- update(fit, interaction=Strauss(13)) fitS <- update(fit, Strauss(13)) # refit using a different edge correction fitS <- update(fitS, correction="isotropic") # re-fit the model to a subset # of the original point pattern nzw <- owin(c(0,148),c(0,95)) nzsub <- nztrees[,nzw] fut <- update(fitS, Q=nzsub) fut <- update(fitS, nzsub) # WARNING: the point pattern argument is called 'Q' ranfit <- ppm(rpoispp(42), ~1, Poisson()) ranfit # different random data! update(ranfit) # the original data update(ranfit, use.internal=TRUE)
update
method for class "rppm"
.
## S3 method for class 'rppm' update(object, ..., envir=environment(terms(object)))
## S3 method for class 'rppm' update(object, ..., envir=environment(terms(object)))
object |
Fitted recursively partitioned point process model.
An object of class |
... |
Arguments passed to |
envir |
Environment in which to re-evaluate the call to |
object
should be a fitted recursively partitioned point process model,
obtained from the model-fitting function rppm
.
The model will be updated according to the new arguments provided.
Another fitted recursively partitioned point process model (object of
class "rppm"
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
rppm
.
fit <- rppm(nztrees ~ x) newfit <- update(fit, . ~ x + y)
fit <- rppm(nztrees ~ x) newfit <- update(fit, . ~ x + y)
Determines whether a point process model object corresponds to a valid point process.
valid(object, ...)
valid(object, ...)
object |
Object of some class, describing a point process model. |
... |
Additional arguments passed to methods. |
The function valid
is generic,
with methods for the classes "ppm"
and "dppmodel"
.
An object representing a point process is called valid if
all its parameter values are known (for example, no parameter
takes the value NA
or NaN
) and the parameter values
correspond to a well-defined point process (for example, the
parameter values satisfy all the constraints that are imposed by
mathematical theory.)
See the methods for further details.
A logical value, or NA
.
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
valid.ppm
,
valid.detpointprocfamily
Checks the validity of a determinantal point process model.
## S3 method for class 'detpointprocfamily' valid(object, ...)
## S3 method for class 'detpointprocfamily' valid(object, ...)
object |
Model of class |
... |
Ignored. |
Logical
Adrian Baddeley [email protected]
Rolf Turner [email protected]
and Ege Rubak [email protected]
model1 <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) valid(model1) model2 <- dppMatern(lambda=100, alpha=1, nu=1, d=2) valid(model2)
model1 <- dppMatern(lambda=100, alpha=.01, nu=1, d=2) valid(model1) model2 <- dppMatern(lambda=100, alpha=1, nu=1, d=2) valid(model2)
Determines whether a fitted point process model satisfies the integrability conditions for existence of the point process.
## S3 method for class 'ppm' valid(object, warn=TRUE, ...)
## S3 method for class 'ppm' valid(object, warn=TRUE, ...)
object |
Fitted point process model (object of class |
warn |
Logical value indicating whether to issue a warning if the validity of the model cannot be checked (due to unavailability of the required code). |
... |
Ignored. |
This is a method for the generic function valid
for Poisson and Gibbs point process models (class "ppm"
).
The model-fitting function ppm
fits Gibbs point process models to point pattern data.
By default, ppm
does not check whether the
fitted model actually exists as a point process. This checking
is done by valid.ppm
.
Unlike a regression model, which is well-defined for any values
of the fitted regression coefficients, a Gibbs point process model
is only well-defined if the fitted interaction parameters
satisfy some constraints.
A famous example is the Strauss process (see Strauss
)
which exists only when the interaction parameter
is less than or equal to 1. For values
,
the probability density is not integrable and the process does not
exist (and cannot be simulated).
By default, ppm
does not enforce the constraint that
a fitted Strauss process (for example) must satisfy
.
This is because a fitted parameter value of
could be useful information for data analysis, as it indicates that
the Strauss model is not appropriate, and suggests a clustered model should be
fitted.
The function valid.ppm
checks whether the fitted model
object
specifies a well-defined point process. It returns
TRUE
if the model is well-defined.
Another possible reason for invalid models is that the data may not
be adequate for estimation of the model parameters. In this case,
some of the fitted coefficients could be NA
or infinite
values. If this happens
then valid.ppm
returns FALSE
.
Use the function project.ppm
to force the fitted model
to be valid.
A logical value, or NA
.
Adrian Baddeley [email protected]
and Rolf Turner [email protected]
fit1 <- ppm(cells, ~1, Strauss(0.1)) valid(fit1) fit2 <- ppm(redwood, ~1, Strauss(0.1)) valid(fit2)
fit1 <- ppm(cells, ~1, Strauss(0.1)) valid(fit1) fit2 <- ppm(redwood, ~1, Strauss(0.1)) valid(fit2)
Determines whether a fitted spatial logistic regression model is a well-defined model.
## S3 method for class 'slrm' valid(object, warn=TRUE, ...)
## S3 method for class 'slrm' valid(object, warn=TRUE, ...)
object |
Fitted spatial logistic regression model (object of class |
warn |
Logical value indicating whether to issue a warning if the validity of the model cannot be checked (due to unavailability of the required code). |
... |
Ignored. |
This is a method for the generic function valid
for spatial logistic regression models (class "slrm"
).
In a model fitted by slrm
, some of the fitted
coefficients may be NA
or infinite values.
This can occur if the data are not
adequate for estimation of the model parameters. The model is said
to be unidentifiable or confounded.
The function valid.slrm
checks whether the fitted coefficients of
object
specify a well-defined model. It returns
TRUE
if the model is well-defined, and FALSE
otherwise.
Use the function emend.slrm
to force the fitted model
to be valid.
A logical value, or NA
.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
fit1 <- slrm(cells ~ x) valid(fit1) fit2 <- slrm(cells ~ x + I(x)) valid(fit2)
fit1 <- slrm(cells ~ x) valid(fit1) fit2 <- slrm(cells ~ x + I(x)) valid(fit2)
Given a fitted point process model, calculate the predicted variance
of the number of points in a nominated set B
.
varcount(model, B=Window(model), ..., dimyx = NULL, relative=FALSE)
varcount(model, B=Window(model), ..., dimyx = NULL, relative=FALSE)
model |
A fitted point process model
(object of class |
B |
A window (object of class |
... |
Additional arguments passed to |
dimyx |
Spatial resolution for the calculations.
Argument passed to |
relative |
Logical value specifying whether to divide the variance by the mean value. |
The function varcount
calculates the variance of the number of points
falling in a specified window B
according to the model
.
It can also calculate the variance of a sum of weights attached
to each random point.
If relative=FALSE
(the default), the result is the variance.
If relative=TRUE
, the result is the variance divided by the
mean, which is the overdispersion index (equal to 1 if the number of
points has a Poisson distribution).
The model
should be a fitted point process model
(object of class "ppm"
, "kppm"
or "dppm"
).
If B
is a window, varcount
calculates the variance
of the number of points falling in B
, according to the
fitted model
.
If the model
depends on spatial covariates other than the
Cartesian coordinates, then B
should be a subset of the
domain in which these covariates are defined.
If B
is a pixel image,
varcount
calculates the variance of
,
the sum of the values of
B
over all random points
falling in the domain of the image.
If the model
depends on spatial covariates other than the
Cartesian coordinates, then the domain of the pixel image,
as.owin(B)
, should be a subset of the domain in which these
covariates are defined.
If B
is a function(x,y)
or function(x,y,...)
then varcount
calculates the variance of
,
the sum of the values of
B
over all random points
falling inside the window W=as.owin(model)
, the window
in which the original data were observed.
The variance calculation involves the intensity and the
pair correlation function of the model.
The calculation is exact (up to discretisation error)
for models of class "kppm"
and "dppm"
,
and for Poisson point process models of class "ppm"
.
For Gibbs point process models of class "ppm"
the
calculation depends on the Poisson-saddlepoint approximations
to the intensity and pair correlation function, which are rough
approximations. The approximation is not yet implemented
for some Gibbs models.
A single number.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected]
predict.ppm
,
predict.kppm
,
predict.dppm
fitT <- kppm(redwood ~ 1, "Thomas") B <- owin(c(0, 0.5), c(-0.5, 0)) varcount(fitT, B) fitS <- ppm(swedishpines ~ 1, Strauss(9)) BS <- square(50) varcount(fitS, BS)
fitT <- kppm(redwood ~ 1, "Thomas") B <- owin(c(0, 0.5), c(-0.5, 0)) varcount(fitT, B) fitS <- ppm(swedishpines ~ 1, Strauss(9)) BS <- square(50) varcount(fitS, BS)
Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset by the Method of Minimum Contrast.
vargamma.estK(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
vargamma.estK(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the model. |
nu |
Numerical value controlling the shape of the tail of the clusters.
A number greater than |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
This algorithm fits the Neyman-Scott Cluster point process model
with Variance Gamma kernel (Jalilian et al, 2013)
to a point pattern dataset
by the Method of Minimum Contrast, using the function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The function of the point pattern will be computed
using
Kest
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the function,
and this object should have been obtained by a call to
Kest
or one of its relatives.
The algorithm fits the Neyman-Scott Cluster point process
with Variance Gamma kernel to X
,
by finding the parameters of the model
which give the closest match between the
theoretical function of the model
and the observed
function.
For a more detailed explanation of the Method of Minimum Contrast,
see
mincontrast
.
The Neyman-Scott cluster point process with Variance Gamma
kernel is described in Jalilian et al (2013).
It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
have a common distribution described in Jalilian et al (2013).
The shape of the kernel is determined by the dimensionless
index nu
. This is the parameter
appearing in
equation (12) on page 126 of Jalilian et al (2013).
In previous versions of spatstat instead of specifying
nu
(called nu.ker
at that time) the user could specify
nu.pcf
which is the parameter
appearing in equation (13), page 127 of Jalilian et al (2013).
These are related by
nu.pcf = 2 * nu.ker + 1
and nu.ker = (nu.pcf - 1)/2
. This syntax is still supported but
not recommended for consistency across the package. In that case
exactly one of nu.ker
or nu.pcf
must be specified.
If the argument lambda
is provided, then this is used
as the value of the point process intensity .
Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The corresponding model can be simulated using rVarGamma
.
The parameter eta
appearing in startpar
is equivalent to the
scale parameter omega
used in rVarGamma
.
Homogeneous or inhomogeneous Neyman-Scott/VarGamma models can also be
fitted using the function kppm
and the fitted models
can be simulated using simulate.kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley [email protected].
Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
vargamma.estpcf
,
lgcp.estK
,
thomas.estK
,
cauchy.estK
,
mincontrast
,
Kest
,
Kmodel
.
rVarGamma
to simulate the model.
if(interactive()) { u <- vargamma.estK(redwood) print(u) plot(u) }
if(interactive()) { u <- vargamma.estK(redwood) print(u) plot(u) }
Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
vargamma.estpcf(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
vargamma.estpcf(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL, q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs = list())
X |
Data to which the model will be fitted. Either a point pattern or a summary statistic. See Details. |
startpar |
Vector of starting values for the parameters of the model. |
nu |
Numerical value controlling the shape of the tail of the clusters.
A number greater than |
lambda |
Optional. An estimate of the intensity of the point process. |
q , p
|
Optional. Exponents for the contrast criterion. |
rmin , rmax
|
Optional. The interval of |
... |
Optional arguments passed to |
pcfargs |
Optional list containing arguments passed to |
This algorithm fits the Neyman-Scott Cluster point process model with Variance Gamma kernel (Jalilian et al, 2013) to a point pattern dataset by the Method of Minimum Contrast, using the pair correlation function.
The argument X
can be either
An object of class "ppp"
representing a point pattern dataset.
The pair correlation function of the point pattern will be computed
using pcf
, and the method of minimum contrast
will be applied to this.
An object of class "fv"
containing
the values of a summary statistic, computed for a point pattern
dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to
pcf
or one of its relatives.
The algorithm fits the Neyman-Scott Cluster point process
with Variance Gamma kernel to X
,
by finding the parameters of the model
which give the closest match between the
theoretical pair correlation function of the model
and the observed pair correlation function.
For a more detailed explanation of the Method of Minimum Contrast,
see mincontrast
.
The Neyman-Scott cluster point process with Variance Gamma
kernel is described in Jalilian et al (2013).
It is a cluster process formed by taking a
pattern of parent points, generated according to a Poisson process
with intensity , and around each parent point,
generating a random number of offspring points, such that the
number of offspring of each parent is a Poisson random variable with mean
, and the locations of the offspring points of one parent
have a common distribution described in Jalilian et al (2013).
The shape of the kernel is determined by the dimensionless
index nu
. This is the parameter
appearing in
equation (12) on page 126 of Jalilian et al (2013).
In previous versions of spatstat instead of specifying
nu
(called nu.ker
at that time) the user could specify
nu.pcf
which is the parameter
appearing in equation (13), page 127 of Jalilian et al (2013).
These are related by
nu.pcf = 2 * nu.ker + 1
and nu.ker = (nu.pcf - 1)/2
. This syntax is still supported but
not recommended for consistency across the package. In that case
exactly one of nu.ker
or nu.pcf
must be specified.
If the argument lambda
is provided, then this is used
as the value of the point process intensity .
Otherwise, if
X
is a
point pattern, then
will be estimated from
X
.
If X
is a summary statistic and lambda
is missing,
then the intensity cannot be estimated, and
the parameter
will be returned as
NA
.
The remaining arguments rmin,rmax,q,p
control the
method of minimum contrast; see mincontrast
.
The corresponding model can be simulated using rVarGamma
.
The parameter eta
appearing in startpar
is equivalent to the
scale parameter omega
used in rVarGamma
.
Homogeneous or inhomogeneous Neyman-Scott/VarGamma models can also be
fitted using the function kppm
and the fitted models
can be simulated using simulate.kppm
.
The optimisation algorithm can be controlled through the
additional arguments "..."
which are passed to the
optimisation function optim
. For example,
to constrain the parameter values to a certain range,
use the argument method="L-BFGS-B"
to select an optimisation
algorithm that respects box constraints, and use the arguments
lower
and upper
to specify (vectors of) minimum and
maximum values for each parameter.
An object of class "minconfit"
. There are methods for printing
and plotting this object. It contains the following main components:
par |
Vector of fitted parameter values. |
fit |
Function value table (object of class |
Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley [email protected]
Jalilian, A., Guan, Y. and Waagepetersen, R. (2013) Decomposition of variance for spatial Cox processes. Scandinavian Journal of Statistics 40, 119-137.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous Neyman-Scott processes. Biometrics 63, 252–258.
kppm
,
vargamma.estK
,
lgcp.estpcf
,
thomas.estpcf
,
cauchy.estpcf
,
mincontrast
,
pcf
,
pcfmodel
.
rVarGamma
to simulate the model.
u <- vargamma.estpcf(redwood) u plot(u, legendpos="topright")
u <- vargamma.estpcf(redwood) u plot(u, legendpos="topright")
Returns the variance-covariance matrix of the estimates of the parameters of a fitted cluster point process model.
## S3 method for class 'kppm' vcov(object, ..., what=c("vcov", "corr", "fisher"), fast = NULL, rmax = NULL, eps.rmax = 0.01, verbose = TRUE)
## S3 method for class 'kppm' vcov(object, ..., what=c("vcov", "corr", "fisher"), fast = NULL, rmax = NULL, eps.rmax = 0.01, verbose = TRUE)
object |
A fitted cluster point process model (an object of class
|
... |
Ignored. |
what |
Character string (partially-matched)
that specifies what matrix is returned.
Options are |
fast |
Logical specifying whether tapering (using sparse matrices from Matrix) should be used to speed up calculations. Warning: This is expected to underestimate the true asymptotic variances/covariances. |
rmax |
Optional. The dependence range. Not usually specified by the
user. Only used when |
eps.rmax |
Numeric. A small positive number which is used to determine |
verbose |
Logical value indicating whether to print progress reports during very long calculations. |
This function computes the asymptotic variance-covariance
matrix of the estimates of the canonical (regression) parameters in the
cluster point process model object
. It is a method for the
generic function vcov
.
The result is an n * n
matrix where n =
length(coef(model))
.
To calculate a confidence interval for a regression parameter,
use confint
as shown in the examples.
A square matrix.
Abdollah Jalilian and Rasmus Waagepetersen. Ported to spatstat by Adrian Baddeley [email protected] and Ege Rubak [email protected].
Waagepetersen, R. (2007) Estimating functions for inhomogeneous spatial point processes with incomplete covariate data. Biometrika 95, 351–363.
fit <- kppm(redwood ~ x + y) vcov(fit) vcov(fit, what="corr") # confidence interval confint(fit) # cross-check the confidence interval by hand: sd <- sqrt(diag(vcov(fit))) t(coef(fit) + 1.96 * outer(sd, c(lower=-1, upper=1)))
fit <- kppm(redwood ~ x + y) vcov(fit) vcov(fit, what="corr") # confidence interval confint(fit) # cross-check the confidence interval by hand: sd <- sqrt(diag(vcov(fit))) t(coef(fit) + 1.96 * outer(sd, c(lower=-1, upper=1)))
Given a fitted multiple point process model, calculate the variance-covariance matrix of the parameter estimates.
## S3 method for class 'mppm' vcov(object, ..., what="vcov", err="fatal")
## S3 method for class 'mppm' vcov(object, ..., what="vcov", err="fatal")
object |
A multiple point process model (object of class |
... |
Arguments recognised by |
what |
Character string indicating which quantity should be calculated.
Options include |
err |
Character string indicating what action to take if an error occurs.
Either |
This is a method for the generic function vcov
.
The argument object
should be a fitted multiple point process
model (object of class "mppm"
) generated by mppm
.
The variance-covariance matrix of the parameter estimates is computed using asymptotic theory for maximum likelihood (for Poisson processes) or estimating equations (for other Gibbs models).
If what="vcov"
(the default), the variance-covariance matrix
is returned.
If what="corr"
, the variance-covariance matrix is normalised
to yield a correlation matrix, and this is returned.
If what="fisher"
, the Fisher information matrix is returned instead.
In all three cases, the rows and columns of the matrix correspond
to the parameters (coefficients) in the same order as in
coef{model}
.
If errors or numerical problems occur, the
argument err
determines what will happen. If
err="fatal"
an error will occur. If err="warn"
a warning will be issued and NA
will be returned.
If err="null"
, no warning is issued, but NULL
is returned.
A numeric matrix (or NA
or NULL
).
An error message that reports
system is computationally singular indicates that the
determinant of the Fisher information matrix of one of the models
was either too large or too small for reliable numerical calculation.
See vcov.ppm
for suggestions on how to handle this.
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
fit <- mppm(Wat ~x, data=hyperframe(Wat=waterstriders)) vcov(fit)
fit <- mppm(Wat ~x, data=hyperframe(Wat=waterstriders)) vcov(fit)
Returns the variance-covariance matrix of the estimates of the parameters of a fitted point process model.
## S3 method for class 'ppm' vcov(object, ..., what = c("vcov", "corr", "fisher"), verbose = TRUE, fine=FALSE, gam.action=c("warn", "fatal", "silent"), matrix.action=c("warn", "fatal", "silent"), logi.action=c("warn", "fatal", "silent"), nacoef.action=c("warn", "fatal", "silent"), hessian=FALSE)
## S3 method for class 'ppm' vcov(object, ..., what = c("vcov", "corr", "fisher"), verbose = TRUE, fine=FALSE, gam.action=c("warn", "fatal", "silent"), matrix.action=c("warn", "fatal", "silent"), logi.action=c("warn", "fatal", "silent"), nacoef.action=c("warn", "fatal", "silent"), hessian=FALSE)
object |
A fitted point process model (an object of class |
... |
Ignored. |
what |
Character string (partially-matched)
that specifies what matrix is returned.
Options are |
fine |
Logical value indicating whether to use a quick estimate
( |
verbose |
Logical. If |
gam.action |
String indicating what to do if |
matrix.action |
String indicating what to do if the matrix is ill-conditioned (so that its inverse cannot be calculated). |
logi.action |
String indicating what to do if |
nacoef.action |
String indicating what to do if some of the fitted coefficients
are |
hessian |
Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of the Fisher information. |
This function computes the asymptotic variance-covariance
matrix of the estimates of the canonical parameters in the
point process model object
. It is a method for the
generic function vcov
.
object
should be an object of class "ppm"
, typically
produced by ppm
.
The canonical parameters of the fitted model object
are the quantities returned by coef.ppm(object)
.
The function vcov
calculates the variance-covariance matrix
for these parameters.
The argument what
provides three options:
what="vcov"
return the variance-covariance matrix of the parameter estimates
what="corr"
return the correlation matrix of the parameter estimates
what="fisher"
return the observed Fisher information matrix.
In all three cases, the result is a square matrix.
The rows and columns of the matrix correspond to the canonical
parameters given by coef.ppm(object)
. The row and column
names of the matrix are also identical to the names in
coef.ppm(object)
.
For models fitted by the Berman-Turner approximation (Berman and Turner, 1992;
Baddeley and Turner, 2000) to the maximum pseudolikelihood (using the
default method="mpl"
in the call to ppm
), the implementation works
as follows.
If the fitted model object
is a Poisson process,
the calculations are based on standard asymptotic theory for the maximum
likelihood estimator (Kutoyants, 1998).
The observed Fisher information matrix of the fitted model
object
is first computed, by
summing over the Berman-Turner quadrature points in the fitted model.
The asymptotic variance-covariance matrix is calculated as the
inverse of the
observed Fisher information. The correlation matrix is then obtained
by normalising.
If the fitted model is not a Poisson process (i.e. it is some other Gibbs point process) then the calculations are based on Coeurjolly and Rubak (2012). A consistent estimator of the variance-covariance matrix is computed by summing terms over all pairs of data points. If required, the Fisher information is calculated as the inverse of the variance-covariance matrix.
For models fitted by the Huang-Ogata method (method="ho"
in
the call to ppm
), the implementation uses the
Monte Carlo estimate of the Fisher information matrix that was
computed when the original model was fitted.
For models fitted by the logistic regression approximation to the
maximum pseudolikelihood (method="logi"
in the call to
ppm
),
Calculations are based on Baddeley et al. (2013). A consistent estimator of the variance-covariance matrix is computed by summing terms over all pairs of data points. If required, the Fisher information is calculated as the inverse of the variance-covariance matrix.
The calculations depend on the type of dummy pattern used when the model was fitted:
currently only the dummy types
"stratrand"
(the default), "binomial"
and "poisson"
as
generated by quadscheme.logi
are supported.
For other dummy types the behavior depends on the argument
logi.action
. If logi.action="fatal"
an error is
produced. Otherwise, for dummy types
"grid"
and "transgrid"
the formulas for
"stratrand"
are used which in many cases should be
conservative. For an arbitrary, user-specified dummy pattern (type
"given"
), the formulas for "poisson"
are used which in
many cases should be conservative. If logi.action="warn"
a
warning is issued, otherwise the calculation proceeds without a
warning.
The result of the calculation is random (i.e. not
deterministic) when dummy type is "stratrand"
(the default)
because some of the variance terms are estimated by random sampling.
This can be avoided by specifying
dummytype='poisson'
or dummytype='binomial'
in the
call to ppm
when the model is fitted.
The argument verbose
makes it possible to suppress some
diagnostic messages.
The asymptotic theory is not correct if the model was fitted using
gam
(by calling ppm
with use.gam=TRUE
).
The argument gam.action
determines what to do in this case.
If gam.action="fatal"
, an error is generated.
If gam.action="warn"
, a warning is issued and the calculation
proceeds using the incorrect theory for the parametric case, which is
probably a reasonable approximation in many applications.
If gam.action="silent"
, the calculation proceeds without a
warning.
If hessian=TRUE
then the negative Hessian (second derivative)
matrix of the log pseudolikelihood, and its inverse, will be computed.
For non-Poisson models, this is not a valid estimate of variance,
but is useful for other calculations.
Note that standard errors and 95% confidence intervals for
the coefficients can also be obtained using
confint(object)
or coef(summary(object))
.
A square matrix.
An error message that reports system is computationally singular indicates that the determinant of the Fisher information matrix was either too large or too small for reliable numerical calculation.
If this message occurs, try repeating the calculation
using fine=TRUE
.
Singularity can occur because of numerical overflow or collinearity in the covariates. To check this, rescale the coordinates of the data points and refit the model. See the Examples.
In a Gibbs model, a singular matrix may also occur if the fitted model is a hard core process: this is a feature of the variance estimator.
Original code for Poisson point process was written by Adrian Baddeley [email protected] and Rolf Turner [email protected]. New code for stationary Gibbs point processes was generously contributed by Ege Rubak [email protected] and Jean-Francois Coeurjolly. New code for generic Gibbs process written by Adrian Baddeley [email protected]. New code for logistic method written by Ege Rubak [email protected].
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Coeurjolly, J.-F. and Rubak, E. (2013) Fast covariance estimation for innovations computed from a spatial Gibbs point process. Scandinavian Journal of Statistics 40 669–684.
Kutoyants, Y.A. (1998) Statistical Inference for Spatial Poisson Processes, Lecture Notes in Statistics 134. New York: Springer 1998.
vcov
for the generic,
ppm
for information about fitted models,
confint
for confidence intervals.
X <- rpoispp(42) fit <- ppm(X ~ x + y) vcov(fit) vcov(fit, what="Fish") # example of singular system m <- ppm(demopat ~polynom(x,y,2)) try(v <- vcov(m)) # rescale x, y coordinates to range [0,1] x [0,1] approximately demopatScale <- rescale(demopat, 10000) m <- ppm(demopatScale ~ polynom(x,y,2)) v <- vcov(m) # Gibbs example fitS <- ppm(swedishpines ~1, Strauss(9)) coef(fitS) sqrt(diag(vcov(fitS)))
X <- rpoispp(42) fit <- ppm(X ~ x + y) vcov(fit) vcov(fit, what="Fish") # example of singular system m <- ppm(demopat ~polynom(x,y,2)) try(v <- vcov(m)) # rescale x, y coordinates to range [0,1] x [0,1] approximately demopatScale <- rescale(demopat, 10000) m <- ppm(demopatScale ~ polynom(x,y,2)) v <- vcov(m) # Gibbs example fitS <- ppm(swedishpines ~1, Strauss(9)) coef(fitS) sqrt(diag(vcov(fitS)))
Returns the variance-covariance matrix of the estimates of the parameters of a point process model that was fitted by spatial logistic regression.
## S3 method for class 'slrm' vcov(object, ..., what=c("vcov", "corr", "fisher", "Fisher"))
## S3 method for class 'slrm' vcov(object, ..., what=c("vcov", "corr", "fisher", "Fisher"))
object |
A fitted point process model of class |
... |
Ignored. |
what |
Character string (partially-matched)
that specifies what matrix is returned.
Options are |
This function computes the asymptotic variance-covariance
matrix of the estimates of the canonical parameters in the
point process model object
. It is a method for the
generic function vcov
.
object
should be an object of class "slrm"
, typically
produced by slrm
. It represents a Poisson point process
model fitted by spatial logistic regression.
The canonical parameters of the fitted model object
are the quantities returned by coef.slrm(object)
.
The function vcov
calculates the variance-covariance matrix
for these parameters.
The argument what
provides three options:
what="vcov"
return the variance-covariance matrix of the parameter estimates
what="corr"
return the correlation matrix of the parameter estimates
what="fisher"
return the observed Fisher information matrix.
In all three cases, the result is a square matrix.
The rows and columns of the matrix correspond to the canonical
parameters given by coef.slrm(object)
. The row and column
names of the matrix are also identical to the names in
coef.slrm(object)
.
Note that standard errors and 95% confidence intervals for
the coefficients can also be obtained using
confint(object)
or coef(summary(object))
.
Standard errors for the fitted intensity can be obtained
using predict.slrm
.
A square matrix.
An error message that reports system is computationally singular indicates that the determinant of the Fisher information matrix was either too large or too small for reliable numerical calculation. This can occur because of numerical overflow or collinearity in the covariates.
Adrian Baddeley [email protected]
and Rolf Turner [email protected] .
Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K.,
Schuhmacher, D., Shah, R. and Turner, R. (2010)
Spatial logistic regression and change-of-support
for spatial Poisson point processes.
Electronic Journal of Statistics
4, 1151–1201.
DOI: 10.1214/10-EJS581
vcov
for the generic,
slrm
for information about fitted models,
predict.slrm
for other kinds of calculation about the model,
confint
for confidence intervals.
X <- rpoispp(42) fit <- slrm(X ~ x + y) vcov(fit) vcov(fit, what="corr") vcov(fit, what="f")
X <- rpoispp(42) fit <- slrm(X ~ x + y) vcov(fit) vcov(fit, what="corr") vcov(fit, what="f")
Given a spatial object (such as a point pattern or pixel image) in two dimensions, these functions extract the window in which the object is defined.
## S3 method for class 'ppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'kppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'dppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'slrm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'msr' Window(X, ...)
## S3 method for class 'ppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'kppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'dppm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'slrm' Window(X, ..., from=c("points", "covariates")) ## S3 method for class 'msr' Window(X, ...)
X |
A spatial object. |
... |
Ignored. |
from |
Character string. See Details. |
These are methods for the generic function Window
which extract the spatial window in which the object X
is defined.
The argument from
applies when X
is a fitted
two-dimensional point process model
(object of class "ppm"
, "kppm"
, "slrm"
or "dppm"
).
If from="data"
(the default),
Window
extracts the window of the original point
pattern data to which the model was fitted.
If from="covariates"
then Window
returns the
window in which the spatial covariates of the model were provided.
An object of class "owin"
(see owin.object
)
specifying an observation window.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
Window
,
Window.ppp
,
Window.psp
.
A <- ppm(cells ~ 1) Window(A)
A <- ppm(cells ~ 1) Window(A)
An expression involving the names of components of a measure is evaluated.
## S3 method for class 'msr' with(data, expr, ...)
## S3 method for class 'msr' with(data, expr, ...)
data |
A measure (object of class |
expr |
An expression to be evaluated. |
... |
Ignored. |
This is a method for the generic function
with
for the class "msr"
.
The argument data
should be an object of class "msr"
representing a measure (a function which assigns a value to each
subset of two-dimensional space).
This function can be used to extract the components of the measure, or to perform more complicated manipulations of the components.
The argument expr
should be an un-evaluated expression
in the R language. The expression may involve any of the variable
names listed below with their corresponding meanings.
qlocations |
(point pattern) all quadrature locations |
qweights |
(numeric) all quadrature weights |
density |
(numeric) density value at each quadrature point |
discrete |
(numeric) discrete mass at each quadrature point |
continuous |
(numeric) increment of continuous component |
increment |
(numeric) increment of measure |
is.atom |
(logical) whether quadrature point is an atom |
atoms |
(point pattern) locations of atoms |
atommass |
(numeric) massess of atoms |
The measure is the sum of discrete and continuous components. The discrete component assigns non-zero mass to several points called atoms. The continuous component has a density which should be integrated over a region to determine the value for that region.
An object of class "msr"
approximates the continuous component
by a sum over quadrature points. The quadrature points are chosen
so that they include the atoms of the measure. In the list above,
we have increment = continuous + discrete
,
continuous = density * qweights
,
is.atom = (discrete > 0)
,
atoms = qlocations[is.atom]
and
atommass = discrete[is.atom]
.
The result of evaluating the expression could be an object of any kind.
Adrian Baddeley [email protected], Rolf Turner [email protected] and Ege Rubak [email protected].
msr
,
split.msr
,
measureContinuous
, measurePositive
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") with(rp, atoms) with(rp, qlocations %mark% continuous)
X <- rpoispp(function(x,y) { exp(3+3*x) }) fit <- ppm(X, ~x+y) rp <- residuals(fit, type="pearson") with(rp, atoms) with(rp, qlocations %mark% continuous)
Experimental code. Creates an object representing a cluster point process model. Typically used for theoretical calculations about such a model.
zclustermodel(name = "Thomas", ..., mu, kappa, scale)
zclustermodel(name = "Thomas", ..., mu, kappa, scale)
name |
Name of the cluster process. One of
|
... |
Other arguments needed for the model. |
mu |
Mean cluster size. A single number, or a pixel image. |
kappa |
Parent intensity. A single number. |
scale |
Cluster scale parameter of the model. |
Experimental.
Object of the experimental class "zclustermodel"
.
Adrian Baddeley [email protected]
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1)
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1)
Experimental code. Creates an object representing a Gibbs point process model. Typically used for theoretical calculations about such a model.
zgibbsmodel(beta = 1, interaction = NULL, icoef = NULL)
zgibbsmodel(beta = 1, interaction = NULL, icoef = NULL)
beta |
First order trend term. A numeric value, numeric vector, pixel image, function, or a list of such objects. |
interaction |
Object of class |
icoef |
Numeric vector of coefficients for the interpoint interaction. |
Experimental.
Object belonging to the experimental class zgibbsmodel
.
Adrian Baddeley [email protected].
m <- zgibbsmodel(10, Strauss(0.1), -0.5)
m <- zgibbsmodel(10, Strauss(0.1), -0.5)