Fits a point process model to an observed point pattern.
# S3 method for ppp
ppm(Q, trend=~1, interaction=Poisson(),
…,
covariates=data,
data=NULL,
covfunargs = list(),
subset,
clipwin,
correction="border",
rbord=reach(interaction),
use.gam=FALSE,
method="mpl",
forcefit=FALSE,
emend=project,
project=FALSE,
prior.mean = NULL,
prior.var = NULL,
nd = NULL,
eps = NULL,
gcontrol=list(),
nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
verb=TRUE,
callstring=NULL) # S3 method for quad
ppm(Q, trend=~1, interaction=Poisson(),
…,
covariates=data,
data=NULL,
covfunargs = list(),
subset,
clipwin,
correction="border",
rbord=reach(interaction),
use.gam=FALSE,
method="mpl",
forcefit=FALSE,
emend=project,
project=FALSE,
prior.mean = NULL,
prior.var = NULL,
nd = NULL,
eps = NULL,
gcontrol=list(),
nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
verb=TRUE,
callstring=NULL)
A data point pattern (of class "ppp")
to which the model will be fitted,
or a quadrature scheme (of class "quad")
containing this pattern.
An R formula object specifying the spatial trend to be fitted.
The default formula, ~1, indicates the model is stationary
and no trend is to be fitted.
An object of class "interact"
describing the point process interaction
structure, or a function that makes such an object,
or NULL indicating that a Poisson process (stationary
or nonstationary) should be fitted.
Ignored.
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.
Optional.
An expression (which may involve the names of the
Cartesian coordinates x and y
and the names of entries in data)
defining a subset of the spatial domain,
to which the likelihood or pseudolikelihood should be restricted.
See Details.
The result of evaluating the expression should be either a logical
vector, or a window (object of class "owin")
or a logical-valued pixel image (object of class "im").
Optional. A spatial window (object of class "owin")
to which data will be restricted, before model-fitting
is performed. See Details.
A named list containing the values of any additional arguments required by covariate functions.
The name of the edge correction to be used. The default
is "border" indicating the border correction.
Other possibilities may include "Ripley", "isotropic",
"periodic", "translate" and "none", depending on the
interaction.
If correction = "border"
this argument specifies the distance by which
the window should be eroded for the border correction.
Logical flag; if TRUE then computations are performed
using gam instead of glm.
The method used to fit the model. Options are
"mpl" for the method of Maximum PseudoLikelihood,
"logi" for the Logistic Likelihood method,
"VBlogi" for the Variational Bayes Logistic Likelihood method,
and "ho" for the Huang-Ogata approximate maximum likelihood
method.
Logical flag for internal use.
If forcefit=FALSE, some trivial models will be
fitted by a shortcut. If forcefit=TRUE,
the generic fitting method will always be used.
(These are equivalent: project is an older name for
emend.)
Logical value. Setting emend=TRUE will ensure that the
fitted model is always a valid point process by
applying emend.ppm.
Optional vector of prior means for canonical parameters (for
method="VBlogi"). See Details.
Optional prior variance covariance matrix for canonical parameters (for method="VBlogi"). See Details.
Optional. Integer or pair of integers.
The dimension of the grid of dummy points (nd * nd
or nd[1] * nd[2])
used to evaluate the integral in the pseudolikelihood.
Incompatible with 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 nd.
Optional. List of parameters passed to glm.control
(or passed to gam.control if use.gam=TRUE)
controlling the model-fitting algorithm.
Number of simulated realisations
to generate (for method="ho")
Number of Metropolis-Hastings iterations
for each simulated realisation (for method="ho")
Arguments passed to rmh controlling the behaviour
of the Metropolis-Hastings algorithm (for method="ho")
Logical flag indicating whether to print progress reports
(for method="ho")
Internal use only.
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 \(\phi\) is called regular if the log likelihood is a linear function of \(\theta\) where \(\theta = \theta(\psi)\) is some transformation of \(\psi\). [Then \(\theta\) 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 \(\gamma\) controlling the strength of interpoint inhibition, and an irregular parameter \(r\) 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 \(\gamma\) 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 Fisher information matrix of the fitted model has a
determinant close to zero, so that the matrix cannot be inverted,
and the software cannot calculate standard errors or confidence intervals.
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.
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 \(\gamma\)
is less than or equal to \(1\),
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, \(\beta\) 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 \(0\).
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.
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 |
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 \(x\) and \(y\),
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.
The first two arguments of the function should be the
Cartesian coordinates \(x\) and \(y\). 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.
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 \(i\)th row of covariates should contain the values of
spatial variables which have been observed
at the \(i\)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.
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 ith 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.
the model will be 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.
method='logi', method='VBlogi' and
method='ho' involve randomisation, so that the results are
subject to random variation.If method="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 \(\theta_0\).
From this initial model, nsim simulated
realisations are generated. The score and Fisher information of
the model at \(\theta=\theta_0\)
are estimated from the simulated realisations. Then one step
of the Fisher scoring algorithm is taken, yielding an updated
estimate \(\theta_1\). 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.
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.
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. http://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.
# NOT RUN {
# fit the stationary Poisson process
# to point pattern 'nztrees'
ppm(nztrees)
ppm(nztrees ~ 1)
# }
# NOT RUN {
Q <- quadscheme(nztrees)
ppm(Q)
# equivalent.
# }
# NOT RUN {
# }
# NOT RUN {
ppm(nztrees, nd=128)
# }
# NOT RUN {
# }
# NOT RUN {
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))
# }
# NOT RUN {
ppm(nztrees, ~ polynom(x,2))
# }
# NOT RUN {
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx + cx^2)
# }
# NOT RUN {
library(splines)
ppm(nztrees, ~ bs(x,df=3))
# }
# NOT RUN {
# WARNING: do not use predict.ppm() on this result
# Fits the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(B(x))
# where B is a B-spline with df = 3
# }
# NOT RUN {
ppm(nztrees, ~1, Strauss(r=10), rbord=10)
# }
# NOT RUN {
# Fit the stationary Strauss process with interaction range r=10
# using the border method with margin rbord=10
# }
# NOT RUN {
ppm(nztrees, ~ x, Strauss(13), correction="periodic")
# }
# NOT RUN {
# 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 VB fits:
# }
# NOT RUN {
ppm(swedishpines, ~1, Strauss(9))
# }
# NOT RUN {
# }
# NOT RUN {
ppm(swedishpines, ~1, Strauss(9), method="ho")
# }
# NOT RUN {
ppm(swedishpines, ~1, Strauss(9), method="VBlogi")
# COVARIATES
#
X <- rpoispp(42)
weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) }
#
# (a) covariate values as function
ppm(X, ~ y + Z, covariates=list(Z=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, covariates=data.frame(Z=Zvalues))
# Note Q not X
# COVARIATE FUNCTION WITH EXTRA ARGUMENTS
#
f <- function(x,y,a){ y - a }
ppm(X, ~x + f, covariates=list(f=f), covfunargs=list(a=1/2))
# COVARIATE: inside/outside window
b <- owin(c(0.1, 0.6), c(0.1, 0.9))
ppm(X, ~w, covariates=list(w=b))
## MULTITYPE POINT PROCESSES ###
# fit stationary marked Poisson process
# with different intensity for each species
# }
# NOT RUN {
ppm(lansing, ~ marks, Poisson())
# }
# NOT RUN {
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species
# }
# NOT RUN {
ppm(lansing, ~ marks * polynom(x,y,3), Poisson())
# }
# NOT RUN {
# }
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