# ppm.ppp

0th

Percentile

##### Fit Point Process Model to Point Pattern Data

Fits a point process model to an observed point pattern.

Keywords
models, spatial
##### Usage
# S3 method for ppp
ppm(Q, trend=~1, interaction=Poisson(),
…,
covariates=data,
data=NULL,
covfunargs = list(),
subset,
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,
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)
##### Arguments
Q

A data point pattern (of class "ppp") to which the model will be fitted, or a quadrature scheme (of class "quad") containing this pattern.

trend

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.

interaction

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.

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 x and y and the names of entries in data) defining a subset of the spatial domain, to which the model-fitting should be restricted. 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").

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 "border" indicating the border correction. Other possibilities may include "Ripley", "isotropic", "periodic", "translate" and "none", depending on the interaction.

rbord

If correction = "border" this argument specifies the distance by which the window should be eroded for the border correction.

use.gam

Logical flag; if TRUE then computations are performed using gam instead of glm.

method

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.

forcefit

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.

emend,project

(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.

prior.mean

Optional vector of prior means for canonical parameters (for method="VBlogi"). See Details.

prior.var

Optional prior variance covariance matrix for canonical parameters (for method="VBlogi"). See Details.

nd

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.

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.

gcontrol

Optional. List of parameters passed to glm.control (or passed to gam.control if use.gam=TRUE) controlling the model-fitting algorithm.

nsim

Number of simulated realisations to generate (for method="ho")

nrmh

Number of Metropolis-Hastings iterations for each simulated realisation (for method="ho")

start,control

Arguments passed to rmh controlling the behaviour of the Metropolis-Hastings algorithm (for method="ho")

verb

Logical flag indicating whether to print progress reports (for method="ho")

callstring

Internal use only.

##### Details

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.

basic use:

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
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, Hybrid, LennardJones, 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.

Models with covariates:

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

a pixel image,

giving the values of a spatial covariate at a fine grid of locations. It should be an object of class "im", see im.object.

a function,

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.

a window,

interpreted as a logical variable which is TRUE inside the window and FALSE outside it. This should be an object of class "owin".

a tessellation,

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".

a single number,

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 $i$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).

Model-fitting technique:

There are several choices for the technique used to fit the model.

method="mpl"

(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.

method="logi":

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.

method="VBlogi":

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.

method="ho":

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.

Note that method='logi', method='VBlogi' and method='ho' involve randomisation, so that the results are subject to random variation.

Huang-Ogata method:

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

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".

##### Value

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.

##### Interaction parameters

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:

regular parameters:

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.]

irregular parameters

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.

##### Error and Warning Messages

Some common error messages and warning messages are listed below, with explanations.

“System is computationally singular”

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.

“Covariate values were NA or undefined at X% (M out of N) of the quadrature points”

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.

“Model is unidentifiable”

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.

“N data points are illegal (zero conditional intensity)”

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.

##### Warnings

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.

##### References

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, Hybrid, LennardJones, 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.

• ppm.ppp
##### Examples
# NOT RUN {
# fit the stationary Poisson process
# to point pattern 'nztrees'

ppm(nztrees)
ppm(nztrees ~ 1)

# }
# NOT RUN {
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
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 {
# }
`
Documentation reproduced from package spatstat, version 1.52-1, License: GPL (>= 2)

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