ppm: Fit Point Process Model to Data

Description

Fits a point process model to an observed point pattern.

Usage

ppm(Q, …)
   # S3 method for formula
ppm(Q, interaction=NULL, …, data=NULL, subset)

Arguments

A formula in the R language describing the model 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.

…

Arguments passed to ppm.ppp or ppm.quad to control the model-fitting process.

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 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").

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.

Technical Warnings and Error Messages

See ppm.ppp for some technical warnings about the weaknesses of the algorithm, and explanation of some common error messages.

Details

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, Hybrid, LennardJones, 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 \(x\) and \(y\), 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

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 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.

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. (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.

Examples

Run this code

# NOT RUN {
 # fit the stationary Poisson process
 # to point pattern 'nztrees'

 ppm(nztrees ~ 1)

 
# }
# NOT RUN {
 Q <- quadscheme(nztrees) 
 ppm(Q ~ 1) 
 # equivalent.
 
# }
# 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 Variational Bayes 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 + 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: inside/outside window
 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
# }
# 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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