ppm: Fit Point Process Model to Data

Description

Fits a point process model to an observed point pattern

Usage

ppm(Q, trend=~1, interaction=Poisson(),
       ...,
       covariates=NULL,
       correction="border",
       rbord=reach(interaction),
       use.gam=FALSE,
       method="mpl",
       forcefit=FALSE,
       nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
       verb=TRUE)

Arguments

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 Rformula 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 NULL indicating that a Poisson process (stationary or nonstationary) should be fitted.

...

Ignored.

covariates

The values of any spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list of images. See Details.

correction

The name of the edge correction to be used. The default is "border" indicating the border correction. Other possibilities may include "Ripley", "isotropic", "translate" and "none"

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, 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.

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")

Value

An object of class "ppm" describing a fitted point process model. The fitted parameters can be obtained just by printing this object. Fitted spatial trends can be extracted using the predict method for this object (see predict.ppm).
See ppm.object for details of the format of this object.

Warnings

The implementation of the Huang-Ogata method is experimental. 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. The current version of ppm maximises the pseudolikelihood without constraining the parameters, and does not apply any checks for sanity after fitting the model. 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.

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.

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

References

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.

Examples

Run this code

data(nztrees)
 ppm(nztrees)
 # fit the stationary Poisson process
 # to point pattern 'nztrees'

 Q <- quadscheme(nztrees) 
 ppm(Q) 
 # equivalent.

 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

 ppm(nztrees, ~ polynom(x,2))
 # fit the nonstationary Poisson process 
 # with intensity function lambda(x,y) = exp(a + bx + cx^2)

 library(splines)
 ppm(nztrees, ~ bs(x,df=3))
 #       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
 
 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.

# Huang-Ogata fit:
ppm(nztrees, ~1, Strauss(r=10), method="ho")
<testonly>ppm(nztrees, ~1, Strauss(r=10), method="ho", nsim=10)</testonly>


 # COVARIATES
 #
 X <- rpoispp(42)
 weirdfunction <- function(x,y){ 10 * x^2 + runif(length(x))}
 Zimage <- as.im(weirdfunction, unit.square())
 #
 # (a) covariate values in pixel image
 ppm(X, ~ y + Z, covariates=list(Z=Zimage))
 #
 # (b) 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


 ## MULTITYPE POINT PROCESSES ### 
 data(lansing)
 # Multitype point pattern --- trees marked by species
<testonly># equivalent functionality - smaller dataset
 data(betacells)</testonly>

 # fit stationary marked Poisson process
 # with different intensity for each species
ppm(lansing, ~ marks, Poisson())
<testonly>ppm(betacells, ~ marks, Poisson())</testonly>

 # fit nonstationary marked Poisson process
 # with different log-cubic trend for each species
ppm(lansing, ~ marks * polynom(x,y,3), Poisson())
<testonly>ppm(betacells, ~ marks * polynom(x,y,2), Poisson())</testonly>

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