Gibbs point process models with mixed effects

Abstract

SUMMARYWe consider spatial point patterns that have been observed repeatedly in the same area at several points in time. We take a maximum pseudolikelihood approach (Besag, 1976) to parameter estimation in the context of Gibbs processes (Stoyan et al., 1995;Illian et al., 2008). More specifically, we discuss pair-wise interaction processes where the conditional intensity has a log-linear form and extend existing models by expressing the intensity and the interaction terms in the pseudolikelihood as a sum of fixed and random effects, where the latter accounts for variation over time. We initially derive a Strauss process model with mixed effects. As this model is too simplistic in the given context, we further consider a more general model that allows for inter-group differences in intensity and interaction strength and has a more flexible interaction function. We apply the approximate Berman-Turner device (Baddeley and Turner, 2000) to a generalised linear mixed model with log link and Poisson outcome rather than a simple generalised linear model. Estimates are obtained using existing software for generalised linear mixed models based on penalised quasi-likelihood methods (Bresow and Clayton, 1993).The approach is applied to a data set detailing the spatial locations of different types of muskoxen herds in a fixed area in Greenland at different points in time within several years (Meltofte and Berg, 2004).

A spatio-temporal Poisson hurdle point process to model wildfires

“…To treat this kind of problems, Cox processes are widely used. In particular, Log Gaussian Cox processes (LGCP), which define a class of flexible models are particularly useful in the context of modelling aggregation relative to some underlying unobserved environmental field ( [22]; [41]) and they are characterised by their intensity surface being modeled as (3.9) log ( ) = ( ) where ( ) is a Gaussian random field.…”

“…For instance, the work by [4] has facilitated the routine fitting of point processes based on an approximation of the pseudolikelihood to avoid the issue of intractable normalizing constants ( [5]) through the use of the library spatstat for R ( [4]). In the same way, ( [22]) consider hierarchical models able to analyse a wide variety of point process models, for example those appearing in fire problems.…”

A Bayesian Approach to Fitting Gibbs Processes with Temporal Random Effects

“…We set an interaction radius for the discs centred on each point to be R = 125. This is slightly larger than the interaction radius used by Illian and Hendrichsen (2010), who use 2R = 200 and consider a pairwise interaction function. The specification of the slightly larger radius here is to ensure there are observed overlaps between at least two herds within each dataset.…”

“…These can be of interest in themselves, for example, in identifying which datasets have a positive/negative temporal random effect component within the intensity function and/or comparing the magnitude of the random effects between datasets to investigate the strongest deviation from the underlying mean. In addition, and in contrast to the previous approach proposed by Illian and Hendrichsen (2010), we are able to easily obtain posterior credible intervals for each of the parameters providing a measure of the precision of the parameters of interest.…”

“…We follow Illian and Hendrichsen (2010) who propose a mixed effects model resulting from replicated spatial point patterns, allowing for a temporal random effect component within the model combined with a fixed effects environmental component. In particular, they re-express the conditional likelihood so that it is equivalent to the log-likelihood of independent weighted Poisson variables using the Berman-Turner device.…”