Gibbs point process models with mixed effects

• Published in: Environmetrics (London, Ont.) Ont.), 2010-05, Vol.21 (3-4), p.341-353
• Main Authors: Illian, Janine B., Hendrichsen, Ditte K.
• Format: Article
• Language: English
• Subjects: Berman-Turner device; glmm; mixed effects; replicated spatial patterns; spatial point process models
• ISSN: 1180-4009; 1099-095X
• DOI: 10.1002/env.1008

We 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). Copyright © 2009 John Wiley & Sons, Ltd.