Gibbs point process models with mixed effects

Authors

  • Janine B. Illian,

    Corresponding author
    1. CREEM, School of Mathematics and Statistics, University of St Andrews, Buchanan Gardens, KY16 9LZ, Scotland, UK
    • CREEM, School of Mathematics and Statistics, University of St Andrews, Buchanan Gardens, KY16 9LZ, Scotland UK.
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  • Ditte K. Hendrichsen

    1. Department of Population Biology, Institute of Biology, University of Copenhagen, Universitetsparken 15, DK-2100 Copenhagen East, Denmark
    2. Department of Arctic Environment, National Environmental Research Institute, University of Århus, Frederiksborgvej 399, DK-4000 Roskilde, Denmark
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Abstract

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.

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