Gibbs point process models with mixed effects

Illian, J. B. and Hendrichsen, D. K. (2010) Gibbs point process models with mixed effects. Environmetrics, 21(3-4), pp. 341-353. (doi:10.1002/env.1008)

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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).

Item Type:Articles
Additional Information:Special Issue: Spatio‐Temporal Stochastic Modelling: Environmental and Health Processes.
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Illian, Professor Janine
Authors: Illian, J. B., and Hendrichsen, D. K.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Statistics
Journal Name:Environmetrics
Publisher:Wiley
ISSN:1180-4009
ISSN (Online):1099-095X
Published Online:18 August 2009

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