Gibbs point process models with mixed effects

• Published in: Environmetrics
• Main Authors: Illian, Janine B., Hendrichsen, Ditte K.
• Format: Article in Journal/Newspaper
• Language: English
• Published: Wiley 2009
• Subjects: Clayton; Greenland; Strauss
• Online Access: http://dx.doi.org/10.1002/env.1008; https://api.wiley.com/onlinelibrary/tdm/v1/articles/10.1002%2Fenv.1008; https://onlinelibrary.wiley.com/doi/pdf/10.1002/env.1008

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.