# A Bayesian approach to fitting Gibbs processes with temporal random effects

### Ruth King, Janine B. Illian, Stuart E. King, Glenna F. Evans and Ditte K. Hendrichsen

### University of St. Andrews

## Summary

We consider spatial point pattern data that have been observed repeatedly over a period of
time in an inhomogeneous environment. Each spatial point pattern can be regarded as a "snapshot" of the underlying point process at a series of times. Thus, the number of points and
corresponding locations of points differ for each snapshot. Each snapshot can be analysed independently, but in many cases, there may be little information in the data relating to model parameters, particularly parameters relating to the interaction between points. Thus, we develop an integrated approach, simultaneously analysing all snapshots within a single analysis. We assume that sufficient time has passed between observation dates (here approximately one year) so that each spatial point pattern can be regarded as independent replicates, given
spatial covariates. We develop a joint mixed effects Gibbs point process model for the
replicates of spatial point patterns by considering environmental covariates in the analysis as fixed effects, to model the heterogeneous environment, with a random effect component to account for the different observation days for the intensity function. We demonstrate how the model can be fitted within a Bayesian framework using an auxiliary variable approach to deal with the issue of the random effects component. We apply the methods to a dataset of muskoxen herds and demonstrate the increased precision of the parameter estimates when considering all available data within a single integrated analysis.

### Keywords:

Data augmentation; Markov chain Monte Carlo; Mixed effects model; Muskoxen data; Spatial and temporal point processes