A Bayesian Approach to Fitting Gibbs Processes with Temporal Random Effects

Abstract

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 robust and consistent analysis. We assume that sufficient time has passed between observation dates so that the spatial point patterns 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 (or hierarchical) 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.

A method for analysing replicated point patterns in ecology

“…For example, King et al . () used a Bayesian approach to model replicated point patterns of muskoxen locations as realizations of a Gibbs process and showed that combining data in a single integrated analysis increased the precision of parameter estimation. Illian et al .…”

“…Although replicated point patterns are considerably more robust to departures from homogeneity than individual analyses (Diggle, Lange & Benes 1991), accounting for heterogeneity through the use of environmental covariates and methods for the analysis of inhomogeneous point processes (Baddeley, Moller & Waagepetersen 2000) may increase the power of the methods and would provide an obvious next step. In addition to the nonparametric approaches to summarizing spatial structure described here, several parametric approaches are being developed (Baddeley & Turner 2000;King et al 2012;Illian, Sørbye & Rue 2013). Bayesian approaches through the use of Markov chain Monte Carlo (MCMC) methods or the integrated nested Laplace approximation (INLA, Rue, Martino & Chopin 2009) for model fitting may provide options for parametric analysis of replicated point patterns.…”

“…Such parametric approaches to modelling variation in spatial structure due to covariates may lead to advances in power and performance, at least when the models are well specified (Diggle, Mateu & Clough 2000). For example, King et al (2012) used a Bayesian approach to model replicated point patterns of muskoxen locations as realizations of a Gibbs process and showed that combining data in a single integrated analysis increased the precision of parameter estimation. considered replicated point pattern models using marked Cox process models fitting these with the INLA approach.…”

“…) and parametric models (King et al . ; Illian et al . ) have all been used to analyse spatial patterns in ecology.…”

“…Computer software to analyse spatial patterns is widely available, with the R-package spatstat (Baddeley & Turner 2005) and the stand-alone software Programita (Wiegand & Moloney 2004, 2014 particularly popular. Nearest-neighbour distances (Hubbell 1979), Ripley's K-function and summary statistics related to it, like the pair correlation function and L-function (Condit et al 2000;Bagchi et al 2011), spectral analyses (Detto & Muller-Landau 2013;Harrison et al 2013) and parametric models (King et al 2012;) have all been used to analyse spatial patterns in ecology.…”