Approximate Bayesian Computation and Model Assessment for Repulsive Spatial Point Processes

ABSTRACT

In many applications involving spatial point patterns, we find evidence of inhibition or repulsion. The most commonly used class of models for such settings are the Gibbs point processes. A recent alternative, at least to the statistical community, is the determinantal point process. Here, we examine model fitting and inference for both of these classes of processes in a Bayesian framework. While usual MCMC model fitting can be available, the algorithms are complex and are not always well behaved. We propose using approximate Bayesian computation (ABC) for such fitting. This approach becomes attractive because, though likelihoods are very challenging to work with for these processes, generation of realizations given parameter values is relatively straightforward. As a result, the ABC fitting approach is well-suited for these models. In addition, such simulation makes them well-suited for posterior predictive inference as well as for model assessment. We provide details for all of the above along with some simulation investigation and an illustrative analysis of a point pattern of tree data exhibiting repulsion. R code and datasets are included in the supplementary material.

KEYWORDS:

• Determinantal point processes
• Gibbs point processes
• Inefficiency factors
• Model checking
• Summary statistics

Acknowledgments

The work of the first author was supported in part by the Nakajima Foundation. The authors thank James Clark for providing the Duke Forest dataset.

Notes

1 Fearnhead and Prangle (2012) implemented linear regression for each component of θ. Since we have a small number of parameters, we keep the notation as linear regression for multivariate responses.

2 In the simulation examples, we considered only an interaction radius. However, with real data, often we find hardcore repulsion with a very small radius or moderate repulsion with a larger radius.