Bayesian inference for Mat\'ern repulsive processes

TL;DR

This paper develops a Bayesian inference framework for Matérn type-III repulsive point processes, providing flexible models and efficient algorithms for analyzing spatial data with regular patterns.

Contribution

It introduces a generalized Matérn type-III process, derives its probability density, and develops a novel MCMC algorithm for inference in both homogeneous and inhomogeneous cases.

Findings

Effective modeling of spatial point patterns with repulsion.

Successful application to real datasets of trees, nerve cells, and bus stations.

Enhanced understanding of spatial regularity in various applications.

Abstract

In many applications involving point pattern data, the Poisson process assumption is unrealistic, with the data exhibiting a more regular spread. Such a repulsion between events is exhibited by trees for example, because of competition for light and nutrients. Other examples include the locations of biological cells and cities, and the times of neuronal spikes. Given the many applications of repulsive point processes, there is a surprisingly limited literature developing flexible, realistic and interpretable models, as well as efficient inferential methods. We address this gap by developing a modelling framework around the Mat\'ern type-III repulsive process. We consider a number of extensions of the original Mat\'ern type-III process for both the homogeneous and inhomogeneous cases. We also derive the probability density of this generalized Mat\'ern process. This allows us to characterize the posterior distribution of the various latent variables, and leads to a novel and efficient Markov chain Monte Carlo algorithm. We apply our ideas to datasets involving the spatial locations of trees, nerve fiber cells and Greyhound bus stations.