Bayesian inference for Matérn repulsive processes

In many applications involving point pattern data, the Poisson process assumption is unrealistic, with the data exhibiting a more regular spread. Such 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 Matérn type III repulsive process. We consider some extensions of the original Matérn type III process for both the homogeneous and the inhomogeneous cases. We also derive the probability density of this generalized Matérn process, allowing us to characterize the conditional distribution of the various latent variables, and leading to a novel and efficient Markov chain Monte Carlo algorithm. We apply our ideas to data sets of spatial locations of trees, nerve fibre cells and Greyhound bus stations.