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 Matérn type-III repulsive process. We consider a number of extensions of the original Matérn type-III process for both the homogeneous and inhomogeneous cases. We also derive the probability density of this generalized Matérn 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.
@article{rao2016matern,
year = {2016},
author = {Rao, Vinayak and Adams, Ryan P. and Dunson, David B.},
title = {Bayesian Inference for Mat\'{e}rn Repulsive Processes},
journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
volume = {79},
number = {3},
pages = {877--897},
note = {arXiv:1308.1136 [stat.ME]},
keywords = {Bayesian methods, Matern processes, Markov chain Monte Carlo}
}