Cluster analysis of spatial point patterns: posterior distribution of parents inferred from offspring

This manuscript demonstrates an empirical Bayesian estimation of cluster centers (parents) in spatial point pattern. We use the Thomas models on a plane, namely, Neyman–Scott process of a two-dimensional Gaussian-type or its extension to the multi-type species. This manuscript firstly confirms restoration method; namely, the numbers of parents and their locations are estimated from simulated datasets of Thomas process or its multi-type process; and the Metropolis’ simulation of parent locations are well performed from their posterior distributions. Then, we further apply the models to three real datasets of plant ecology, volcano swarms, and inland shallow micro-earthquakes. However, in the latter two cases, a preliminary analysis by the pairwise correlation suggests that the classical second-order moment relationship between the ordinary Thomas process and the Palm intensity cannot be applicable. Therefore, alternative method is implemented by applying a Bayesian selection of the optimal number of parents; which is searched by maximizing the marginal likelihood (integrated posterior) by numerically performing the high-dimensional integration with respective to the location coordinates of the parents. Then, the optimal posterior distribution of the selected model provides the image of likely locations of parents.