Bayesian inference for Gaussian excursion set generated Cox processes with set-marking

Abstract

This work considers spatial Cox point processes where the random intensity is defined by a random closed set such that different point intensities appear in the two phases formed by the random set and its complement. The point pattern is observed as a set of point coordinates in a bounded region W⊂ℝ^d together with the information on the phase of the location of each point. This phase information, called set-marking, is not a representative sample from the random set, and hence it cannot be directly used for deducing properties of the random set. Excursion sets of continuous-parameter Gaussian random fields are applied as a flexible model for the random set. Fully Bayesian method and Markov chain Monte Carlo (MCMC) simulation is adopted for inferring the parameters of the model and estimating the random set. The performance of the new approach is studied by means of simulation experiments. Further, two forestry data sets on point patterns of saplings are analysed. The saplings grow in a clear-cut forest area where, before planting and natural seeding, the soil has been mounded forming a blotched soil structure. The tree densities tend to be different in the tilled patches and in the area outside the patches. The coordinates of each sapling have been measured and it is known whether this location is in a patch or outside. This example has been a motivation for the study.