Bayesian Estimation and Segmentation of Spatial Point Processes Using Voronoi Tilings
In this chapter we consider Bayesian estimation and segmentation for nonhomogeneous Poisson point processes in two dimensions. This work turns out to be easily generalizable to higher dimensions and to have similarities with work done in one dimension. The principal motivation for the methods considered here can be found in
Muise and Smith (1992), Dasgupta and Raftery (1998), Byers and Raftery (1998) and Allard and Fraley (1997). The simplest case is the segmentation of a point process of inhomogeneous rate, for instance the isolation of regions of high density in a point process in the plane, or in a higher dimensional space. Estimation of the rate is also an issue, one that might be intimately bound up with the segmentation as the segmentation methods might serve as a (semi-)parametric method of rate estimation. In Dasgupta and Raftery (1998) the assumption is made that the regions
of high density are formed by a mixture of Gaussian distributions on a background Poisson process and model-based clustering Banﬁeld and Raftery (1993) is used to segment the data. This method is particularly suited to ﬁnding very linear features or those approximated by linear forms. In Byers and Raftery (1998) and Allard and Fraley (1997) the point process is assumed to be one of piecewise constant rate, with only two distinct rates being present. The related methods of kth nearest neighbors and Voronoi tile areas, respectively, are used to segment the data. All of these methods provide some form of classiﬁcation of the events in
the process into high or low density regions. The aim of this paper is to explore methods of gaining more information, such as a region-based estimator with uncertainties and other complex posterior probabilities. The Voronoi tile area based method provides a region but this can be quite crude, and has no uncertainties. Fully Bayesian extensions to the Gaussian cluster approach with solutions via MCMC were considered in Bensmail
et al. (1997). We intend to provide the analogous exploration for the approaches in Byers and Raftery (1998) and Allard and Fraley (1997). Figure 6.1 shows some point process data from a mineﬁeld detection
problem that might be segmented into two parts, high density and low density. The second panel shows the data that are estimated to be in the higher intensity region on the basis of their posterior probabilities. Section 6.5 discusses this example further and compares other methods of segmentation.