The Poisson Processes in Cluster Analysis

This paper aims to review some use of the point processes in cluster analysis. The homogeneous Poisson process is, in many ways, the simplest point process, and it plays a role in point process theory in most respects analogous to the normal distribution in the study of random variables. We first propose a statistical model for cluster analysis based on the homogeneous Poisson process. The clustering criterion is extracted from that model thanks to maximum likelihood estimation. It consists in minimizing the sum of the Lebesgue measures of the convex hulls of the clusters. We also present a generalization of that model to the non-stationary Poisson process, as well as some monothetic divisive clustering methods also based on the Poisson processes. On the other hand, it is usually considered that the central problem of cluster validation is the determination of the best number of natural clusters. We present two likelihood ratio tests for the number of clusters based on the Poisson processes. Most of these clustering methods and tests for the number of clusters have been extended to symbolic data.