Introduction

In this chapter, we introduce four additional point process models: cluster processes, hard-core processes, Cox processes, and Gibbs processes. Poisson point processes exhibit complete spatial randomness due to their independence property. Cox processes model less regular spatial distributions – they are overdispersed relative to PPPs, which means that the ratio of the variance of the number of nodes in a set to its mean is larger than 1. Gibbs processes, on the other hand, may be overdispersed or underdispersed.

Figure 3.1 illustrates the two directions along which a point process model may depart from the point of complete spatial randomness, the PPP. In terms of the J function introduced in Definition 2.40, regular processes have J values larger than 1, since the probability of having a nearby neighbor is small; conversely, clustered processes have J values smaller than 1.

General finite point processes

A general finite point process is a generalization of the binomial point process to a process where the total number of points is itself a random variable. Compared with a random vector, there are three differences: The randomness of the number of points and the facts that the point process is unordered and simple and thus is better represented as a set.