Homogeneous Poisson Processes

 

 

Figure 1:  Hom. Poisson process

The simplest stochastic model for a planar point pattern is the homogeneous Poisson process. A realization of a homogeneous Poisson process is given in Figure 1. The idea of this model is that the point events of interest occur completely independently of each other. This lack of interaction between points is called complete spatial randomness by some authors; see e.g. Diggle (1996). There are various books where Poisson processes on the plane and in higher-dimensional spaces are studied; see e.g. Cox and Isham (1980), Daley and Vere-Jones (1988), Karr (1991), Kendall et al. (1998), Kingman (1993), Serfozo (1990), Snyder and Miller (1991), Stoyan et al. (1995), Stoyan and Stoyan (1995).

For many years, almost all of the available methods for statistical analysis of planar point patterns were based on the assumption that the points in question form a realization of a planar homogeneous Poisson process. Although nowadays there exists a large variety of alternative point process models, the homogeneous Poisson process still provides a basic reference model against which to compare other models where effects of clustering or regularity (repulsion, inhibition) of points are incorporated into the model, either because of some feature (heterogeneity) of the underlying environment, or because of interactions between the points themselves.