Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

Abstract

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable functionβdefined on the space for the points of the process. It is possible to superpose a locally stable spatial point processXwith a complementary spatial point processYto obtain a Poisson processX⋃Ywith intensity functionβ. Underlying this is a bivariate spatial birth-death process (Xt,Yt) which converges towards the distribution of (X,Y). We study the joint distribution ofXandY, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure forYconditional onX. This may be used for model checking: given a model for the Papangelou intensity of the original spatial point process, this model is used to generate the complementary process, and the resulting superposition is a Poisson process with intensity functionβif and only if the true Papangelou intensity is used. Whether the superposition is actually such a Poisson process can easily be examined using well-known results and fast simulation procedures for Poisson processes. We illustrate this approach to model checking in the case of a Strauss process.