Transforming spatial point processes into Poisson processes using random superposition

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process, which we refer to as the target point process, with a complementary spatial point process so that to a Poisson process is obtained. Underlying this is a bivariate spatial birth-death process which converges in distribution to the joint distribution of the target point process and its socalled complementary point process.
We study this joint distribution and its marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for conditional distribution of the complementary point process given the target point process. This may be used for model checking: given a fitted model for the Papangelou intensity of the target point process, this model is used to generate the complementary process, and the resulting su- perposition is a Poisson process if 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.