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March 2012 Transforming spatial point processes into Poisson processes using random superposition
Jesper Møller, Kasper K. Berthelsen
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Adv. in Appl. Probab. 44(1): 42-62 (March 2012). DOI: 10.1239/aap/1331216644

Abstract

Most finite spatial point process models specified by a density are locally stable, implying that the Papangelou intensity is bounded by some integrable function β defined on the space for the points of the process. It is possible to superpose a locally stable spatial point process X with a complementary spatial point process Y to obtain a Poisson process XY with intensity function β. Underlying this is a bivariate spatial birth-death process (Xt, Yt) which converges towards the distribution of (X, Y). We study the joint distribution of X and Y, and their marginal and conditional distributions. In particular, we introduce a fast and easy simulation procedure for Y conditional on X. 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 function β 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.

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Jesper Møller. Kasper K. Berthelsen. "Transforming spatial point processes into Poisson processes using random superposition." Adv. in Appl. Probab. 44 (1) 42 - 62, March 2012. https://doi.org/10.1239/aap/1331216644

Information

Published: March 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1239.60034
MathSciNet: MR2951546
Digital Object Identifier: 10.1239/aap/1331216644

Subjects:
Primary: 60G55 , 62M99
Secondary: 62H11 , 62M30 , 66C60

Keywords: Complementary point process , coupling , local stability , model checking , Papangelou conditional intensity , spatial birth-death process , Strauss process

Rights: Copyright © 2012 Applied Probability Trust

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Vol.44 • No. 1 • March 2012
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