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Transforming Spatial Point Processes into Poisson Processes Using Random Superposition

Part of: Inference from stochastic processes Stochastic processes Multivariate analysis

Published online by Cambridge University Press:  04 January 2016

Jesper Møller  and
Kasper K. Berthelsen
Jesper Møller*
Affiliation:
Aalborg University
Kasper K. Berthelsen*
Affiliation:
Aalborg University
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg Øst, Denmark.
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg Øst, Denmark.
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Abstract

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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 X ⋃ Y 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.

Keywords

Complementary point processcouplinglocal stabilitymodel checkingPapangelou conditional intensityspatial birth-death processStrauss process

MSC classification

Primary: 60G55: Point processes 62M99: None of the above, but in this section
Secondary: 62M30: Spatial processes 62H11: Directional data; spatial statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 44 , Issue 1 , March 2012 , pp. 42 - 62
Copyright
© Applied Probability Trust 

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