Advances in Applied Probability

Transforming spatial point processes into Poisson processes using random superposition

Jesper Møller and Kasper K. Berthelsen

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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.

Article information

Source
Adv. in Appl. Probab., Volume 44, Number 1 (2012), 42-62.

Dates
First available in Project Euclid: 8 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.aap/1331216644

Digital Object Identifier
doi:10.1239/aap/1331216644

Mathematical Reviews number (MathSciNet)
MR2951546

Zentralblatt MATH identifier
1239.60034

Subjects
Primary: 60G55: Point processes 62M99: None of the above, but in this section
Secondary: 62M30: Spatial processes 66C60 62H11: Directional data; spatial statistics

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

Citation

Møller, Jesper; Berthelsen, Kasper K. Transforming spatial point processes into Poisson processes using random superposition. Adv. in Appl. Probab. 44 (2012), no. 1, 42--62. doi:10.1239/aap/1331216644. https://projecteuclid.org/euclid.aap/1331216644


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