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Thinning spatial point processes into Poisson processes

Part of: Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Frederic Paik Schoenberg
Jesper Møller*
Affiliation:
Aalborg University
Frederic Paik Schoenberg*
Affiliation:
University of California
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark.
∗∗ Postal address: Department of Statistics, 8125 Math-Science Building, University of California, Los Angeles, CA 90095-1554, USA. Email address: [email protected]
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Abstract

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In this paper we describe methods for randomly thinning certain classes of spatial point processes. In the case of a Markov point process, the proposed method involves a dependent thinning of a spatial birth-and-death process, where clans of ancestors associated with the original points are identified, and where we simulate backwards and forwards in order to obtain the thinned process. In the case of a Cox process, a simple independent thinning technique is proposed. In both cases, the thinning results in a Poisson process if and only if the true Papangelou conditional intensity is used, and, thus, can be used as a graphical exploratory tool for inspecting the goodness-of-fit of a spatial point process model. Several examples, including clustered and inhibitive point processes, are considered.

Keywords

Area-interaction point processclans of ancestorscouplingCox processdependent and independent thinningMarkov point processPapangelou conditional intensityPoisson processspatial birth-and-death processThomas process

MSC classification

Primary: 60G55: Point processes 62M30: Spatial processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 347 - 358
Copyright
Copyright © Applied Probability Trust 2010 

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