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Inhomogeneous spatial point processes by location-dependent scaling

Part of: Inference from stochastic processes Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Ute Hahn ,
Eva B. Vedel Jensen ,
Marie-Colette van Lieshout  and
Linda Stougaard Nielsen
Ute Hahn*
Affiliation:
University of Aarhus
Eva B. Vedel Jensen*
Affiliation:
University of Aarhus
Marie-Colette van Lieshout*
Affiliation:
CWI, Amsterdam
Linda Stougaard Nielsen*
Affiliation:
University of Aarhus
*
∗ Current address: Department of Mathematics, Augsburg University, D-86135 Augsburg, Germany. Email address: hahn@math.uni-augsburg.de
∗∗ Postal address: Laboratory for Computational Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
∗∗∗ Postal address: CWI, PO Box 94079, 1090 GB Amsterdam, The Netherlands.
∗∗ Postal address: Laboratory for Computational Stochastics, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark.
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Abstract

A new class of models for inhomogeneous spatial point processes is introduced. These locally scaled point processes are modifications of homogeneous template point processes, having the property that regions with different intensities differ only by a scale factor. This is achieved by replacing volume measures used in the density with locally scaled analogues defined by a location-dependent scaling function. The new approach is particularly appealing for modelling inhomogeneous Markov point processes. Distance-interaction and shot noise weighted Markov point processes are discussed in detail. It is shown that the locally scaled versions are again Markov and that locally the Papangelou conditional intensity of the new process behaves like that of a global scaling of the homogeneous process. Approximations are suggested that simplify calculation of the density, for example, in simulation. For sequential point processes, an alternative and simpler definition of local scaling is proposed.

Keywords

Point processinhomogeneitylocal scalingsequential point process

MSC classification

Primary: 60G55: Point processes
Secondary: 62M30: Spatial processes 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 319 - 336
Copyright
Copyright © Applied Probability Trust 2003 

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