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Generalised shot noise Cox processes

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

Published online by Cambridge University Press:  01 July 2016

Jesper Møller  and
Giovanni Luca Torrisi
Jesper Møller*
Affiliation:
Aalborg University
Giovanni Luca Torrisi*
Affiliation:
CNR-Istituto per le Applicazioni del Calcolo ‘M. Picone’
*
∗ Postal address: Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg, Denmark. Email address: jm@math.auc.dk
∗ Postal address: CNR-Istituto per le Applicazioni del Calcolo ‘M. Picone’, Viale del Policlinico 137, I-00161 Rome, Italy. Email address: torrisi@iac.rm.cnr.it
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Abstract

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We introduce a class of Cox cluster processes called generalised shot noise Cox processes (GSNCPs), which extends the definition of shot noise Cox processes (SNCPs) in two directions: the point process that drives the shot noise is not necessarily Poisson, and the kernel of the shot noise can be random. Thereby, a very large class of models for aggregated or clustered point patterns is obtained. Due to the structure of GSNCPs, a number of useful results can be established. We focus first on deriving summary statistics for GSNCPs and, second, on how to simulate such processes. In particular, results on first- and second-order moment measures, reduced Palm distributions, the J-function, simulation with or without edge effects, and conditional simulation of the intensity function driving a GSNCP are given. Our results are exemplified in important special cases of GSNCPs, and we discuss their relation to the corresponding results for SNCPs.

Keywords

Cluster processconditional simulationCox processedge effectgeneralised shot noise G Cox processgeometric ergodicityMarkov point processMetropolis-Hastings algorithmPalm distributionperfect simulationshot noise Cox processspatial point processsummary statistic

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes 62M30: Spatial processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 68U20: Simulation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 37 , Issue 1 , March 2005 , pp. 48 - 74
Copyright
Copyright © Applied Probability Trust 2005 

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