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Generalised shot noise Cox processes

Published online by Cambridge University Press:  01 July 2016

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: [email protected]
Postal address: CNR-Istituto per le Applicazioni del Calcolo ‘M. Picone’, Viale del Policlinico 137, I-00161 Rome, Italy. Email address: [email protected]
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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.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2005 

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