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A new metric between distributions of point processes

Part of:Stochastic processesLimit theoremsInference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Dominic Schuhmacher and
Aihua Xia
Dominic Schuhmacher*
Affiliation:
The University of Western Australia
Aihua Xia*
Affiliation:
The University of Melbourne
*
∗ Current address: Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, CH-3012 Bern, Switzerland. Email address: dominic.schuhmacher@stat.unibe.ch
∗∗ Postal address: Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia. Email address: xia@ms.unimelb.edu.au
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Abstract

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Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

Keywords

Wasserstein metricpoint processPoisson point processStein's methoddistributional approximationstatistical analysis of point pattern data

MSC classification

Primary:60G55: Point processes
Secondary:60F05: Central limit and other weak theorems62M30: Spatial processes

Information

Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 40 , Issue 3 , September 2008, pp. 651 - 672
Copyright
Copyright © Applied Probability Trust 2008 

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