A new metric between distributions of point processes

Abstract

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.