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 $\bar{d}_1$ 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 $\bar{d}_1$ and its induced Wasserstein metric $\bar{d}_2$ for point process distributions are given, including examples of useful $\bar{d}_1$-Lipschitz continuous functions, $\bar{d}_2$ upper bounds for Poisson process approximation, and $\bar{d}_2$ upper and lower bounds between distributions of point processes of i.i.d. points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of $\bar{d}_1$ in applications.