A new metric between distributions of point processes

TL;DR

This paper introduces a new metric for comparing distributions of point processes that effectively accounts for differences in total mass and point locations, improving upon existing metrics.

Contribution

The paper proposes the $ar{d}_1$ metric that combines positional and mass differences, along with theoretical properties and practical applications for point process distributions.

Findings

$ar{d}_1$ effectively handles total mass differences.

Provides bounds for Poisson process approximation.

Demonstrates a statistical test for point pattern data.

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.