Metrics and barycenters for point pattern data

TL;DR

This paper introduces new metrics for finite point patterns, focusing on barycenters as summary statistics, with algorithms and applications demonstrating their effectiveness in geospatial data analysis.

Contribution

It proposes the transport-transform and relative transport-transform metrics, along with a heuristic algorithm for barycenter computation, unifying and extending previous point pattern metrics.

Findings

The algorithms are fast and reliable in simulations.

Barycenters effectively summarize geocoded crime data.

The framework generalizes existing point pattern metrics.

Abstract

We introduce the transport-transform (TT) and the relative transport-transform (RTT) metrics between finite point patterns on a general space, which provide a unified framework for earlier point pattern metrics, in particular the generalized spike time and the normalized and unnormalized OSPA metrics. Our main focus is on barycenters, i.e. minimizers of a $q$-th order Fr\'echet functional with respect to these metrics. We present a heuristic algorithm that terminates in a local minimum and is shown to be fast and reliable in a simulation study. The algorithm serves as an umbrella method that can be applied on any state space where an appropriate algorithm for solving the location problem for individual points is available. We present applications to geocoded data of crimes in Euclidean space and on a street network, illustrating that barycenters serve as informative summary statistics. Our work is a first step towards statistical inference in covariate-based models of repeated point pattern observations.