Metrics and barycenters for point pattern data

Abstract

We introduce the transport–transform and the relative transport–transform metrics between finite point patterns on a generalspace, which provide a unified framework for earlier point pattern metrics, in particular the generalized spike time and thenormalized and unnormalized optimal subpattern assignment metrics. Our main focus is on barycenters, i.e., minimizersof aq-th-order Fréchet functional with respect to these metrics. We present a heuristic algorithm that terminates in a localminimum and is shown to be fast and reliable in a simulation study. The algorithm serves as a general plug-in method that canbe applied to point patterns on any state space where an appropriate algorithm for solving the location problem for individualpoints is available. We present applications to geocoded data of crimes in Euclidean space and on a street network, illustratingthat barycenters serve as informative summary statistics. Our work is a first step toward statistical inference in covariate-basedmodels of repeated point pattern observations.