Metrics and barycenters for point pattern data

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 Fréchet 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. MSC2010 Subject Classification: Primary 65C60; Secondary 62-07, 90B80.

ANOVA for Data in Metric Spaces, with Applications to Spatial Point Patterns

“…Although the methods described are applicable in general metric spaces, our central goal in undertaking this research was to be able to perform ANOVA for point pattern data, see also the discussion section of Müller et al (2020). Among all the metric spaces mentioned above, we therefore focus in the later part of the present paper on the space of finite point patterns equipped with the TT-metric from Müller et al (2020).…”

“…Although the methods described are applicable in general metric spaces, our central goal in undertaking this research was to be able to perform ANOVA for point pattern data, see also the discussion section of Müller et al (2020). Among all the metric spaces mentioned above, we therefore focus in the later part of the present paper on the space of finite point patterns equipped with the TT-metric from Müller et al (2020). As in many other spaces, exact Fréchet means can be computed within reasonable time only for (very) small data sets and one typically has to resort to a heuristic algorithm that finds only local minima of the Fréchet functional.…”

“…In Sections 6 and 7 we apply the four statistics from Table 1 for the space of finite point patterns equipped with the metric introduced in Müller et al (2020). For self-containedness we give a short summary of the relevant concepts and results, referring to the paper as MSM20.…”

“…The computation of the Fréchet T and T L depend on the calculation of a barycenter. For this we used the heuristic algorithm presented in Müller et al (2020). The calculation of an exact barycenter is computationally infeasible for this kind of data.…”

“…A common feature of the underlying spaces is that there is typically a more or less natural concept of distance between data points available. In addition to the more obvious choices of distances on function spaces and Riemannian manifolds, suitable metrics for tree spaces, graph spaces and point pattern spaces can be found in Billera et al (2001), Ginestet et al (2017) and Müller et al (2020), respectively.…”