The Statistics of Circular Optimal Transport

TL;DR

This paper establishes a central limit theorem for empirical optimal transport distances on circular data, enabling advanced statistical inference like goodness-of-fit tests, with broad applicability and strong performance for unimodal distributions.

Contribution

It introduces a novel CLT for circular OT distances, including mild assumptions and bootstrap methods, facilitating statistical inference for circular distributions.

Findings

CLT for empirical OT on circular data derived

Bootstrap principle established for inference tasks

OT-based goodness-of-fit test performs well for unimodal distributions

Abstract

Empirical optimal transport (OT) plans and distances provide effective tools to compare and statistically match probability measures defined on a given ground space. Fundamental to this are distributional limit laws and we derive a central limit theorem for the empirical OT distance of circular data. Our limit results require only mild assumptions in general and include prominent examples such as the von Mises or wrapped Cauchy family. Most notably, no assumptions are required when data are sampled from the probability measure to be compared with, which is in strict contrast to the real line. A bootstrap principle follows immediately as our proof relies on Hadamard differentiability of the OT functional. This paves the way for a variety of statistical inference tasks and is exemplified for asymptotic OT based goodness of fit testing for circular distributions. We discuss numerical implementation, consistency and investigate its statistical power. For testing uniformity, it turns out that this approach performs particularly well for unimodal alternatives and is almost as powerful as Rayleigh's test, the most powerful invariant test for von Mises alternatives. For regimes with many modes the circular OT test is less powerful which is explained by the shape of the corresponding transport plan.