Nonparametric Measure-Transportation-Based Methods for Directional Data

TL;DR

This paper introduces nonparametric, measure-transportation-based tools for analyzing directional data on the hypersphere, including new distribution and quantile functions, and develops distribution-free tests for uniformity and MANOVA with superior performance.

Contribution

It develops novel nonparametric methods based on optimal transport for directional data, including distribution functions, quantiles, and distribution-free tests, with theoretical guarantees and practical applications.

Findings

Empirical distribution functions satisfy Glivenko-Cantelli property.

New tests for uniformity and MANOVA outperform existing methods in simulations.

Real-data analysis of sunspots demonstrates practical utility.

Abstract

This paper proposes various nonparametric tools based on measure transportation for directional data. We use optimal transports to define new notions of distribution and quantile functions on the hypersphere, with meaningful quantile contours and regions and closed-form formulas under the classical assumption of rotational symmetry. The empirical versions of our distribution functions enjoy the expected Glivenko-Cantelli property of traditional distribution functions. They provide fully distribution-free concepts of ranks and signs and define data-driven systems of (curvilinear) parallels and (hyper)meridians. Based on this, we also construct a universally consistent test of uniformity and a class of fully distribution-free and universally consistent tests for directional MANOVA which, in simulations, outperform all their existing competitors. A real-data example involving the analysis of sunspots concludes the paper.