Harmonic analysis and distribution-free inference for spherical distributions

TL;DR

This paper develops a harmonic analysis framework for spherical distributions, enabling model-free inference on 3D directional data, including estimation and symmetry testing, with applications demonstrated on sunspot activity data.

Contribution

It introduces the first harmonic analysis approach for spherical distributions, generalizing Fourier analysis to 3D, and provides distribution-free inference methods for symmetry and other properties.

Findings

Harmonic representations for many spherical models are derived.

A general family of spherical distributions is proposed, encompassing known models.

Distribution-free tests for symmetry and inference methods are developed.

Abstract

Fourier analysis and representation of circular distributions in terms of their Fourier coefficients, is quite commonly discussed and used for model-free inference such as testing uniformity and symmetry etc. in dealing with 2-dimensional directions. However a similar discussion for spherical distributions, which are used to model 3-dimensional directional data, has not been fully developed in the literature in terms of their harmonics. This paper, in what we believe is the first such attempt, looks at the probability distributions on a unit sphere, through the perspective of spherical harmonics, analogous to the Fourier analysis for distributions on a unit circle. Harmonic representations of many currently used spherical models are presented and discussed. A very general family of spherical distributions is then introduced, special cases of which yield many known spherical models. Through the prism of harmonic analysis, one can look at the mean direction, dispersion, and various forms of symmetry for these models in a generic setting. Aspects of distribution free inference such as estimation and large-sample tests for these symmetries, are provided. The paper concludes with a real-data example analyzing the longitudinal sunspot activity.