Cartesian Statistics on Spheres

TL;DR

This paper develops nonparametric methods for estimating and comparing functionals of directional and axial data distributions on spheres, using Cartesian coordinate representations to overcome algebraic complexities.

Contribution

It introduces nonparametric estimators and bootstrap confidence sets for directional/axial functionals based on Cartesian coordinates, enabling more flexible analysis.

Findings

Provides nonparametric estimators for directional/axial means and dispersions.

Develops bootstrap confidence sets for these functionals.

Facilitates comparison and trend analysis of mean directions in spherical data.

Abstract

Directional data consists of unit vectors in q-dimensions that can be described in polar or Cartesian coordinates. Axial data can be viewed as a pair of directions pointed in opposite directions or as a projection matrix of rank 1. Historically, their statistical analysis has largely been based on a few low-order exponential family models of distributions for random directions or axes. A lack of tractable algebraic forms for the normalizing constants has hindered the use of higher-order exponential families for less constrained modeling. Of interest are functionals of the unknown distribution of the directional/axial data, such as the directional/axial mean, dispersion, or distribution itself. This paper outlines nonparametric estimators and bootstrap confidence sets for such functionals. The procedures are based on the empirical distribution of the directional/axial sample expressed in Cartesian coordinates. Sketched as well are nonparametric comparisons among multiple mean directions or axes, estimation of trend in mean directions, and analysis of q-dimensional observations restricted to lie in a specified compact subset.