Robust Comparison of Kernel Densities on Spherical Domains

TL;DR

This paper introduces a spectral framework for comparing spherical probability densities that is robust to kernel choices, bandwidths, and sample sizes, enabling fairer population comparisons in directional data analysis.

Contribution

The authors develop a novel spectral method that characterizes spherical densities by smoothness and derives a distance measure that is invariant to kernel and sample size variations.

Findings

Framework effectively compares spherical densities across different sample sizes.

Spectral representation captures density smoothness via Laplacian eigenfunctions.

Method demonstrates improved robustness and performance in simulations and real data.

Abstract

While spherical data arises in many contexts, including in directional statistics, the current tools for density estimation and population comparison on spheres are quite limited. Popular approaches for comparing populations (on Euclidean domains) mostly involvea two-step procedure: (1) estimate probability density functions (pdfs) from their respective samples, most commonly using the kernel density estimator, and, (2) compare pdfs using a metric such as the L2 norm. However, both the estimated pdfs and their differences depend heavily on the chosen kernels, bandwidths, and sample sizes. Here we develop a framework for comparing spherical populations that is robust to these choices. Essentially, we characterize pdfs on spherical domains by quantifying their smoothness. Our framework uses a spectral representation, with densities represented by their coefficients with respect to the eigenfunctions of the Laplacian operator on a sphere. The change in smoothness, akin to using different kernel bandwidths, is controlled by exponential decays in coefficient values. Then we derive a proper distance for comparing pdf coefficients while equalizing smoothness levels, negating influences of sample size and bandwidth. This signifies a fair and meaningful comparisons of populations, despite vastly different sample sizes, and leads to a robust and improved performance. We demonstrate this framework using examples of variables on S1 and S2, and evaluate its performance using a number of simulations and real data experiments.