Robust Comparison of Kernel Densities on Spherical Domains

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 involve a two-step procedure: (1) estimate probability density functions (pdf s) from their respective samples, most commonly using the kernel density estimator, and (2) compare pdf s using a metric such as the $\mathbb {L}^{2}$ norm. However, both the estimated pdf s 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 pdf s 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 $\mathbb {S}^{1}$ and $\mathbb {S}^{2}$ , and evaluate its performance using a number of simulations and real data experiments.