On the spherical Laplace distribution

TL;DR

This paper introduces the spherical Laplace distribution as a robust alternative to the von Mises-Fisher distribution for modeling directional data, especially in the presence of outliers, with new estimation methods and clustering applications.

Contribution

The paper proposes the spherical Laplace distribution, providing a novel robust model for directional data, along with sampling, estimation algorithms, and clustering methods.

Findings

Validated parameter estimation methods through simulations.

Demonstrated robustness to outliers in real data.

Showed improved clustering performance with the new model.

Abstract

The von Mises-Fisher (vMF) distribution has long been a mainstay for inference with data on the unit hypersphere in directional statistics. The performance of statistical inference based on the vMF distribution, however, may suffer when there are significant outliers and noise in the data. Based on an analogy of the median as a robust measure of central tendency and its relationship to the Laplace distribution, we proposed the spherical Laplace (SL) distribution, a novel probability measure for modelling directional data. We present a sampling scheme and theoretical results on maximum likelihood estimation. We derive efficient numerical routines for parameter estimation in the absence of closed-form formula. An application of model-based clustering is considered under the finite mixture model framework. Our numerical methods for parameter estimation and clustering are validated using simulated and real data experiments.