Von Mises-Fisher Elliptical Distribution

TL;DR

This paper introduces a new approach using the von Mises-Fisher distribution to model skewed elliptical distributions, making them easier to estimate and more applicable to real-world, non-symmetric data in probabilistic learning systems.

Contribution

The paper proposes a simple, explicit representation of skewed elliptical distributions using the von Mises-Fisher distribution, improving estimation and applicability.

Findings

The vMF-based distribution is easy to generate.

It is stable to estimate both theoretically and empirically.

The approach generalizes symmetric distributions while maintaining key properties.

Abstract

A large class of modern probabilistic learning systems assumes symmetric distributions, however, real-world data tend to obey skewed distributions and are thus not always adequately modelled through symmetric distributions. To address this issue, elliptical distributions are increasingly used to generalise symmetric distributions, and further improvements to skewed elliptical distributions have recently attracted much attention. However, existing approaches are either hard to estimate or have complicated and abstract representations. To this end, we propose to employ the von-Mises-Fisher (vMF) distribution to obtain an explicit and simple probability representation of the skewed elliptical distribution. This is shown not only to allow us to deal with non-symmetric learning systems, but also to provide a physically meaningful way of generalising skewed distributions. For rigour, our extension is proved to share important and desirable properties with its symmetric counterpart. We also demonstrate that the proposed vMF distribution is both easy to generate and stable to estimate, both theoretically and through examples.