Von Mises-Fisher Elliptical Distribution
This work addresses the challenge of modeling non-symmetric data in probabilistic learning systems, offering a more practical alternative to existing complex methods, though it appears incremental as it builds on elliptical distributions.
The authors tackled the problem of modeling real-world skewed data by proposing a von-Mises-Fisher (vMF) distribution to create an explicit and simple representation for skewed elliptical distributions, which is shown to be easy to generate and stable to estimate.
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.