The Multivariate Generalised von Mises distribution: Inference and applications

TL;DR

This paper introduces the multivariate Generalised von Mises distribution for circular data, extending probabilistic models like Gaussian processes and PCA to the circular domain, enabling efficient inference and learning.

Contribution

It presents a new multivariate distribution for circular variables and develops circular regression and PCA models with efficient inference methods.

Findings

The mGvM distribution generalizes previous circular distributions.

Proposed models leverage standard covariance functions and relevance determination.

Efficient variational inference schemes are developed for these models.

Abstract

Circular variables arise in a multitude of data-modelling contexts ranging from robotics to the social sciences, but they have been largely overlooked by the machine learning community. This paper partially redresses this imbalance by extending some standard probabilistic modelling tools to the circular domain. First we introduce a new multivariate distribution over circular variables, called the multivariate Generalised von Mises (mGvM) distribution. This distribution can be constructed by restricting and renormalising a general multivariate Gaussian distribution to the unit hyper-torus. Previously proposed multivariate circular distributions are shown to be special cases of this construction. Second, we introduce a new probabilistic model for circular regression, that is inspired by Gaussian Processes, and a method for probabilistic principal component analysis with circular hidden variables. These models can leverage standard modelling tools (e.g. covariance functions and methods for automatic relevance determination). Third, we show that the posterior distribution in these models is a mGvM distribution which enables development of an efficient variational free-energy scheme for performing approximate inference and approximate maximum-likelihood learning.