Directional Statistics - Distributions On Higher Dimensional Manifolds

Distributions On Higher Dimensional Manifolds

There also exist distributions on the two-dimensional sphere (such as the Kent distribution), the N-dimensional sphere (the Von Mises-Fisher distribution) or the torus (the bivariate von Mises distribution).

The Von Mises–Fisher distribution is a distribution on the Stiefel manifold, and can be used to construct probability distributions over rotation matrices.

The Bingham distribution is a distribution over axes in N dimensions, or equivalently, over points on the (N − 1)-dimensional sphere with the antipodes identified. For example, if N = 2, the axes are undirected lines through the origin in the plane. In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For N = 4, the Bingham distribution is a distribution over the space of unit quaternions. Since a unit quaternion corresponds to a rotation matrix, the Bingham distribution for N = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution.

These distributions are for example used in geology, crystallography and bioinformatics.