**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.

Read more about this topic: Directional Statistics

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