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
Famous quotes containing the words higher and/or dimensional:
“I remain just one thing, and one thing only—and that is a clown. It places me on a far higher plane than any politician.”
—Charlie Chaplin (1889–1977)
“I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.”
—Kristin Hunter (b. 1931)