Bingham distribution explained

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.^[1] It is a generalization of the Watson distribution and a special case of the Kent and Fisher–Bingham distributions.

The Bingham distribution is widely used in paleomagnetic data analysis,^[2] and has been used in the field of computer vision.^{[3] [4] [5]}

Its probability density function is given by

f( x ;M,Z) dS ^{n-1} = {} _{1} F _{1} \left( \tfrac 12 ; \tfrac n 2 ; Z \right) ^{-1} ⋅ \exp \left( \operatorname{tr} Z M ^{T} x x ^{T} M \right) dS ^{n-1} which may also be writtenf( x ;M,Z) dS ^{n-1} = {} _{1} F _{1} \left( \tfrac 12 ; \tfrac n 2 ;Z \right) ^{-1} ⋅ \exp\left( x ^{T} M Z M ^{T} x \right) dS ^{n-1}

where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and

{} _{1} F _{1} ( ⋅ ; ⋅ , ⋅ ) is a confluent hypergeometric function of matrix argument. The matrices M and Z are the result of diagonalizing the positive-definite covariance matrix of the Gaussian distribution that underlies the Bingham distribution.

See also

• Directional statistics
• von Mises–Fisher distribution
• Kent distribution

Notes and References

1. Bingham, Ch. (1974) "An antipodally symmetric distribution on the sphere". Annals of Statistics, 2(6):1201–1225.
2. Onstott, T.C. (1980) "Application of the Bingham distribution function in paleomagnetic studies". Journal of Geophysical Research, 85:1500–1510.
3. S. Teller and M. Antone (2000). Automatic recovery of camera positions in Urban Scenes
4. Book: Springer. 2008. 10.1007/978-3-540-88690-7_58. Computer Vision – ECCV 2008. 5304. 780–791. Lecture Notes in Computer Science. Haines. Tom S. F.. Wilson. Richard C.. 978-3-540-88689-1. 15488343 .
5. Web site: Better robot vision: A neglected statistical tool could help robots better understand the objects in the world around them.. MIT News. October 7, 2013. October 7, 2013.