Bingham distribution

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 reported as being of use in the field of computer vision.[3][4][5]

Its probability density function is given by

${\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle {\frac {1}{2}}};{\textstyle {\frac {n}{2}}};Z)^{-1}\;\cdot \;\exp \left({{\textrm {tr}}\;ZM^{T}\mathbf {x} \mathbf {x} ^{T}M}\right)\;dS^{n-1}}$

which may also be written

${\displaystyle f(\mathbf {x} \,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle {\frac {1}{2}}};{\textstyle {\frac {n}{2}}};Z)^{-1}\;\cdot \;\exp \left({\mathbf {x} ^{T}MZM^{T}\mathbf {x} }\right)\;dS^{n-1}}$

where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and ${\displaystyle {}_{1}F_{1}(\cdot ;\cdot ,\cdot )}$ 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.

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. Haines, Tom S. F.; Wilson, Richard C. (2008). Computer Vision – ECCV 2008 (PDF). Lecture Notes in Computer Science. 5304. Springer. pp. 780–791. doi:10.1007/978-3-540-88690-7_58. ISBN 978-3-540-88689-1.
5. "Better robot vision: A neglected statistical tool could help robots better understand the objects in the world around them". MIT News. October 7, 2013. Retrieved October 7, 2013.
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