Bingham distribution

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere.^[1]

It 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

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

which may also be written

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

where x is an axis, M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, ${}_{1}F_{1}(\cdot;\cdot,\cdot)$ is a confluent hypergeometric function of matrix argument.

See also

• Directional statistics
• Kent 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. "Belief Propagation with Directional Statistics for Solving the Shape-from-Shading Problem". Springer. 2008. Retrieved November 29, 2013.
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.