Английская Википедия:Bingham distribution
Шаблон:Short description 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
- <math>
f(\mathbf{x}\,;\,M,Z) \; dS^{n-1} = {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ; Z \right)^{-1} \cdot \exp \left( \operatorname{tr} Z M^T \mathbf{x} \mathbf{x}^T M \right)\; dS^{n-1} </math> which may also be written
- <math>
f(\mathbf{x}\,;\,M,Z)\; dS^{n-1} \;=\; {}_1 F_1 \left( \tfrac12 ; \tfrac n2 ;Z \right)^{-1} \cdot \exp\left( \mathbf{x}^T M Z M^T \mathbf{x} \right)\; dS^{n-1} </math>
where x is an axis (i.e., a unit vector), M is an orthogonal orientation matrix, Z is a diagonal concentration matrix, and <math>{}_{1}F_{1}(\cdot;\cdot,\cdot)</math> 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
References
- ↑ Bingham, Ch. (1974) "An antipodally symmetric distribution on the sphere". Annals of Statistics, 2(6):1201–1225.
- ↑ Onstott, T.C. (1980) "Application of the Bingham distribution function in paleomagnetic studiesШаблон:Dead link". Journal of Geophysical Research, 85:1500–1510.
- ↑ S. Teller and M. Antone (2000). Automatic recovery of camera positions in Urban Scenes
- ↑ Шаблон:Cite book
- ↑ Шаблон:Cite web
