Bingham distribution

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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[edit]

References[edit]

  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.