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

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

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 ${\displaystyle {}_{1}F_{1}(\cdot$ 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

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 studies". Journal of Geophysical Research, 85:1500–1510.
S. Teller and M. Antone (2000). Automatic recovery of camera positions in Urban Scenes
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.
"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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[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.

Probability distributions (List)
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher soliton discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Flory–Schulz Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular continuous Bernoulli Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Fréchet gamma gamma/Gompertz generalized gamma generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared noncentral F Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized chi-squared generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Bingham distribution

See also

References