Bingham distribution - Misplaced Pages

Antipodally symmetric probability distribution on the n-sphere

In statistics, the Bingham distribution, named after Christopher Bingham, is an antipodally symmetric probability distribution on the n-sphere. 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, and has been used in the field of computer vision.

Its probability density function is given by

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

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. Vol. 5304. Springer. pp. 780–791. doi:10.1007/978-3-540-88690-7_58. ISBN 978-3-540-88689-1. S2CID 15488343.
"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 Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda

Mixed
univariate

continuous- discrete	Rectified Gaussian

Multivariate
(joint)

Discrete:
Ewens
Multinomial
- Dirichlet
- Negative
Continuous:
Dirichlet
- Generalized
Multivariate Laplace
Multivariate normal
Multivariate stable
Multivariate t
Normal-gamma
- Inverse
Matrix-valued:
LKJ
Matrix beta
Matrix normal
Matrix t
Matrix gamma
- Inverse
Wishart
- Normal
- Inverse
- Normal-inverse
- Complex

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

Categories:

See also

References