Von Mises–Fisher distribution

In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), is a probability distribution on the $(p-1)$ -dimensional sphere in $\mathbb {R} ^{p}$ . If $p=2$ the distribution reduces to the von Mises distribution on the circle.

The probability density function of the von Mises–Fisher distribution for the random p-dimensional unit vector $\mathbf {x} \,$ is given by:

f_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=C_{p}(\kappa )\exp \left({\kappa {\boldsymbol {\mu }}^{T}\mathbf {x} }\right)

where $\kappa \geq 0,\left\Vert {\boldsymbol {\mu }}\right\Vert =1\,$ and the normalization constant $C_{p}(\kappa )\,$ is equal to

C_{p}(\kappa )={\frac {\kappa ^{p/2-1}}{(2\pi )^{p/2}I_{p/2-1}(\kappa )}}.\,

where $I_{v}$ denotes the modified Bessel function of the first kind at order $v$ . If $p=3$ , the normalization constant reduces to

C_{3}(\kappa )={\frac {\kappa }{4\pi \sinh \kappa }}={\frac {\kappa }{2\pi (e^{\kappa }-e^{-\kappa })}}.\,

The parameters $\mu \,$ and $\kappa \,$ are called the mean direction and concentration parameter, respectively. The greater the value of $\kappa \,$ , the higher the concentration of the distribution around the mean direction $\mu \,$ . The distribution is unimodal for $\kappa >0\,$ , and is uniform on the sphere for $\kappa =0\,$ .

The von Mises–Fisher distribution for $p=3$ , also called the Fisher distribution, was first used to model the interaction of electric dipoles in an electric field (Mardia, 2000). Other applications are found in geology, bioinformatics, and text mining.

Estimation of parameters

A series of N independent measurements $x_{i}$ are drawn from a von Mises–Fisher distribution. Define

A_{p}(\kappa)=\frac {I_{p/2}(\kappa)} {I_{p/2-1}(\kappa)} . \,

Then (Sra, 2011) the maximum likelihood estimates of $\mu \,$ and $\kappa \,$ are given by

\mu = \frac{\sum_i^N x_i}{\|\sum_i^N x_i\|} ,

\kappa = A_p^{-1}(\bar{R}) .

Thus $\kappa \,$ is the solution to

A_p(\kappa) = \frac{\|\sum_i^N x_i\|}{N} = \bar{R} .

A simple approximation to $\kappa$ is

\hat{\kappa} = \frac{\bar{R}(p-\bar{R}^2)}{1-\bar{R}^2} ,

but a more accurate measure can be obtained by iterating the Newton method a few times

\hat{\kappa}_1 = \hat{\kappa} - \frac{A_p(\hat{\kappa})-\bar{R}}{1-A_p(\hat{\kappa})^2-\frac{p-1}{\hat{\kappa}}A_p(\hat{\kappa})} ,

\hat{\kappa}_2 = \hat{\kappa}_1 - \frac{A_p(\hat{\kappa}_1)-\bar{R}}{1-A_p(\hat{\kappa}_1)^2-\frac{p-1}{\hat{\kappa}_1}A_p(\hat{\kappa}_1)} .

For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as^[1]

\hat{\sigma} = \left(\frac{d}{N\bar{R}^2}\right)^{1/2}

where

d = 1 - \frac{1}{N}\sum_i^N (\mu^Tx_i)^2

It's then possible to approximate a $100(1-\alpha)\%$ confidence cone about $\mu$ with semi-vertical angle

q = \arcsin(e_\alpha^{1/2}\hat{\sigma}) ,

where

e_\alpha = -\ln(\alpha).

For example, for a 95% confidence cone, $\alpha = 0.05, e_\alpha = -\ln(0.05) = 2.996,$ and thus $q = \arcsin(1.731\hat{\sigma}).$

Generalizations

The matrix von Mises-Fisher distribution has the density

f_{n,p}(\mathbf {X} ;\mathbf {F} )\propto \exp(\operatorname {tr} (\mathbf {F} ^{T}\mathbf {X} ))

supported on the Stiefel manifold of $n\times p$ orthonormal p-frames $\mathbf {X}$ , where $\mathbf {F}$ is an arbitrary $n\times p$ real matrix.^[2]^[3]

References

↑ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.
↑ Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
↑ Downs (1972). "Orientational statistics". Biometrika. 59: 665–676. doi:10.1093/biomet/59.3.665.

Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
Banerjee, A., Dhillon, I. S., Ghosh, J., & Sra, S. (2005). "Clustering on the unit hypersphere using von Mises-Fisher distributions". Journal of Machine Learning Research, 6(Sep), 1345-1382.
Fisher, RA, "Dispersion on a sphere'". (1953) Proc. Roy. Soc. London Ser. A., 217: 295–305
Mardia, Kanti; Jupp, P. E. (1999). Directional Statistics. Wiley. ISBN 978-0-471-95333-3.
Sra, S. (2011). "A short note on parameter approximation for von Mises-Fisher distributions: And a fast implementation of I s (x)". Computational Statistics. 27: 177–190. CiteSeerX 10.1.1.186.1887. doi:10.1007/s00180-011-0232-x.

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[1] Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.

[2] Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.

[3] Downs (1972). "Orientational statistics". Biometrika. 59: 665–676. doi:10.1093/biomet/59.3.665.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial 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 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 Flory–Schulz Fréchet gamma gamma/Gompertz 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 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 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

Von Mises–Fisher distribution

Estimation of parameters

Generalizations

See also

References