Von Mises–Fisher distribution

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Points sampled from three von Mises-Fisher distributions on the sphere (blue: \kappa=1, green: \kappa=10, red: \kappa=100). The mean directions \mu are shown with arrows.

In directional statistics, the von Mises–Fisher distribution 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}; \mu, \kappa)=C_{p}(\kappa)\exp \left( {\kappa \mu^T \mathbf{x} } \right)

where  \kappa \ge 0, \left \Vert \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 and 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})}. \,

Note that the equations above apply for polar coordinates only.

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 dipoles in an electric field (Mardia, 2000). Other applications are found in geology, bioinformatics, and text mining.

Estimation of parameters[edit]

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}).

See also[edit]

References[edit]

  1. ^ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1. 
  • Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
  • 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. doi:10.1007/s00180-011-0232-x. CiteSeerX: 10.1.1.186.1887.  edit