von Mises–Fisher distribution

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

Definition[]

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

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

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

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

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

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

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

The von Mises–Fisher distribution for ${\displaystyle p=3}$ is also called the Fisher distribution.^[1][2] It was first used to model the interaction of electric dipoles in an electric field.^[3] Other applications are found in geology, bioinformatics, and text mining.

Relation to normal distribution[]

Starting from a normal distribution

${\displaystyle G_{p}(\mathbf {x} ;{\boldsymbol {\mu }},\kappa )=\left({\sqrt {\frac {\kappa }{2\pi }}}\right)^{p}\exp \left(-\kappa {\frac {(\mathbf {x} -{\boldsymbol {\mu }})^{2}}{2}}\right),}$

the von Mises-Fisher distribution is obtained by expanding

${\displaystyle (\mathbf {x} -{\boldsymbol {\mu }})^{2}=\mathbf {x} ^{2}+{\boldsymbol {\mu }}^{2}-2{\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} ,}$

using the fact that ${\displaystyle \mathbf {x} }$ and ${\displaystyle {\boldsymbol {\mu }}}$ are unit vectors, and recomputing the normalization constant by integrating ${\displaystyle \mathbf {x} }$ over the unit sphere.

Estimation of parameters[]

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

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

Then^[3] the maximum likelihood estimates of ${\displaystyle \mu }$ and ${\displaystyle \kappa }$ are given by the sufficient statistic

${\displaystyle {\bar {x}}={\frac {1}{N}}\sum _{i}^{N}x_{i},}$

as

${\displaystyle \mu ={\bar {x}}/{\bar {R}},{\text{where }}{\bar {R}}=\|{\bar {x}}\|,}$

and

${\displaystyle \kappa =A_{p}^{-1}({\bar {R}}).}$

Thus ${\displaystyle \kappa }$ is the solution to

${\displaystyle A_{p}(\kappa )={\frac {\left\|\sum _{i}^{N}x_{i}\right\|}{N}}={\bar {R}}.}$

A simple approximation to ${\displaystyle \kappa }$ is (Sra, 2011)

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

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

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

${\displaystyle {\hat {\kappa }}_{2}={\hat {\kappa }}_{1}-{\frac {A_{p}({\hat {\kappa }}_{1})-{\bar {R}}}{1-A_{p}({\hat {\kappa }}_{1})^{2}-{\frac {p-1}{{\hat {\kappa }}_{1}}}A_{p}({\hat {\kappa }}_{1})}}.}$

For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as^[4]

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

where

${\displaystyle d=1-{\frac {1}{N}}\sum _{i}^{N}\left(\mu ^{T}x_{i}\right)^{2}}$

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

${\displaystyle q=\arcsin \left(e_{\alpha }^{1/2}{\hat {\sigma }}\right),}$ where ${\displaystyle e_{\alpha }=-\ln(\alpha ).}$

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

Generalizations[]

The matrix von Mises-Fisher distribution (Also Known as ^[5]) has the density

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

supported on the Stiefel manifold of ${\displaystyle n\times p}$ orthonormal p-frames ${\displaystyle \mathbf {X} }$, where ${\displaystyle \mathbf {F} }$ is an arbitrary ${\displaystyle n\times p}$ real matrix.^[6][7]

Distribution of polar angle[]

For ${\displaystyle p=3}$, the angle θ between ${\displaystyle \mathbf {x} }$ and ${\displaystyle {\boldsymbol {\mu }}}$ satisfies ${\displaystyle \cos \theta ={\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} }$. It has the distribution

${\displaystyle p(\theta )=\int d^{2}xf(x;{\boldsymbol {\mu }},\kappa )\,\delta \left(\theta -{\text{arc cos}}({\boldsymbol {\mu }}^{\mathsf {T}}\mathbf {x} )\right)}$,

which can be easiliy evaluated as

${\displaystyle p(\theta )=2\pi C_{3}(\kappa )\,\sin \theta \,e^{\kappa \cos \theta }}$.

See also[]

• Kent distribution, a related distribution on the two-dimensional unit sphere
• von Mises distribution, von Mises–Fisher distribution where p = 2, the one-dimensional unit circle
• Bivariate von Mises distribution
• Directional statistics
• , also known as Matrix von Mises-Fisher distribution.

References[]

Further reading[]

• 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.
• 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. S2CID 3654195.