von Mises–Fisher 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

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

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.

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 (Mardia&Jupp, 1999) 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

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

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[1]

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

where

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

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

${\displaystyle q=\arcsin(e_{\alpha }^{1/2}{\hat {\sigma }}),}$ 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 ${\displaystyle q=\arcsin(1.731{\hat {\sigma }}).}$

The matrix von Mises-Fisher distribution has the density

${\displaystyle f_{n,p}(\mathbf {X}$ ;\mathbf {F} )\propto \exp(\operatorname {tr} (\mathbf {F} ^{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.[2][3]

• 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

• 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. John Wiley & Sons Ltd. 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.