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| [[File:von mises fisher.png|thumb|Points sampled from three von Mises-Fisher distributions on the sphere (blue: <math>\kappa=1</math>, green: <math>\kappa=10</math>, red: <math>\kappa=100</math>). The mean directions <math>\mu</math> are shown with arrows.]]
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| In [[directional statistics]], the '''von Mises–Fisher distribution''' is a
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| [[probability distribution]] on the <math>(p-1)</math>-dimensional [[sphere]] in <math>\mathbb{R}^{p}</math>. If <math>p=2</math>
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| the distribution reduces to the [[von Mises distribution]] on the [[circle]].
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| The probability density function of the von Mises-Fisher distribution for the random ''p''-dimensional unit vector <math>\mathbf{x}\,</math> is given by:
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| :<math>
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| f_{p}(\mathbf{x}; \mu, \kappa)=C_{p}(\kappa)\exp \left( {\kappa \mu^T \mathbf{x} } \right)
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| </math>
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| where <math> \kappa \ge 0, \left \Vert \mu \right \Vert =1 \,</math> and
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| the normalization constant <math>C_{p}(\kappa)\, </math> is equal to
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| : <math>
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| C_{p}(\kappa)=\frac {\kappa^{p/2-1}} {(2\pi)^{p/2}I_{p/2-1}(\kappa)}. \,
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| </math>
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| where <math> I_{v}</math> denotes the modified [[Bessel function]] of the first kind and order <math>v</math>. If <math>p=3</math>, the normalization constant reduces to
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| : <math> | |
| C_{3}(\kappa)=\frac {\kappa} {4\pi\sinh \kappa}=\frac {\kappa} {2\pi(e^{\kappa}-e^{-\kappa})}. \,
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| </math>
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| Note that the equations above apply for polar coordinates only.
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| The parameters <math>\mu\,</math> and <math>\kappa\,</math> are called the ''mean direction'' and ''[[concentration parameter]]'', respectively. The greater the value of <math>\kappa\,</math>, the higher the concentration of the distribution around the mean direction <math>\mu\,</math>. The distribution is [[unimodal]] for <math>\kappa>0\,</math>, and is uniform on the sphere for <math>\kappa=0\,</math>.
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| The von Mises-Fisher distribution for <math>p=3</math>, 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]].
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| ==Estimation of parameters==
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| A series of <math>N</math> [[independence (probability theory)|independent]] measurements <math>x_i</math> are drawn from a von Mises-Fisher distribution. Define
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| : <math> | |
| A_{p}(\kappa)=\frac {I_{p/2}(\kappa)} {I_{p/2-1}(\kappa)} . \,
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| </math>
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| Then (Sra, 2011) the [[maximum likelihood]] estimates of <math>\mu\,</math> and <math>\kappa\,</math> are given by
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| :<math>
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| \mu = \frac{\sum_i^N x_i}{||\sum_i^N x_i||} ,
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| </math>
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| :<math> | |
| \kappa = A_p^{-1}(\bar{R}) .
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| </math>
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| Thus <math>\kappa\,</math> is the solution to
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| :<math>
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| A_p(\kappa) = \frac{||\sum_i^N x_i||}{N} = \bar{R} .
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| </math>
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| A simple approximation to <math>\kappa</math> is
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| :<math>
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| \hat{\kappa} = \frac{\bar{R}(p-\bar{R}^2)}{1-\bar{R}^2} ,
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| </math>
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| but a more accurate measure can be obtained by iterating the Newton method a few times
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| :<math>
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| \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})} ,
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| </math>
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| :<math>
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| \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)} .
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| </math>
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| ==See also==
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| * [[Kent distribution]], a related distribution on the two-dimensional unit sphere
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| * [[von Mises distribution]], von Mises–Fisher distribution where p=2, the one-dimensional unit circle
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| * [[Bivariate von Mises distribution]]
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| * [[Directional statistics]]
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| ==References==
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| * Dhillon, I., Sra, S. (2003) "Modeling Data using Directional Distributions". Tech. rep., University of Texas, Austin.
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| * Fisher, RA, "Dispersion on a sphere'". (1953) ''Proc. Roy. Soc. London Ser. A.'', 217: 295-305
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| * {{cite book |title=Directional Statistics |last=Mardia |first=Kanti |authorlink=Kantilal Mardia |coauthors=Jupp, P. E.|year=1999 |publisher=Wiley |isbn=978-0-471-95333-3}}
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| * {{cite doi| 10.1007/s00180-011-0232-x}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.186.1887&rep=rep1&type=pdf Preprint]
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| {{ProbDistributions|directional}}
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| {{DEFAULTSORT:Von Mises-Fisher distribution}}
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| [[Category:Probability distributions]]
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| [[Category:Directional statistics]]
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Continuous distributions]]
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Greetings. Allow me start by telling you the writer's title - Phebe. My day occupation is a librarian. North Dakota is our beginning location. To gather cash is 1 of the things I adore most.
Here is my web blog :: http://nuvem.tk/altergalactica/MarissaepDonnt