Seymour Ginsburg: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Omnipaedista
 
en>Tassedethe
No edit summary
Line 1: Line 1:
Hello and welcome. My title is Numbers Wunder. What I love doing is performing ceramics but I haven't produced a dime with it. Years ago we moved to North Dakota. For many years I've been working as a payroll clerk.<br><br>Here is my webpage; [http://www.Dailyroads.com/profile_info.php?ID=2363798 over the counter std test]
[[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.]]
 
In [[directional statistics]], the '''von Mises–Fisher distribution''' is a
[[probability distribution]] on the <math>(p-1)</math>-dimensional [[sphere]] in <math>\mathbb{R}^{p}</math>. If <math>p=2</math>
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 <math>\mathbf{x}\,</math> is given by:
 
:<math>
 
f_{p}(\mathbf{x}; \mu, \kappa)=C_{p}(\kappa)\exp \left( {\kappa \mu^T \mathbf{x} } \right)
 
</math>
 
where <math> \kappa \ge 0, \left \Vert \mu \right \Vert =1 \,</math> and 
the normalization constant <math>C_{p}(\kappa)\, </math> is equal to
 
: <math>
C_{p}(\kappa)=\frac {\kappa^{p/2-1}} {(2\pi)^{p/2}I_{p/2-1}(\kappa)}. \,
</math>
 
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
: <math>
C_{3}(\kappa)=\frac {\kappa} {4\pi\sinh \kappa}=\frac {\kappa} {2\pi(e^{\kappa}-e^{-\kappa})}. \,
</math>
 
Note that the equations above apply for polar coordinates only.
 
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>.
 
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]].
 
==Estimation of parameters==
A series of <math>N</math> [[independence (probability theory)|independent]] measurements <math>x_i</math> are drawn from a von Mises-Fisher distribution. Define
 
: <math>
A_{p}(\kappa)=\frac {I_{p/2}(\kappa)} {I_{p/2-1}(\kappa)} . \,
</math>
 
Then (Sra, 2011) the [[maximum likelihood]] estimates of <math>\mu\,</math> and <math>\kappa\,</math> are given by
 
:<math>
\mu = \frac{\sum_i^N x_i}{||\sum_i^N x_i||} ,
</math>
:<math>
\kappa = A_p^{-1}(\bar{R}) .
</math>
Thus <math>\kappa\,</math> is the solution to
:<math>
A_p(\kappa) = \frac{||\sum_i^N x_i||}{N} = \bar{R} .
</math>
A simple approximation to <math>\kappa</math> is
:<math>
\hat{\kappa} = \frac{\bar{R}(p-\bar{R}^2)}{1-\bar{R}^2} ,
</math>
but a more accurate measure can be obtained by iterating the Newton method a few times
:<math>
\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})} ,
</math>
:<math>
\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)} .
</math>
 
==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]]
 
==References==
* 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
* {{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}}
* {{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]
 
{{ProbDistributions|directional}}
 
{{DEFAULTSORT:Von Mises-Fisher distribution}}
[[Category:Probability distributions]]
[[Category:Directional statistics]]
[[Category:Multivariate continuous distributions]]
[[Category:Exponential family distributions]]
[[Category:Continuous distributions]]

Revision as of 01:41, 14 October 2013

File:Von mises fisher.png
Points sampled from three von Mises-Fisher distributions on the sphere (blue: , green: , red: ). The mean directions are shown with arrows.

In directional statistics, the von Mises–Fisher distribution is a probability distribution on the -dimensional sphere in . If 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 is given by:

where and the normalization constant is equal to

where denotes the modified Bessel function of the first kind and order . If , the normalization constant reduces to

Note that the equations above apply for polar coordinates only.

The parameters and are called the mean direction and concentration parameter, respectively. The greater the value of , the higher the concentration of the distribution around the mean direction . The distribution is unimodal for , and is uniform on the sphere for .

The von Mises-Fisher distribution for , 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

A series of independent measurements are drawn from a von Mises-Fisher distribution. Define

Then (Sra, 2011) the maximum likelihood estimates of and are given by

Thus is the solution to

A simple approximation to is

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

See also

References

  • 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
  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Template:Cite doi Preprint

55 yrs old Metal Polisher Records from Gypsumville, has interests which include owning an antique car, summoners war hack and spelunkering. Gets immense motivation from life by going to places such as Villa Adriana (Tivoli).

my web site - summoners war hack no survey ios