To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Von Mises–Fisher distribution

From Wikipedia, the free encyclopedia

In directional statistics, the von Mises–Fisher distribution (named after Ronald Fisher and Richard von Mises), 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 at order . If , the normalization constant reduces to

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

YouTube Encyclopedic

  • 1/5
    Views:
    773
    8 591
    2 044
    12 338
    19 016
  • ✪ Moving object detection, tracking and following using an omnidirectional camera on a mobile robot
  • ✪ A Mixed Distribution Example
  • ✪ The Time Preference Theory of Interest and Its Critics | Jeffrey M. Herbener
  • ✪ Friedrich von Hayek and Axel Leijonhufvud - Nov 1978
  • ✪ Factor of Safety Analysis of a Bike Frame - SolidWorks Simulation

Transcription

Contents

Estimation of parameters

A series of N 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

For N ≥ 25, the estimated spherical standard error of the sample mean direction can be computed as[1]

where

It's then possible to approximate a confidence cone about with semi-vertical angle

where

For example, for a 95% confidence cone, and thus

Generalizations

The matrix von Mises-Fisher distribution has the density

supported on the Stiefel manifold of orthonormal p-frames , where is an arbitrary real matrix.[2][3]

See also

References

  1. ^ Embleton, N. I. Fisher, T. Lewis, B. J. J. (1993). Statistical analysis of spherical data (1st pbk. ed.). Cambridge: Cambridge University Press. pp. 115–116. ISBN 0-521-45699-1.
  2. ^ Jupp (1979). "Maximum likelihood estimators for the matrix von Mises-Fisher and Bingham distributions". The Annals of Statistics. 7 (3): 599–606. doi:10.1214/aos/1176344681.
  3. ^ Downs (1972). "Orientational statistics". Biometrika. 59: 665–676. doi:10.1093/biomet/59.3.665.
  • 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. 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. CiteSeerX 10.1.1.186.1887. doi:10.1007/s00180-011-0232-x.
This page was last edited on 31 January 2019, at 13:48
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.