Von Mises distribution

Probability distribution
name =von Mises
type =density
pdf_

The support is chosen to be [-π,π] with μ=0
cdf_

The support is chosen to be [-π,π] with μ=0
parameters =$mu$ real
$kappa>0$
support =$xin$ any interval of length 2π
pdf =$frac\{e^\{kappacos(x-mu)${2pi I_0(kappa)}
cdf =(not analytic - see text)
mean =$mu$
median =$mu$
mode =$mu$
variance =$extrm\{var\}(z)=1-I\_1(kappa)^2/I\_0(kappa)^2$ (circular)
skewness =
kurtosis =
entropy =$-kappafrac\{I\_1(kappa)\}\{I\_0(kappa)\}+ln\; [2pi\; I\_0(kappa)]$ (differential)
mgf =
char =

In probability theory and statistics, the von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the circle. It may be thought of as the circular analogue of the normal distribution. It is used in applications of directional statistics, where a distribution of angles are found which are the result of the addition of many small independent angular deviations, such as target sensing, or grain orientation in a granular material. If "x" is the angular random variable, it is often useful to think of the von Mises distribution as a distribution of complex numbers "z" = "e"^"ix" rather than the real numbers "x". The von Mises distribution is a special case of the von Mises-Fisher distribution on the "N"-dimensional sphere.

The von Mises probability density function for the angle "x" is given by:

:$f(x|mu,kappa)=frac\{e^\{kappacos(x-mu)\{2pi\; I\_0(kappa)\}$

where "I"₀("x") is the modified Bessel function of order 0.

The parameters μ and κ are analogous to μ and σ (the mean and variance) in the normal distribution:
* μ is a measure of location (the distribution is clustered around μ), and
* κ is a measure of concentration (an inverse measure of dispersion, so 1/κ is analogous to σ).
** If κ is zero, the distribution is uniform, and for small κ, it is close to uniform.
** If κ is large, the distribution becomes very concentrated about the angle μ with κ being a measure of the concentration. In fact, as κ increases, the distribution approaches a normal distribution in "x" with mean μ and variance 1/κ.

The probability density can be expressed as a series of Bessel functions (see Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_376.htm §9.6.34] )

:$f(x|mu,kappa)=,$:$frac\{1\}\{2pi\}left(1!+!frac\{2\}\{I\_0(kappa)\}sum\_\{j=1\}^infty\; I\_j(kappa)cos\; [j(x!-!mu)]\; ight)$

where "I"_"j"("x") is the modified Bessel function of order "j". The cumulative distribution function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:

:$Phi(x|mu,kappa)=int\; f(t|mu,kappa),dt=$:$frac\{1\}\{2pi\}left(x!+!frac\{2\}\{I\_0(kappa)\}sum\_\{j=1\}^infty\; I\_j(kappa)frac\{sin\; [j(x!-!mu)]\; \}\{j\}\; ight)$

The cumulative distribution function will be a function of the lower limit ofintegration "x"₀:

:$F(x|mu,kappa)=Phi(x|mu,kappa)-Phi(x\_0|mu,kappa),$

Moments

The moments of the von Mises distribution are usually calculated as the moments of "z" = "e"^"ix" rather than the angle "x" itself. These moments are referred to as "circular moments". The variance calculated from these moments is referred to as the "circular variance". The one exception to this is that the "mean" usually refers to the argument of the circular mean, rather than the circular mean itself.

The "n"th raw moment of "z" is:

:$m\_n=langle\; z^n\; angle=oint\; z^n,f(x|mu,kappa),dx$:$=\; frac\{I\_n(kappa)\}\{I\_0(kappa)\}e^\{i\; n\; mu\}$

where the integral is over any interval of length 2π. In calculating the above integral, we use the fact that "z"^"n" = cos("n"x) + i sin("nx") and the Bessel function identity (See Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_376.htm §9.6.19] ):

:$I\_n(kappa)=frac\{1\}\{pi\}int\_0^pi\; e^\{kappacos(x)\}cos(nx),dx.$

The mean of "z" is then just

:$m\_1=\; frac\{I\_1(kappa)\}\{I\_0(kappa)\}e^\{imu\}$

and the "mean" value of "x" is then taken to be the argument μ. This is the "average" direction of the angular random variables. The variance of "z", or the circular variance of "x" is:

:$extrm\{var\}(z)=langle\; |z|^2\; angle-|langle\; z\; angle|^2=\; 1-frac\{I\_1(kappa)^2\}\{I\_0(kappa)^2\}.$

Limiting behavior

In the limit of large κ the distribution becomes a normal distribution

:$lim\_\{kappa\; ightarrowinfty\}f(x|mu,kappa)=frac\{exp\; [frac\{-(x-mu)^2\}\{2sigma^2\}]\; \}\{sigmasqrt\{2pi$

where σ² = 1/κ. In the limit of small κ it becomes a uniform distribution:

:$lim\_\{kappa\; ightarrow\; 0\}f(x|mu,kappa)=mathrm\{U\}(x)$

where the interval for the uniform distribution "U"("x") is the chosen interval of length 2π.

ee also

* The SOCR Resource provides [http://socr.ucla.edu/htmls/SOCR_Distributions.html interactive interface to Von Mises distribution] .

References

* Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. ISBN 0-486-61272-4
* “Algorithm AS 86: The von Mises Distribution Function,” Mardia, Applied Statistics, 24, 1975 (pp. 268-272).
* “Algorithm 518, Incomplete Bessel Function I0: The von Mises Distribution,” Hill, ACM Transactions on Mathematical Software, Vol. 3, No. 3, September 1977, Pages 279-284.
* Best, D. and Fisher, N. (1979). Efficient simulation of the von Mises distribution. Applied Statistics, 28, 152–157.
* Evans, M., Hastings, N., and Peacock, B., "von Mises Distribution." Ch. 41 in Statistical Distributions, 3rd ed. New York. Wiley 2000.
* Fisher, Nicholas I., Statistical Analysis of Circular Data. New York. Cambridge 1993.
* Mardia, Kanti V., and Jupp, Peter E., Directional Statistics. New York. Wiley 1999.
* “Statistical Distributions,” 2nd. Edition, Evans, Hastings, and Peacock, John Wiley and Sons, 1993, (chapter 39). ISBN 0-471-55951-2