Von Mises distribution

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von Mises
Probability density function
Plot of the von Mises PMF
The support is chosen to be [-π,π] with μ=0
Cumulative distribution function
Plot of the von Mises CMF
The support is chosen to be [-π,π] with μ=0
Parameters μ real
κ > 0
Support x\in any interval of length 2π
Probability density function (pdf) \frac{e^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}
Cumulative distribution function (cdf) (not analytic - see text)
Mean μ
Median μ
Mode μ
Variance var(z) = 1 − I1(κ)2 / I0(κ)2 (circular)
Skewness
Excess Kurtosis
Entropy -\kappa\frac{I_1(\kappa)}{I_0(\kappa)}+\ln[2\pi I_0(\kappa)] (differential)
mgf
Char. func.

In probability theory and statistics, the von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution. It may be thought of as the circular version of the normal distribution, since it describes the distribution of a random variate with period 2π. It is used in applications of Circular 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=eix 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^{\kappa\cos(x-\mu)}}{2\pi I_0(\kappa)}

where I0(x) is the modified Bessel function of order 0. The parameters μ and κ can be understood by considering the case where 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 §9.6.34)

f(x|\mu,\kappa)=\,
\frac{1}{2\pi}\left(1\!+\!\frac{2}{I_0(\kappa)} \sum_{j=1}^\infty I_j(\kappa)\cos[j(x\!-\!\mu)]\right)

where Ij(x) is the modified Bessel function of order j. The cumulative density 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}{2\pi}\left(x\!+\!\frac{2}{I_0(\kappa)} \sum_{j=1}^\infty I_j(\kappa)\frac{\sin[j(x\!-\!\mu)]}{j}\right)

The cumulative distribution function will be a function of the lower limit of integration x0:

F(x|\mu,\kappa)=\Phi(x|\mu,\kappa)-\Phi(x_0|\mu,\kappa)\,

[edit] Moments

The moments of the von Mises distribution are usually calculated as the moments of z=eix 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\rangle=\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 zn=cos(nx)+i sin(nx) and the Bessel function identity (See Abramowitz and Stegun §9.6.19):

I_n(\kappa)=\frac{1}{\pi}\int_0^\pi e^{\kappa\cos(x)}\cos(nx)\,dx

The mean of z  is then just

m_1= \frac{I_1(\kappa)}{I_0(\kappa)}e^{i\mu}

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:

\textrm{var}(z)=\langle |z|^2\rangle-|\langle z \rangle|^2 = 1-\frac{I_1(\kappa)^2}{I_0(\kappa)^2}

[edit] Limiting behavior

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

\lim_{\kappa\rightarrow\infty} f(x|\mu,\kappa)=\frac{\exp[\frac{-(x-\mu)^2}{2\sigma^2}]}{\sigma\sqrt{2\pi}}

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

\lim_{\kappa\rightarrow 0}f(x|\mu,\kappa)=\mathrm{U}(x)

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

[edit] 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, 24, 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
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