Von Mises Distribution: Simple Definition & Examples

Probability Distributions > Von Mises Distribution

What is the Von Mises Distribution?

The Von Mises distribution (also called the circular normal or Tikhonov distribution) is a continuous probability distribution with a range from 0 to 2π. It is similar to the normal distribution, except coordinates are placed on a circular plane. The von Mises distribution can be thought of as a special case of the Von Mises-Fisher distribution, which is an extension of the distribution to multi-dimensional spheres.

The distribution was first described by Richard von Mises in 1918 [1] as a way to model the distribution of atomic weights. It’s now used to model a variety of phenomena including:

  • Brownian motion (Physics),
  • Epidemiology (spread of disease),
  • Interference Alignment (Signal Processing),
  • Privacy-preserving algorithms (in Machine Learning).

Von Mises distribution properties

von mises distribution
PDF for μ = π and κ values of 1 and 4.


The general form of the Von Mises probability density function (PDF) is:
von-mises-formula
Where:

  • μ is the mean direction of the distribution (the location parameter). μ can be any real number.
  • κ is a shape parameter called the “concentration” — which is reciprocal to dispersion; 1/κ is comparable to the variance — high concentrations imply low variance and vice-versa. Values for κ must be non-negative real numbers. If κ = 0, the distribution is uniform; if values for κ are large, the distribution becomes a normal distribution.
  • I00(κ) = modified Bessel function with order zero. Defined for all κ as:
    bessel

The shorthand X ~ von Mises (κ, μ) or X ~ CN(κ, μ) is used to indicate that random variable X has a von Mises distribution with shape parameter κ and location parameter μ.

The cumulative distribution function for this distribution is:
cdf

The median and mean of X = zero.

The following are intractable (difficult or near-impossible to solve for):

Software

  • Dataplot (open-source) has a built-in VONPPF command.
  • Oriana for Windows. “…calculates the special forms of sample and inter-sample statistics required for circular data (e.g. angles or directions measured in degrees, time of day, day of week, month of year, etc.).” Current retail price is around $400.
  • Circstats is an add-on package for R (open-source).
  • Circstat is a free add on for MatLab.

References

  1. Von Mises, R. (1981): Uber die Ganzzahligkeit der Atomgewichte und verwandte Gragen. Phys. Z. 19 (1918), 490-500.

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