# 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 as a way to model the distribution of atomic weights. It’s now used to model a variety of phenomena including:

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

## Von Mises distribution properties

• PDF for μ = π and κ values of 1 and 4.

The general form of the Von Mises PDF is:

Where:

• μ is the mean direction of the distribution. 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) is the modified Bessel function with order zero. It is defined for all κ as:

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:

The median and mean of X is zero.

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

• Characteristic function,
• Hazard function,
• Inverse distribution function,
• Kurtosis,
• Population variance,
• Skewness.

## Software options

The Von Mises distribution isn’t part of most popular statistics packages. However, you do have options like add-ons or a specialized directional statistics package.

• 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

Grafarend, E. & Awange. J. (2012) Applications of Linear and Nonlinear Models: Fixed Effects, Random Effects. Springer Science & Business Media, Aug 15.
Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.
Von Mises, R. (1981): Uber die Ganzzahligkeit der Atomgewichte und verwandte Gragen. Phys. Z. 19 (1918), 490-500.