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

The general form of the Von Mises probability density function (PDF) is:

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.
- I
_{0}0(κ) = modified Bessel function with order zero. 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* = zero.

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

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

## 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

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