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 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).

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.
- I
_{0}0(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) are used to indicate that random variable *X* has a von Mises distribution with shape parameter κ and location parameter μ.

## CDF

The cumulative distribution function for this distribution is:

## Other Characteristics

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,
- Moment generating function,
- Population variance,
- Skewness.

## Software

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.

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

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