What is the von Mises distribution?

As the name suggests, circular data is data that is measured on a circle. One example of circular data is direction (e.g. how many degrees from True North); another is time. (Note: Time is interesting because we can think of it as linear (Jan 2019 -> … -> Dec 2019 -> Jan 2020) or circular (Jan -> … -> Dec -> Jan).)

A common probability distribution we see with circular data is the von Mises distribution (named after Richard von Mises, an early 20th-century Austrian mathematician). It is a close approximation to the wrapped normal distribution which is the normal distribution “wrapped” around a unit circle. The von Mises distribution is simpler and more tractable, and hence is more commonly used.

The von Mises distribution has two parameters: $\mu \in \mathbb{R}$ , which is a measure of location, and $\kappa \geq 0$ , which is a measure of concentration. For any $x \in [\mu - \pi, \mu + \pi]$ , the probability density function (PDF) is given by

$\begin{aligned} f(x) = \dfrac{\exp [\kappa \cos (x - \mu)]}{2\pi I_0 (\kappa)}, \end{aligned}$

where $I_0$ is the modified Bessel function of order 0. Below is a picture of the PDF for different values of $\kappa$ (taken from Wikipedia). You can see that it looks very much like the normal distribution. As $\kappa$ increases, the distribution approaches the $\mathcal{N}(\mu, 1/\kappa)$ distribution. For small $\kappa$ , the distribution is very close to uniform, with $\kappa = 0$ corresponding exactly to the uniform distribution over the interval.

Below is a really nice visualization of circular data overlaid with the estimated von Mises distribution, taken from this blog post:

The mean, median and mode of this distribution is $\mu$ . The variance is equal to $1 - I_1(\kappa)/I_0(\kappa)$ , where $I_j$ is the modified Bessel function of order $j$ .