Von Mises distribution

circular normal distribution

A unimodal probability distribution on the circle with probability density $$ p(\phi) = \frac{1}{2\pi I_0(\kappa)} \exp(\kappa \cos(\phi-\theta_1)) $$ with two parameters, $\kappa$ and $\theta_1$. This function takes its maximum value at $\phi = \theta_1$, so that $\theta_1$ is the mode; $\kappa$ is a concentration parameter. The normalising factor $I_0(\kappa)$ is an incomplete Bessel function.

The von Mises distribution is commonly used in the statistical analysis of directions. It may be obtained as the hitting density of two-dimensional Brownian motion with constant drift.

References

• Gordon, Louis; Hudson, Malcolm, A characterization of the von Mises distribution Ann. Stat. 5 (1977) DOI 10.1214/aos/1176343906 Zbl 0378.62012
• Kendall, David G., Pole-seeking Brownian motion and bird navigation J. R. Stat. Soc., Ser. B 36 (1974) [1] Zbl 0291.92005