Von Mises Distribution

Defined for $x\in\left[-\pi,\pi\right]$ with shape parameter $\kappa>0$ . Note, the PDF and CDF functions are periodic and are always defined over $x\in\left[-\pi,\pi\right]$ regardless of the location parameter. Thus, if an input beyond this range is given, it is converted to the equivalent angle in this range. For values of $\kappa<100$ the PDF and CDF formulas below are used. Otherwise, a normal approximation with variance $1/\kappa$ is used.

$$\begin{eqnarray*} f\left(x;\kappa\right) & = & \frac{e^{\kappa\cos x}}{2\pi I_{0}\left(\kappa\right)}\\ F\left(x;\kappa\right) & = & \frac{1}{2}+\frac{x}{2\pi}+\sum_{k=1}^{\infty}\frac{I_{k}\left(\kappa\right)\sin\left(kx\right)}{I_{0}\left(\kappa\right)\pi k}\\ G\left(q; \kappa\right) & = & F^{-1}\left(x;\kappa\right)\end{eqnarray*}$$

$$\begin{eqnarray*} \mu & = & 0\\ \mu_{2} & = & \int_{-\pi}^{\pi}x^{2}f\left(x;\kappa\right)dx\\ \gamma_{1} & = & 0\\ \gamma_{2} & = & \frac{\int_{-\pi}^{\pi}x^{4}f\left(x;\kappa\right)dx}{\mu_{2}^{2}}-3\end{eqnarray*}$$

This can be used for defining circular variance.

Implementation: scipy.stats.vonmises