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Von Mises Distribution

There is one shape parameter \(\kappa>0\), with support \(x\in\left[-\pi,\pi\right]\). For values of \(\kappa<100\) the PDF and CDF formulas below are used. Otherwise, a normal approximation with variance \(1/\kappa\) is used. [Note that the PDF and CDF functions below are periodic with period \(2\pi\). If an input outside \(x\in\left[-\pi,\pi\right]\) is given, it is converted to the equivalent angle in this range.]

\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*}

where \(I_{k}(\kappa)\) is a modified Bessel function of the first kind.

\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