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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