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.. _continuous-vonmises:
Von Mises Distribution
======================
There is one shape parameter :math:`\kappa>0`, with support :math:`x\in\left[-\pi,\pi\right]`.
For values of :math:`\kappa<100` the PDF and CDF formulas below are used. Otherwise, a normal
approximation with variance :math:`1/\kappa` is used.
[Note that the PDF and CDF functions below are periodic with period :math:`2\pi`.
If an input outside :math:`x\in\left[-\pi,\pi\right]` is given, it is converted
to the equivalent angle in this range.]
.. math::
:nowrap:
\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 :math:`I_{k}(\kappa)` is a modified Bessel function of the first kind.
.. math::
:nowrap:
\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`
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