.. _continuous-vonmises:

Von Mises Distribution
======================

Defined for :math:`x\in\left[-\pi,\pi\right]` with shape parameter :math:`\kappa>0` . Note, the PDF and CDF functions are periodic and are always defined
over :math:`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 :math:`\kappa<100` the PDF and CDF formulas below are used. Otherwise, a normal
approximation with variance :math:`1/\kappa` is used.

.. 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*}

.. 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`