Von Mises distribution

Parameters Probability density functionPlot of the von Mises PMFThe support is chosen to be [-π,π] with μ=0 Cumulative distribution functionPlot of the von Mises CMFThe support is chosen to be [-π,π] with μ=0 ${\displaystyle \mu }$ real${\displaystyle \kappa >0}$ ${\displaystyle x\in }$ any interval of length 2π ${\displaystyle {\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}}$ (not analytic - see text) ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle {\textrm {var}}(z)=1-I_{1}(\kappa )^{2}/I_{0}(\kappa )^{2}}$ (circular) ${\displaystyle -\kappa {\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}+\ln[2\pi I_{0}(\kappa )]}$ (differential)

In probability theory and statistics, the von Mises distribution (also known as the circular normal distribution) is a continuous probability distribution on the circle. It may be thought of as the circular analogue of the normal distribution. It is used in applications of directional statistics, where a distribution of angles are found which are the result of the addition of many small independent angular deviations, such as target sensing, or grain orientation in a granular material. If x is the angular random variable, it is often useful to think of the von Mises distribution as a distribution of complex numbers z=eix rather than the real numbers x. The von Mises distribution is a special case of the von Mises-Fisher distribution on the N-dimensional sphere.

The von Mises probability density function for the angle x is given by:

${\displaystyle f(x|\mu ,\kappa )={\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}}$

where I0(x) is the modified Bessel function of order 0. The parameters μ and κ can be understood by considering the case where κ is large. The distribution becomes very concentrated about the angle μ with κ being a measure of the concentration. In fact, as κ increases, the distribution approaches a normal distribution in x  with mean μ and variance 1/κ.

The probability density can be expressed as a series of Bessel functions (see Abramowitz and Stegun §9.6.34)

${\displaystyle f(x|\mu ,\kappa )=\,}$
${\displaystyle {\frac {1}{2\pi }}\left(1\!+\!{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa )\cos[j(x\!-\!\mu )]\right)}$

where Ij(x) is the modified Bessel function of order j. The cumulative density function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:

${\displaystyle \Phi (x|\mu ,\kappa )=\int f(t|\mu ,\kappa )\,dt=}$
${\displaystyle {\frac {1}{2\pi }}\left(x\!+\!{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa ){\frac {\sin[j(x\!-\!\mu )]}{j}}\right)}$

The cumulative distribution function will be a function of the lower limit of integration x0:

${\displaystyle F(x|\mu ,\kappa )=\Phi (x|\mu ,\kappa )-\Phi (x_{0}|\mu ,\kappa )\,}$

Moments

The moments of the von Mises distribution are usually calculated as the moments of z=eix rather than the angle x itself. These moments are referred to as "circular moments". The variance calculated from these moments is referred to as the "circular variance". The one exception to this is that the "mean" usually refers to the argument of the circular mean, rather than the circular mean itself.

The n-th raw moment of z is:

${\displaystyle m_{n}=\langle z^{n}\rangle =\oint z^{n}\,f(x|\mu ,\kappa )\,dx}$
${\displaystyle ={\frac {I_{n}(\kappa )}{I_{0}(\kappa )}}e^{in\mu }}$

where the integral is over any interval of length 2π. In calculating the above integral, we use the fact that zn=cos(nx)+i sin(nx) and the Bessel function identity (See Abramowitz and Stegun §9.6.19):

${\displaystyle I_{n}(\kappa )={\frac {1}{\pi }}\int _{0}^{\pi }e^{\kappa \cos(x)}\cos(nx)\,dx}$

The mean of z  is then just

${\displaystyle m_{1}={\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}e^{i\mu }}$

and the "mean" value of x is then taken to be the argument μ. This is the "average" direction of the angular random variables. The variance of z, or the circular variance of x is:

${\displaystyle {\textrm {var}}(z)=\langle |z|^{2}\rangle -|\langle z\rangle |^{2}=1-{\frac {I_{1}(\kappa )^{2}}{I_{0}(\kappa )^{2}}}}$

Limiting behavior

In the limit of large κ the distribution becomes a normal distribution

${\displaystyle \lim _{\kappa \rightarrow \infty }f(x|\mu ,\kappa )={\frac {\exp[{\frac {-(x-\mu )^{2}}{2\sigma ^{2}}}]}{\sigma {\sqrt {2\pi }}}}}$

where σ2=1/κ. In the limit of small κ it becomes a uniform distribution:

${\displaystyle \lim _{\kappa \rightarrow 0}f(x|\mu ,\kappa )=\mathrm {U} (x)}$

where the interval for the uniform distribution U(x) is the chosen interval of length 2π