Von Mises distribution

Von Mises distribution
Density function
Distribution function
parameter	${\displaystyle \mu \in \mathbb {R} }$, ${\displaystyle \kappa >0}$
carrier	any interval of length 2π
Density function	${\displaystyle {\frac {1}{2\pi I_{0}(\kappa )}}e^{\kappa \cos(x-\mu )}}$
Distribution function	not analytical , see text
Expected value	${\displaystyle \mu }$
Variance	${\displaystyle 1-I_{1}(\kappa )/I_{0}(\kappa )}$ (circular)
entropy	${\displaystyle \ln(2\pi I_{0}(\kappa ))-\kappa {\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}}$
Characteristic function	${\displaystyle {\frac {I_{|t|}(\kappa )}{I_{0}(\kappa )}}e^{it\mu }}$

The Von Mises distribution is the equivalent of the normal distribution for periodic functions. It has two parameters, for the position of the maximum and for the sharpness of the distribution. The function is symmetrical about its maximum at the point . For it goes over into the equal distribution . For large it goes over into the (periodically continued) normal distribution with variance . ${\displaystyle \mu }$ ${\displaystyle \kappa }$ ${\displaystyle \mu }$ ${\displaystyle \kappa =0}$ ${\displaystyle \kappa }$ ${\displaystyle \sigma ^{2}=1/\kappa }$

The Von Mises distribution is considered to be the easiest distribution function to use in the field of circular statistics . It has the property of maximizing the entropy if only the circular mean and the circular variance are given.

It is named after the Austrian mathematician Richard von Mises .

Sums of several Von Mises components are preferred over Fourier analysis if the function to be approximated disappears over large parts of the period, see the last two examples:

• Frequency distribution of the departure direction of carrier pigeons,
• Probability distribution for the main wind direction as part of a forecast,
• Intensity curve of the radio signal of a pulsar ,

• Bayesian estimate with ambiguous measurements.

definition

The following is either the geometric angle in the plane or the phase angle of a signal. The density function of the Von Mises distribution is: ${\displaystyle x}$

${\displaystyle f(x\mid \mu ,\kappa )={\frac {1}{2\pi I_{0}(\kappa )}}e^{\kappa \cos(x-\mu )}}$

The factor in front of the exponential function normalizes the integral of the function over a period to the value 1. This contains the modified Bessel function of the zeroth order. ${\displaystyle I_{0}(\kappa )}$

Individual evidence

1. ^ S. Rao Jammalamadaka, A. Sengupta: Topics in circular statistics. World Scientific 2001, ISBN 981-02-3778-2 , limited preview in Google Book Search
2. ↑ R. von Mises: On the “integer number” of atomic weights and related questions. Physikalische Zeitung 19, 1918, pp. 490-500.
3. ^ RN Manchester et al .: The Parkes Pulsar Timing Array Project. Publications of the Astronomical Society of Australia (PASA) 30, 2013, doi: 10.1017 / pasa.2012.017 (Open access).
4. ^ Ivan Marković, Ivan Petrović: Bearing-Only Tracking with a Mixture of von Mises Distributions. Contribution to the 2012 IEEE / RSJ International Conference on Intelligent Robots and Systems (IROS), doi: 10.1109 / IROS.2012.6385600 , ( online ).