von Mises distribution

The von Mises probability density function for the angle x is given by:{{cite book |title=Directional Statistics |last=Mardia |first=Kantilal |author-link=Kantilal Mardia |author2=Jupp, Peter E. |year=1999|publisher=Wiley |isbn=978-0-471-95333-3 }}

:$f(x\backslash mid\backslash mu,\backslash kappa)=\backslash frac\{\backslash exp(\backslash kappa\backslash cos(x-\backslash mu))\}\{2\backslash pi\; I\_0(\backslash kappa)\}$

where I₀($\backslash kappa$) is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity: $\backslash int\_\{-\backslash pi\}^\backslash pi\; \backslash exp(\backslash kappa\backslash cos\; x)dx\; =\; \{2\backslash pi\; I\_0(\backslash kappa)\}.$

The parameters μ and 1/$\backslash kappa$ are analogous to μ and σ{{i sup|2}} (the mean and variance) in the normal distribution:

• μ is a measure of location (the distribution is clustered around μ), and
• $\backslash kappa$ is a measure of concentration (a reciprocal measure of dispersion, so 1/$\backslash kappa$ is analogous to σ{{i sup|2}}).
• If $\backslash kappa$ is zero, the distribution is uniform, and for small $\backslash kappa$, it is close to uniform.
• If $\backslash kappa$ is large, the distribution becomes very concentrated about the angle μ with $\backslash kappa$ being a measure of the concentration. In fact, as $\backslash kappa$ increases, the distribution approaches a normal distribution in x  with mean μ and variance 1/$\backslash kappa$.

The probability density can be expressed as a series of Bessel functionssee Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_376.htm §9.6.34]

: $f(x\backslash mid\backslash mu,\backslash kappa)\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash left(1+\backslash frac\{2\}\{I\_0(\backslash kappa)\}\; \backslash sum\_\{j=1\}^\backslash infty\; I\_j(\backslash kappa)\; \backslash cos[j(x-\backslash mu)]\backslash right)$

where I_j(x) is the modified Bessel function of order j.

The cumulative distribution function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:

:$\backslash Phi(x\backslash mid\backslash mu,\backslash kappa)=\backslash int\; f(t\backslash mid\backslash mu,\backslash kappa)\backslash ,dt\; =\backslash frac\{1\}\{2\backslash pi\}\backslash left(x\; +\; \backslash frac\{2\}\{I\_0(\backslash kappa)\}\; \backslash sum\_\{j=1\}^\backslash infty\; I\_j(\backslash kappa)\; \backslash frac\{\backslash sin[j(x-\backslash mu)]\}\{j\}\backslash right).$

The cumulative distribution function will be a function of the lower limit of

integration x₀:

:$F(x\backslash mid\backslash mu,\backslash kappa)=\backslash Phi(x\backslash mid\backslash mu,\backslash kappa)-\backslash Phi(x\_0\backslash mid\backslash mu,\backslash kappa).\backslash ,$

{{further|Circular mean}}

The moments of the von Mises distribution are usually calculated as the moments of the complex exponential z = e{{i sup|ix}} 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 complex mean.

The nth raw moment of z is:

:$m\_n=\backslash langle\; z^n\backslash rangle=\backslash int\_\backslash Gamma\; z^n\backslash ,f(x|\backslash mu,\backslash kappa)\backslash ,dx$

:$=\; \backslash frac\{I\_$

where the integral is over any interval $\backslash Gamma$ of length 2π. In calculating the above integral, we use the fact that z{{i sup|n}} = cos(nx) + i sin(nx) and the Bessel function identity:See Abramowitz and Stegun [http://www.math.sfu.ca/~cbm/aands/page_376.htm §9.6.19]

:$I\_n(\backslash kappa)=\backslash frac\{1\}\{\backslash pi\}\backslash int\_0^\backslash pi\; e^\{\backslash kappa\backslash cos(x)\}\backslash cos(nx)\backslash ,dx.$

The mean of the complex exponential z  is then just

:$m\_1=\; \backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}e^\{i\backslash mu\}$

and the circular mean value of the angle x is then taken to be the argument μ. This is the expected or preferred direction of the angular random variables. The circular variance of x is:

:$V\; =\; 1\; -\; |E[e^\{ix\}]|\; =\; 1\; -\; \backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}.$

A notable advancement in generating Tikhonov (or von Mises) random variates was introduced by Abreu in 2008.{{Cite journal |last=de Abreu |first=Giuseppe Thadeu Freitas |title=On the Generation of Tikhonov Variates |journal=IEEE Transactions on Communications |volume=56 |issue=7 |year=2008 |pages=1157–1168 |doi=10.1109/TCOMM.2008.060510}} This method, termed the "random mixture" (RM) technique, offers a simple and efficient alternative to traditional approaches like the accept-reject (AR) algorithm, which often suffer from inefficiency due to sample rejection and computational complexity. The RM method generates Tikhonov variates by randomly selecting samples from a predefined set of Cauchy and Gaussian generators, followed by a straightforward transformation. Specifically, it uses a bank of $K$ distinct generators (e.g., one Cauchy and two Gaussian processes), with mixture probabilities derived from the characteristic functions of the Cauchy, Gaussian, and Tikhonov distributions, all of which are available in closed form.

The technique leverages the circular moment-determinance property of the Tikhonov distribution, where the distribution is uniquely defined by its circular moments. By ensuring that the first $N$ dominant circular moments of the generated variates closely match the theoretical Tikhonov moments, the method achieves high accuracy. The mixture probabilities and parameters (e.g., variance for Gaussian and half-width for Cauchy) can be computed using either least squares (LS) optimization or a simpler Moore-Penrose pseudo-inverse approach, with the latter offering a practical trade-off between complexity and precision. Unlike AR methods, the RM technique consumes only one pair of uniform random numbers per Tikhonov sample, regardless of the concentration parameter $\backslash alpha$, and avoids sample rejection or repetitive evaluation of complex functions.{{Cite AV media

| title = [On the Generation of Tikhonov Random Variates]

| url = https://www.youtube.com/watch?v=7irV9Yng2mM

| publisher = [The Wireless Channel]

| date = 2025-03-19

| access-date = 2025-03-21

| medium = Video

| website = YouTube

}}

When $\backslash kappa$ is large, the distribution resembles a normal distribution. Mardia, K. V.; Jupp, P. E. (2000). "Directional Statistics". Wiley Series in Probability and Statistics. Chichester: John Wiley & Sons. ISBN 978-0-471-95333-3. p. 36. More specifically, for large positive real numbers $\backslash kappa$,

:$f(x\backslash mid\backslash mu,\backslash kappa)\; \backslash approx\; \backslash frac\; 1\; \{\backslash sigma\backslash sqrt\{2\backslash pi\}\}\; \backslash exp\backslash left[\backslash dfrac\{-(x-\backslash mu)^2\}\{2\backslash sigma^2\}\backslash right]$

where σ² = 1/$\backslash kappa$ and the difference between the left hand side and the right hand side of the approximation converges uniformly to zero as $\backslash kappa$ goes to infinity. Also, when $\backslash kappa$ is small, the probability density function resembles a uniform distribution:

:$\backslash lim\_\{\backslash kappa\backslash rightarrow\; 0\}f(x\backslash mid\backslash mu,\backslash kappa)=\backslash mathrm\{U\}(x)$

where the interval for the uniform distribution $\backslash mathrm\{U\}(x)$ is the chosen interval of length $2\backslash pi$ (i.e. $\backslash mathrm\{U\}(x)\; =\; 1/(2\backslash pi)$ when $x$ is in the interval and $\backslash mathrm\{U\}(x)=0$ when $x$ is not in the interval).

A series of N measurements $z\_n=e^\{i\backslash theta\_n\}$ drawn from a von Mises distribution may be used to estimate certain parameters of the distribution.{{cite book |title=Statistics of earth science data : their distribution in time, space, and orientation |last=Borradaile |first=G. J. |year=2003 |publisher=Springer |isbn=978-3-662-05223-5 }} The average of the series $\backslash overline\{z\}$ is defined as

:$\backslash overline\{z\}=\backslash frac\{1\}\{N\}\backslash sum\_\{n=1\}^N\; z\_n$

and its expectation value will be just the first moment:

:$\backslash langle\backslash overline\{z\}\backslash rangle=\backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}e^\{i\backslash mu\}.$

In other words, $\backslash overline\{z\}$ is an unbiased estimator of the first moment. If we assume that the mean $\backslash mu$ lies in the interval $[-\backslash pi,\backslash pi]$, then Arg$(\backslash overline\{z\})$ will be a (biased) estimator of the mean $\backslash mu$.

Viewing the $z\_n$ as a set of vectors in the complex plane, the $\backslash bar\{R\}^\; 2$ statistic is the square of the length of the averaged vector:

:$\backslash bar\{R\}^\; 2=\backslash overline\{z\}\backslash ,\backslash overline\{z^*\}=\backslash left(\backslash frac\{1\}\{N\}\backslash sum\_\{n=1\}^N\; \backslash cos\backslash theta\_n\backslash right)^2+\backslash left(\backslash frac\{1\}\{N\}\backslash sum\_\{n=1\}^N\; \backslash sin\backslash theta\_n\backslash right)^2$

and its expectation value is {{cite journal |last=Kutil |first=Rade |title=Biased and unbiased estimation of the circular mean resultant length and its variance.|date=August 2012 |url=https://www.researchgate.net/publication/233474043 |journal=Statistics: A Journal of Theoretical and Applied Statistics |volume= 46 |issue=4 |pages=549–561 |doi=10.1080/02331888.2010.543463 |s2cid=7045090 |citeseerx=10.1.1.302.8395 }}

:$\backslash langle\; \backslash bar\{R\}^2\backslash rangle=\backslash frac\{1\}\{N\}+\backslash frac\{N-1\}\{N\}\backslash ,\backslash frac\{I\_1(\backslash kappa)^2\}\{I\_0(\backslash kappa)^2\}.$

In other words, the statistic

:$R\_e^2=\backslash frac\{N\}\{N-1\}\backslash left(\backslash bar\{R\}^2-\backslash frac\{1\}\{N\}\backslash right)$

will be an unbiased estimator of $\backslash frac\{I\_1(\backslash kappa)^2\}\{I\_0(\backslash kappa)^2\}\backslash ,$ and solving the equation $R\_e=\backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}\backslash ,$ for $\backslash kappa\backslash ,$ will yield a (biased) estimator of $\backslash kappa\backslash ,$. In analogy to the linear case, the solution to the equation $\backslash bar\{R\}=\backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}\backslash ,$ will yield the maximum likelihood estimate of $\backslash kappa\backslash ,$ and both will be equal in the limit of large N. For approximate solution to $\backslash kappa\backslash ,$ refer to von Mises–Fisher distribution.

The distribution of the sample mean $\backslash overline\{z\}\; =\; \backslash bar\{R\}e^\{i\backslash overline\{\backslash theta\}\}$ for the von Mises distribution is given by:{{cite book |title=Topics in Circular Statistics |last=Jammalamadaka |first=S. Rao |author2=Sengupta, A. |year=2001 |publisher=World Scientific Publishing Company |isbn=978-981-02-3778-3 }}

:$$

P(\bar{R},\bar{\theta})\,d\bar{R}\,d\bar{\theta}=\frac{1}{ (2\pi I_0(\kappa))^N}\int_\Gamma \prod_{n=1}^N \left( e^{\kappa\cos(\theta_n-\mu)} d\theta_n\right) = \frac{e^{\kappa N\bar{R}\cos(\bar{\theta}-\mu)}}{I_0(\kappa)^N}\left(\frac{1}{(2\pi)^N}\int_\Gamma \prod_{n=1}^N d\theta_n\right)

where N is the number of measurements and $\backslash Gamma\backslash ,$ consists of intervals of $2\backslash pi$ in the variables, subject to the constraint that $\backslash bar\{R\}$ and $\backslash bar\{\backslash theta\}$ are constant, where $\backslash bar\{R\}$ is the mean resultant:

:$$

\bar{R}^2=|\bar{z}|^2= \left(\frac{1}{N}\sum_{n=1}^N \cos(\theta_n) \right)^2 + \left(\frac{1}{N}\sum_{n=1}^N \sin(\theta_n) \right)^2

and $\backslash overline\{\backslash theta\}$ is the mean angle:

:$$

\overline{\theta}=\mathrm{Arg}(\overline{z}). \,

Note that the product term in parentheses is just the distribution of the mean for a circular uniform distribution.

This means that the distribution of the mean direction $\backslash mu$ of a von Mises distribution $VM(\backslash mu,\; \backslash kappa)$ is a von Mises distribution $VM(\backslash mu,\; \backslash bar\{R\}N\backslash kappa)$, or, equivalently, $VM(\backslash mu,\; R\backslash kappa)$.

By definition, the information entropy of the von Mises distribution is

:$H\; =\; -\backslash int\_\backslash Gamma\; f(\backslash theta;\backslash mu,\backslash kappa)\backslash ,\backslash ln(f(\backslash theta;\backslash mu,\backslash kappa))\backslash ,d\backslash theta\backslash ,$

where $\backslash Gamma$ is any interval of length $2\backslash pi$. The logarithm of the density of the Von Mises distribution is straightforward:

:$\backslash ln(f(\backslash theta;\backslash mu,\backslash kappa))=-\backslash ln(2\backslash pi\; I\_0(\backslash kappa))+\; \backslash kappa\; \backslash cos(\backslash theta)\backslash ,$

The characteristic function representation for the Von Mises distribution is:

:$f(\backslash theta;\backslash mu,\backslash kappa)\; =\backslash frac\{1\}\{2\backslash pi\}\backslash left(1+2\backslash sum\_\{n=1\}^\backslash infty\backslash phi\_n\backslash cos(n\backslash theta)\backslash right)$

where $\backslash phi\_n=\; I\_$

:$H\; =\; \backslash ln(2\backslash pi\; I\_0(\backslash kappa))-\backslash kappa\backslash phi\_1\; =\; \backslash ln(2\backslash pi\; I\_0(\backslash kappa))-\backslash kappa\backslash frac\{I\_1(\backslash kappa)\}\{I\_0(\backslash kappa)\}$

For $\backslash kappa=0$, the von Mises distribution becomes the circular uniform distribution and the entropy attains its maximum value of $\backslash ln(2\backslash pi)$.

Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified{{cite book |title=Topics in circular statistics |last=Jammalamadaka |first=S. Rao |author2=SenGupta, A.|year=2001 |publisher=World Scientific |location=New Jersey |isbn=981-02-3778-2 |url=https://books.google.com/books?id=sKqWMGqQXQkC&q=Jammalamadaka+Topics+in+circular |access-date=2011-05-15}} or, equivalently, the circular mean and circular variance are specified.

• Bivariate von Mises distribution
• Directional statistics
• Von Mises–Fisher distribution
• Kent distribution

• Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. {{isbn|0-486-61272-4}}

{{ProbDistributions|directional}}

{{DEFAULTSORT:Von Mises Distribution}}

Category:Continuous distributions

Category:Directional statistics

Category:Exponential family distributions