In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
directional statistics
Directional statistics (also circular statistics or spherical statistics) is the subdiscipline of statistics that deals with directions (unit vectors in Euclidean space, R''n''), axes ( lines through the origin in R''n'') or rotations in R''n''. ...
, the
von Mises
The Mises family or von Mises is the name of an Austrian noble family. Members of the family excelled especially in mathematics and economy.
Notable members
* Ludwig von Mises, an Austrian-American economist of the Austrian School, older bro ...
distribution (also known as the circular normal distribution or the
Tikhonov distribution) is a continuous
probability distribution
In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
on the
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
. It is a close approximation to the
wrapped normal distribution
In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Browni ...
, which is the circular analogue of the
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
. A freely diffusing angle
on a circle is a wrapped normally distributed random variable with an
unwrapped
''Unwrapped'', also known as ''Unwrapped with Marc Summers'', is an American television program on Food Network that reveals the origins of sponsored foods. It first aired in June 2001 and is hosted by Marc Summers. The show leads viewers on t ...
variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation.
The von Mises distribution is the
maximum entropy distribution for circular data when the real and imaginary parts of the first
circular moment are specified. The von Mises distribution is a special case of the
von Mises–Fisher distribution
In directional statistics, the von Mises–Fisher distribution (named after Richard von Mises and Ronald Fisher), is a probability distribution on the (p-1)-sphere in \mathbb^. If p=2
the distribution reduces to the von Mises distribution on the c ...
on the ''N''-dimensional sphere.
Definition
The von Mises probability density function for the angle ''x'' is given by:
:
where ''I''
0(
) is the modified
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity:
The parameters ''μ'' and 1/
are analogous to ''μ'' and ''σ'' (the mean and variance) in the normal distribution:
* ''μ'' is a measure of location (the distribution is clustered around ''μ''), and
*
is a measure of concentration (a reciprocal measure of
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
* Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variat ...
, so 1/
is analogous to ''σ'').
** If
is zero, the distribution is uniform, and for small
, it is close to uniform.
** If
is large, the distribution becomes very concentrated about the angle ''μ'' with
being a measure of the concentration. In fact, as
increases, the distribution approaches a normal distribution in ''x'' with mean ''μ'' and variance 1/
.
The probability density can be expressed as a series of Bessel functions
:
where ''I''
''j''(''x'') is the modified
Bessel function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation
x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0
for an arbitrary complex ...
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:
:
The cumulative distribution function will be a function of the lower limit of
integration ''x''
0:
:
Moments
The moments of the von Mises distribution are usually calculated as the moments of the complex exponential ''z'' = ''e'' 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
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
of the complex mean.
The ''n''th raw moment of ''z'' is:
:
:
where the integral is over any interval
of length 2π. In calculating the above integral, we use the fact that ''z'' = cos(''n''x) + i sin(''nx'') and the Bessel function identity:
:
The mean of the complex exponential ''z'' is then just
:
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:
:
Generation of von Mises Variates
A notable advancement in generating Tikhonov (or von Mises) random variates was introduced by Abreu in 2008. 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
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
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
, and avoids sample rejection or repetitive evaluation of complex functions.
Limiting behavior
When
is large, the distribution resembles a
normal distribution
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
f(x) = \frac ...
.
[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
,
: