In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or the Tikhonov distribution) is a continuous probability distribution on the circle.
It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution.
on a circle is a wrapped normally distributed random variable with an unwrapped 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.
[1] 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 on the N-dimensional sphere.
The von Mises probability density function for the angle x is given by:[2] where I0(
) is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity:
are analogous to μ and σ2 (the mean and variance) in the normal distribution: The probability density can be expressed as a series of Bessel functions[3] where Ij(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: The cumulative distribution function will be a function of the lower limit of integration x0: The moments of the von Mises distribution are usually calculated as the moments of the complex exponential z = eix rather than the angle x itself.
The nth raw moment of z is: where the integral is over any interval
In calculating the above integral, we use the fact that zn = cos(nx) + i sin(nx) and the Bessel function identity:[4] 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.
[5] More specifically, for large positive real numbers
and the difference between the left hand side and the right hand side of the approximation converges uniformly to zero as
is small, the probability density function resembles a uniform distribution: where the interval for the uniform distribution
is defined as and its expectation value will be just the first moment: In other words,
as a set of vectors in the complex plane, the
statistic is the square of the length of the averaged vector: and its expectation value is [7] In other words, the statistic will be an unbiased estimator of
In analogy to the linear case, the solution to the equation
will yield the maximum likelihood estimate of
and both will be equal in the limit of large N. For approximate solution to
for the von Mises distribution is given by:[8] where N is the number of measurements and
is the mean angle: Note that the product term in parentheses is just the distribution of the mean for a circular uniform distribution.
By definition, the information entropy of the von Mises distribution is[2] where
The logarithm of the density of the Von Mises distribution is straightforward: The characteristic function representation for the Von Mises distribution is: where
Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written: For
, the von Mises distribution becomes the circular uniform distribution and the entropy attains its maximum value of
Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified[9] or, equivalently, the circular mean and circular variance are specified.