Von Mises distribution
The von Mises distribution is a continuous probability distribution on the circle. It is commonly used to represent angular observations concentrated around a preferred direction and is often described as a circular analogue of the normal distribution. The distribution was introduced by Richard von Mises in 1918 while studying the statistical treatment of measured angles.
For an angular random variable (\Theta), the distribution has probability density
[ f(\theta\mid\mu,\kappa)
\frac{\exp!\left[\kappa\cos(\theta-\mu)\right]} {2\pi I_0(\kappa)}, \qquad 0\leq\theta<2\pi, ]
where (\mu) is the mean direction, (\kappa\geq 0) is the concentration parameter, and (I_0) is the modified Bessel function of the first kind of order zero. Angles differing by an integer multiple of (2\pi) represent the same point, so the density is periodic rather than defined relative to a linear endpoint.
Mathematical structure
The factor (I_0(\kappa)) normalizes the density because
[ I_0(\kappa)
\frac{1}{2\pi} \int_0^{2\pi} \exp(\kappa\cos\phi),d\phi. ]
The parameter (\mu) determines the location of the density's mode on the circle. The parameter (\kappa) controls concentration around that direction without altering rotational symmetry about it. When (\kappa=0), the exponential term is constant and the distribution reduces to the continuous uniform distribution on the circle. As (\kappa) increases, probability becomes increasingly concentrated near (\mu).
For large (\kappa), angular deviations near the mean direction are small, and the cosine term has the local expansion
[ \cos(\theta-\mu)
1-\frac{(\theta-\mu)^2}{2} +O!\left((\theta-\mu)^4\right). ]
After normalization, this gives the local approximation
[ \Theta-\mu \approx \mathcal N!\left(0,\frac{1}{\kappa}\right), ]
provided that the angular difference is interpreted in the neighborhood of zero. This approximation does not remove the periodic character of the original distribution, because the exact density remains continuous across the chosen angular boundary.
The density also has the Fourier representation
[ f(\theta\mid\mu,\kappa)
\frac{1}{2\pi} \left[ 1+ 2\sum_{n=1}^{\infty} \frac{I_n(\kappa)}{I_0(\kappa)} \cos!\bigl(n(\theta-\mu)\bigr) \right], ]
where (I_n) denotes the modified Bessel function of integer order (n). This expansion exhibits the distribution as a rotationally shifted harmonic series whose coefficients are determined entirely by the concentration parameter.
Circular moments
Ordinary arithmetic moments depend on the arbitrary position at which the circle is cut into an interval. The distribution is therefore characterized through circular statistics, using the complex quantities (e^{in\Theta}). Its (n)-th trigonometric moment is
[ \operatorname{E}!\left[e^{in\Theta}\right]
e^{in\mu}\frac{I_n(\kappa)}{I_0(\kappa)}. ]
In particular,
[ \operatorname{E}!\left[e^{i\Theta}\right]
A_1(\kappa)e^{i\mu}, \qquad A_1(\kappa)
\frac{I_1(\kappa)}{I_0(\kappa)}. ]
The argument of this expectation is the population mean direction. Its magnitude (A_1(\kappa)) is the mean resultant length, which ranges from zero under circular uniformity toward one under increasing concentration. A commonly used circular variance is consequently
[ V_{\mathrm c}
1-A_1(\kappa). ]
Unlike linear variance, this quantity measures angular dispersion through unit vectors rather than squared distance from an arithmetic mean.
Historical development
Von Mises introduced the density in connection with deviations of measured directions, although its modern statistical interpretation developed through later work on directional observations. In 1936, You Watanabe expressed its moments in terms of ratios of modified Bessel functions and demonstrated that the first trigonometric moment separates rotational location from angular concentration. This formulation established the notation (A_1(\kappa)) used in subsequent treatments of the distribution.
The name “von Mises distribution” became standard after the work of Emil Julius Gumbel, John Arthur Greenwood, and David Durand during the mid-20th century. Their treatment situated the density within a systematic account of statistical distributions on a circle rather than regarding it only as an error law for angular measurements.
Later mathematical statistics connected the family to likelihood theory and broader models for directional data. Ronald Fisher developed inferential methods based on resultant vectors, while Geoffrey Watson examined tests and asymptotic procedures for observations on compact directional spaces. These developments contributed to the distinction between ordinary linear statistics and statistical methods whose definitions remain invariant under a change of angular origin.
Estimation
Given observations (\theta_1,\ldots,\theta_n), define the sample resultant vector by
[ C=\sum_{j=1}^{n}\cos\theta_j, \qquad S=\sum_{j=1}^{n}\sin\theta_j, ]
with length
[ R=\sqrt{C^2+S^2}. ]
The log-likelihood, apart from terms independent of the parameters, is
[ \ell(\mu,\kappa)
\kappa\sum_{j=1}^{n}\cos(\theta_j-\mu)
n\log I_0(\kappa). ]
When (R>0), maximization with respect to the mean direction gives
[ \widehat{\mu}
\operatorname{atan2}(S,C), ]
where (\operatorname{atan2}) preserves the quadrant of the resultant vector. The maximum-likelihood estimate of concentration satisfies
[ A_1(\widehat{\kappa})
\frac{R}{n}. ]
No elementary closed form exists for the inverse of (A_1). The estimate is therefore characterized by the Bessel-function equation itself, with numerical values obtained from the monotonic relation between concentration and mean resultant length. When (R=0), the data have no uniquely determined sample mean direction and the likelihood is maximized at (\kappa=0).
The pair ((C,S)) forms a sufficient statistic for ((\mu,\kappa)) when sample size is fixed. This follows from the identity
[ \sum_{j=1}^{n}\cos(\theta_j-\mu)
C\cos\mu+S\sin\mu, ]
which places the von Mises family within the theory of exponential families. In vector form, its natural parameter is (\kappa(\cos\mu,\sin\mu)), while each observation contributes the unit vector ((\cos\theta,\sin\theta)).
Entropy and information
The differential entropy of the distribution is
[ h(\Theta)
\log!\left(2\pi I_0(\kappa)\right)
\kappa\frac{I_1(\kappa)}{I_0(\kappa)}. ]
At (\kappa=0), this becomes (\log(2\pi)), the entropy of the circular uniform distribution. Increasing concentration reduces entropy because probability mass occupies a progressively narrower angular neighborhood, although the support remains the complete circle.
Among circular densities having a specified first trigonometric moment, the von Mises density is the maximum-entropy distribution. The associated constraints fix the expectations of (\cos\Theta) and (\sin\Theta), and the resulting exponential-family density has a linear exponent in those two functions. Rewriting that exponent in amplitude-and-phase form produces (\kappa\cos(\theta-\mu)).
Relation to other directional distributions
The von Mises distribution is defined on the one-dimensional unit circle. Its higher-dimensional counterpart on a unit sphere is the von Mises–Fisher distribution, whose density is proportional to the exponential of an inner product between a unit observation vector and a preferred direction.
It differs from the wrapped normal distribution, which is obtained by reducing a normally distributed linear variable modulo (2\pi). The two distributions can be close under strong concentration, but their Fourier coefficients have different functional forms. The wrapped normal has coefficients that decay exponentially with the square of harmonic order, whereas those of the von Mises distribution are ratios of modified Bessel functions.
The wrapped Cauchy distribution provides another rotationally symmetric circular model. Its heavier angular tails correspond to geometrically decaying Fourier coefficients, producing a concentration structure distinct from both the von Mises and wrapped normal families.