Von Mises Distribution - Definition

Definition

The von Mises probability density function for the angle x is given by:

where I0(x) is the modified Bessel function of order 0.

The parameters μ and 1/κ are analogous to μ and σ2 (the mean and variance) in the normal distribution:

  • μ is a measure of location (the distribution is clustered around μ), and
  • κ is a measure of concentration (a reciprocal measure of dispersion, so 1/κ is analogous to σ2).
    • If κ is zero, the distribution is uniform, and for small κ, it is close to uniform.
    • If κ is large, the distribution becomes very concentrated about the angle μ with κ being a measure of the concentration. In fact, as κ increases, the distribution approaches a normal distribution in x with mean μ and variance 1/κ.

The probability density can be expressed as a series of Bessel functions (see Abramowitz and Stegun §9.6.34)

\frac{1}{2\pi}\left(1\!+\!\frac{2}{I_0(\kappa)} \sum_{j=1}^\infty I_j(\kappa)\cos\right)

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:

\frac{1}{2\pi}\left(x\!+\!\frac{2}{I_0(\kappa)} \sum_{j=1}^\infty I_j(\kappa)\frac{\sin}{j}\right).

The cumulative distribution function will be a function of the lower limit of integration x0:

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