Von Mises Distribution - Moments

Moments

The moments of the von Mises distribution are usually calculated as the moments of z = eix 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 circular mean, rather than the circular mean itself.

The nth 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 zn = cos(nx) + i sin(nx) and the Bessel function identity (See Abramowitz and Stegun §9.6.19):

The mean of z is then just

and the "mean" value of x is then taken to be the argument μ. This is the "average" direction of the angular random variables. The variance of z, or the circular variance of x is:

\textrm{var}(x)= 1-E = 1-\frac{I_1(\kappa)}{I_0(\kappa)}.

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