Wrapped Normal Distribution - Moments

Moments

In terms of the circular variable the circular moments of the wrapped Normal distribution are the characteristic function of the Normal distribution evaluated at integer arguments:

where is some interval of length . The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

\langle z \rangle=e^{i\mu-\sigma^2/2}

The mean angle is

\theta_\mu=\mathrm{Arg}\langle z \rangle = \mu

and the length of the mean resultant is

R=|\langle z \rangle| = e^{-\sigma^2/2}

The circular standard deviation, which is a useful measure of dispersion for the wrapped Normal distribution and its close relative, the von Mises distribution is given by:

s=\sqrt{\ln(1/R^2)} = \sigma

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