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

Read more about this topic:  Wrapped Normal Distribution

Famous quotes containing the word moments:

    “There are thoughts which are prayers. There are moments when, whatever the posture of the body, the soul is on its knees.”
    —Victor Hugo (1802–1885)

    “When it looks at great accomplishments, the world, bent on simplifying its images, likes best to look at the dramatic, picturesque moments experienced by its heroes.... But the no less creative years of preparation remain in the shadow.”
    —Stefan Zweig (18811942)

    “There are thoughts which are prayers. There are moments when, whatever the posture of the body, the soul is on its knees.”
    —Victor Hugo (1802–1885)