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:

    “Just as we need to encourage women to test life’s many options, we need to acknowledge real limits of energy and resources. It would be pointless and cruel to prescribe role combination for every woman at each moment of her life. Life has its seasons. There are moments when a woman ought to invest emotionally in many different roles, and other moments when she may need to conserve her psychological energies.”
    —Faye J. Crosby (20th century)

    “There are certain moments when we might wish the future were built by men of the past.”
    —Jean Rostand (1894–1977)

    “Insults from an adolescent daughter are more painful, because they are seen as coming not from a child who lashes out impulsively, who has moments of intense anger and of negative feelings which are not integrated into that large body of responses, impressions and emotions we call ‘our feelings for someone,’ but instead they are coming from someone who is seen to know what she does.”
    —Terri Apter (20th century)