Directional Statistics - Moments


The raw vector (or trigonometric) moments of a circular distribution are defined as

m_n=E(z^n)=\int_\Gamma P(\theta)z^n d\theta\,

where is any interval of length and is the PDF of the circular distribution. Since the integral is unity, and the integration interval is finite, it follows that the moments of any circular distribution are always finite and well defined.

Sample moments are analogously defined:

\overline{m}_n=\frac{1}{N}\sum_{i=1}^N z_i^n.

The population resultant vector, length, and mean angle are defined in analogy with the corresponding sample parameters.




In addition, the lengths of the higher moments are defined as:


while the angular parts of the higher moments are just . The lengths of the higher moments will all lie between 0 and 1.

Read more about this topic:  Directional Statistics

Famous quotes containing the word moments:

    There are moments when the body is as numinous
    as words, days that are the good flesh continuing
    Such tenderness, those afternoons and evenings,
    saying blackberry, blackberry, blackberry.
    Robert Hass (b. 1941)

    I like to compare the holiday season with the way a child listens to a favorite story. The pleasure is in the familiar way the story begins, the anticipation of familiar turns it takes, the familiar moments of suspense, and the familiar climax and ending.
    Fred Rogers (20th century)

    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)