# Directional Statistics - Moments

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. $\rho=m_1\,$ $R=|m_1|\,$ $\theta_\mu=\mathrm{Arg}(m_1).\,$

In addition, the lengths of the higher moments are defined as: $R_n=|m_n|\,$

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

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Robert Hass (b. 1941)

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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)