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.

Read more about this topic:  Directional Statistics

Famous quotes containing the word moments:

    “The essence of the modern state is that the universal be bound up with the complete freedom of its particular members and with private well-being, that thus the interests of family and civil society must concentrate themselves on the state.... It is only when both these moments subsist in their strength that the state can be regarded as articulated and genuinely organized.”
    —Georg Wilhelm Friedrich Hegel (1770–1831)

    “There are moments when very little truth would be enough to shape opinion. One might be hated at extremely low cost.”
    —Jean Rostand (1894–1977)

    “But now moments surround us
    Like a crowd, some inquisitive faces, some hostile ones,
    Some enigmatic or turned away to an anterior form of time
    Given once and for all. The jetstream inscribes a final flourish
    That melts as it stays.”
    —John Ashbery (b. 1927)