Directional Statistics - Circular and Higher Dimensional Distributions

Circular and Higher Dimensional Distributions

Any probability density function on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the pdf of the wrapped variable


\theta = x_w=x \mod 2\pi\ \ \in (-\pi,\pi]

is


p_w(\theta)=\sum_{k=-\infty}^{\infty}{p(\theta+2\pi k)}.

This concept can be extended to the multivariate context by an extension of the simple sum to a number of sums that cover all dimensions in the feature space:


p_w(\vec\theta)=\sum_{k_1=-\infty}^{\infty}\cdots \sum_{k_F=-\infty}^\infty{p(\vec\theta+2\pi k_1\mathbf{e}_1+\dots+2\pi k_F\mathbf{e}_F)}

where is the th Euclidean basis vector.

Read more about this topic:  Directional Statistics

Famous quotes containing the words circular, higher and/or dimensional:

    The night in prison was novel and interesting enough.... I found that even here there was a history and a gossip which never circulated beyond the walls of the jail. Probably this is the only house in the town where verses are composed, which are afterward printed in a circular form, but not published. I was shown quite a long list of verses which were composed by some young men who had been detected in an attempt to escape, who avenged themselves by singing them.
    Henry David Thoreau (1817–1862)

    ... the majority of colored men do not yet think it worth while that women aspire to higher education.... The three R’s, a little music and a good deal of dancing, a first rate dress-maker and a bottle of magnolia balm, are quite enough generally to render charming any woman possessed of tact and the capacity for worshipping masculinity.
    Anna Julia Cooper (1859–1964)

    I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.
    Kristin Hunter (b. 1931)