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:

    “Oh Lolita, you are my girl, as Vee was Poe’s and Bea Dante’s, and what little girl would not like to whirl in a circular skirt and scanties?”
    —Vladimir Nabokov (1899–1977)

    “[O]ur people are steadily increasing their spending for higher standards of living ... the slogan of progress is changing from the full dinner pail to the full garage.”
    —Herbert Hoover (1874–1964)

    “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)