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.