Wrapped distribution

In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit n-sphere. In one dimension, a wrapped distribution consists of points on the unit circle. If ${\displaystyle \phi }$ is a random variate in the interval ${\displaystyle (-\infty ,\infty )}$ with probability density function (PDF) ${\displaystyle p(\phi )}$, then ${\displaystyle z=e^{i\phi }}$ is a circular variable distributed according to the wrapped distribution ${\displaystyle p_{wz}(\theta )}$ and ${\displaystyle \theta =\arg(z)}$ is an angular variable in the interval ${\displaystyle (-\pi ,\pi ]}$ distributed according to the wrapped distribution ${\displaystyle p_{w}(\theta )}$.

Any probability density function ${\displaystyle p(\phi )}$ on the line can be "wrapped" around the circumference of a circle of unit radius. That is, the PDF of the wrapped variable

${\displaystyle \theta =\phi \mod 2\pi }$ in some interval of length ${\displaystyle 2\pi }$

is

${\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}}$

which is a periodic sum of period ${\displaystyle 2\pi }$. The preferred interval is generally ${\displaystyle (-\pi <\theta \leq \pi )}$ for which ${\displaystyle \ln(e^{i\theta })=\arg(e^{i\theta })=\theta }$.