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 will consist of points on the unit circle. If φ is a random variate in the interval (-∞,∞;) with probability density function p(φ), then z = e ^{i φ} will be a circular variable distributed according to the wrapped distribution p_zw(z) and θ=arg(z) will be an angular variable in the interval (-π,π ] distributed according to the wrapped distribution p_w(θ).

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

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

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