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wrapped dis·tri·bu·tion

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 pzw(z) and θ=arg(z) will be an angular variable in the interval (-π,π ] distributed according to the wrapped distribution pw(θ). Any probability density function (pdf) p ( ϕ ) {\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 θ = ϕ mod 2 π {\displaystyle \theta =\phi \mod 2\pi } in some interval of length 2 π {\displaystyle 2\pi } is p w ( θ ) = ∑ k = − ∞ ∞ p ( θ + 2 π k ) . {\displaystyle p_{w}(\theta )=\sum _{k=-\infty }^{\infty }{p(\theta +2\pi k)}.} which is a periodic sum of period 2 π {\displaystyle 2\pi } . The preferred interval is generally ( − π < θ ≤ π ) {\displaystyle (-\pi <\theta \leq \pi )} for which ln ⁡ ( e i θ ) = arg ⁡ ( e i θ ) = θ {\displaystyle \ln(e^{i\theta })=\arg(e^{i\theta })=\theta }

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