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 ϕ {\displaystyle \phi } is a random variate in the interval ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} with probability density function (PDF) p ( ϕ ) {\displaystyle p(\phi )} , then z = e i ϕ {\displaystyle z=e^{i\phi }} is a circular variable distributed according to the wrapped distribution p w z ( θ ) {\displaystyle p_{wz}(\theta )} and θ = arg ( z ) {\displaystyle \theta =\arg(z)} is an angular variable in the interval ( − π , π ] {\displaystyle (-\pi ,\pi ]} distributed according to the wrapped distribution p w ( θ ) {\displaystyle p_{w}(\theta )} . Any probability density function 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 } .
Theory In most situations, a process involving circular statistics produces angles ( ϕ {\displaystyle \phi } ) which lie in the interval ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} , and are described by an "unwrapped" probability density function p ( ϕ ) {\displaystyle p(\phi )} . However, a measurement will yield an angle θ {\displaystyle \theta } which lies in some interval of length 2 π {\displaystyle 2\pi } (for example, 0 to 2 π {\displaystyle 2\pi } ). In other words, a measurement cannot tell whether the true angle ϕ {\displaystyle \phi } or a wrapped angle θ = ϕ + 2 π a {\displaystyle \theta =\phi +2\pi a} , where a {\displaystyle a} is some unknown integer, has been measured. If we wish to calculate the expected value of some function of the measured angle it will be:
⟨ f ( θ ) ⟩ = ∫ − ∞ ∞ p ( ϕ ) f ( ϕ + 2 π a ) d ϕ {\displaystyle \langle f(\theta )\rangle =\int _{-\infty }^{\infty }p(\phi )f(\phi +2\pi a)d\phi } . We can express the integral as a sum of integrals over periods of 2 π {\displaystyle 2\pi } :
⟨ f ( θ ) ⟩ = ∑ k = − ∞ ∞ ∫ 2 π k 2 π ( k + 1 ) p ( ϕ ) f ( ϕ + 2 π a ) d ϕ {\displaystyle \langle f(\theta )\rangle =\sum _{k=-\infty }^{\infty }\int _{2\pi k}^{2\pi (k+1)}p(\phi )f(\phi +2\pi a)d\phi } . Changing the variable of integration to θ ′ = ϕ − 2 π k {\displaystyle \theta '=\phi -2\pi k} and exchanging the order of integration and summation, we have
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