In probability theory and directional statistics, a wrapped normal distribution is a wrapped probability distribution that results from the "wrapping" of the normal distribution around the unit circle. It finds application in the theory of Brownian motion and is a solution to the heat equation for periodic boundary conditions. It is closely approximated by the von Mises distribution, which, due to its mathematical simplicity and tractability, is the most commonly used distribution in directional statistics.
Definition The probability density function of the wrapped normal distribution is
f WN ( θ ; μ , σ ) = 1 σ 2 π ∑ k = − ∞ ∞ exp [ − ( θ − μ + 2 π k ) 2 2 σ 2 ] , {\displaystyle f_{\text{WN}}(\theta ;\mu ,\sigma )={\frac {1}{\sigma {\sqrt {2\pi }}}}\sum _{k=-\infty }^{\infty }\exp \left[{\frac {-(\theta -\mu +2\pi k)^{2}}{2\sigma ^{2}}}\right],}
where μ and σ are the mean and standard deviation of the unwrapped distribution, respectively. Expressing the above density function in terms of the characteristic function of the normal distribution yields:
f WN ( θ ; μ , σ ) = 1 2 π ∑ n = − ∞ ∞ e − σ 2 n 2 / 2 + i n ( θ − μ ) = 1 2 π ϑ ( θ − μ 2 π , i σ 2 2 π ) , {\displaystyle f_{\text{WN}}(\theta ;\mu ,\sigma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{-\sigma ^{2}n^{2}/2+in(\theta -\mu )}={\frac {1}{2\pi }}\vartheta \left({\frac {\theta -\mu }{2\pi }},{\frac {i\sigma ^{2}}{2\pi }}\right),}
where ϑ ( θ , τ ) {\displaystyle \vartheta (\theta ,\tau )} is the Jacobi theta function, given by
ϑ ( θ , τ ) = ∑ n = − ∞ ∞ ( w 2 ) n q n 2 where w ≡ e i π θ {\displaystyle \vartheta (\theta ,\tau )=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}{\text{ where }}w\equiv e^{i\pi \theta }} and q ≡ e i π τ . {\displaystyle q\equiv e^{i\pi \tau }.}
The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:
f WN ( θ ; μ , σ ) = 1 2 π ∏ n = 1 ∞ ( 1 − q n ) ( 1 + q n − 1 / 2 z ) ( 1 + q n − 1 / 2 / z ) . {\displaystyle f_{\text{WN}}(\theta ;\mu ,\sigma )={\frac {1}{2\pi }}\prod _{n=1}^{\infty }(1-q^{n})(1+q^{n-1/2}z)(1+q^{n-1/2}/z).}
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