Wrapped normal distribution

The probability density function of the wrapped normal distribution is[2]

${\displaystyle f_{WN}(\theta$ ;\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:[2]

${\displaystyle f_{WN}(\theta$ ;\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 ${\displaystyle \vartheta (\theta ,\tau )}$ is the Jacobi theta function, given by

${\displaystyle \vartheta (\theta ,\tau )=\sum _{n=-\infty }^{\infty }(w^{2})^{n}q^{n^{2}}{\text{ where }}w\equiv e^{i\pi \theta }}$ and ${\displaystyle q\equiv e^{i\pi \tau }.}$

The wrapped normal distribution may also be expressed in terms of the Jacobi triple product:[3]

${\displaystyle f_{WN}(\theta$ ;\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).}

where ${\displaystyle z=e^{i(\theta -\mu )}\,}$ and ${\displaystyle q=e^{-\sigma ^{2}}.}$

In terms of the circular variable ${\displaystyle z=e^{i\theta }}$ the circular moments of the wrapped normal distribution are the characteristic function of the normal distribution evaluated at integer arguments:

${\displaystyle \langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{WN}(\theta$ ;\mu ,\sigma )\,d\theta =e^{in\mu -n^{2}\sigma ^{2}/2}.}

where ${\displaystyle \Gamma \,}$ is some interval of length ${\displaystyle 2\pi }$. The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector:

${\displaystyle \langle z\rangle =e^{i\mu -\sigma ^{2}/2}}$

The mean angle is

${\displaystyle \theta _{\mu }=\mathrm {Arg} \langle z\rangle =\mu }$

and the length of the mean resultant is

${\displaystyle R=|\langle z\rangle |=e^{-\sigma ^{2}/2}}$

The circular standard deviation, which is a useful measure of dispersion for the wrapped normal distribution and its close relative, the von Mises distribution is given by:

${\displaystyle s=\ln(R^{-2})^{1/2}=\sigma }$

A series of N measurements z_n = e ^iθ_n drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series z is defined as

${\displaystyle {\overline {z}}={\frac {1}{N}}\sum _{n=1}^{N}z_{n}}$

and its expectation value will be just the first moment:

${\displaystyle \langle {\overline {z}}\rangle =e^{i\mu -\sigma ^{2}/2}.\,}$

In other words, z is an unbiased estimator of the first moment. If we assume that the mean μ lies in the interval [−π, π), then Arg z will be a (biased) estimator of the mean μ.

Viewing the z_n as a set of vectors in the complex plane, the R² statistic is the square of the length of the averaged vector:

${\displaystyle {\overline {R}}^{2}={\overline {z}}\,{\overline {z^{*}}}=\left({\frac {1}{N}}\sum _{n=1}^{N}\cos \theta _{n}\right)^{2}+\left({\frac {1}{N}}\sum _{n=1}^{N}\sin \theta _{n}\right)^{2}\,}$

and its expected value is:

${\displaystyle \left\langle {\overline {R}}^{2}\right\rangle ={\frac {1}{N}}+{\frac {N-1}{N}}\,e^{-\sigma ^{2}}\,}$

In other words, the statistic

${\displaystyle R_{e}^{2}={\frac {N}{N-1}}\left({\overline {R}}^{2}-{\frac {1}{N}}\right)}$

will be an unbiased estimator of e^−σ2, and ln(1/R_e²) will be a (biased) estimator of σ²

The information entropy of the wrapped normal distribution is defined as:[2]

${\displaystyle H=-\int _{\Gamma }f_{WN}(\theta$ ;\mu ,\sigma )\,\ln(f_{WN}(\theta ;\mu ,\sigma ))\,d\theta }

where ${\displaystyle \Gamma }$ is any interval of length ${\displaystyle 2\pi }$. Defining ${\displaystyle z=e^{i(\theta -\mu )}}$ and ${\displaystyle q=e^{-\sigma ^{2}}}$, the Jacobi triple product representation for the wrapped normal is:

${\displaystyle f_{WN}(\theta$ ;\mu ,\sigma )={\frac {\phi (q)}{2\pi }}\prod _{m=1}^{\infty }(1+q^{m-1/2}z)(1+q^{m-1/2}z^{-1})}

where ${\displaystyle \phi (q)\,}$ is the Euler function. The logarithm of the density of the wrapped normal distribution may be written:

${\displaystyle \ln(f_{WN}(\theta$ ;\mu ,\sigma ))=\ln \left({\frac {\phi (q)}{2\pi }}\right)+\sum _{m=1}^{\infty }\ln(1+q^{m-1/2}z)+\sum _{m=1}^{\infty }\ln(1+q^{m-1/2}z^{-1})}

Using the series expansion for the logarithm:

${\displaystyle \ln(1+x)=-\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\,x^{k}}$

the logarithmic sums may be written as:

${\displaystyle \sum _{m=1}^{\infty }\ln(1+q^{m-1/2}z^{\pm 1})=-\sum _{m=1}^{\infty }\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\,q^{mk-k/2}z^{\pm k}=-\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\,{\frac {q^{k/2}}{1-q^{k}}}\,z^{\pm k}}$

so that the logarithm of density of the wrapped normal distribution may be written as:

${\displaystyle \ln(f_{WN}(\theta$ ;\mu ,\sigma ))=\ln \left({\frac {\phi (q)}{2\pi }}\right)-\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{\frac {q^{k/2}}{1-q^{k}}}\,(z^{k}+z^{-k})}

which is essentially a Fourier series in ${\displaystyle \theta \,}$. Using the characteristic function representation for the wrapped normal distribution in the left side of the integral:

${\displaystyle f_{WN}(\theta$ ;\mu ,\sigma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }q^{n^{2}/2}\,z^{n}}

the entropy may be written:

${\displaystyle H=-\ln \left({\frac {\phi (q)}{2\pi }}\right)+{\frac {1}{2\pi }}\int _{\Gamma }\left(\sum _{n=-\infty }^{\infty }\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}{\frac {q^{(n^{2}+k)/2}}{1-q^{k}}}\left(z^{n+k}+z^{n-k}\right)\right)\,d\theta }$

which may be integrated to yield:

${\displaystyle H=-\ln \left({\frac {\phi (q)}{2\pi }}\right)+2\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}\,{\frac {q^{(k^{2}+k)/2}}{1-q^{k}}}}$

• Wrapped distribution
• Dirac comb
• Wrapped Cauchy distribution
• Von Mises distribution

• Borradaile, Graham (2003). Statistics of Earth Science Data. Springer. ISBN 978-3-540-43603-4. Retrieved 31 Dec 2009.
• Fisher, N. I. (1996). Statistical Analysis of Circular Data. Cambridge University Press. ISBN 978-0-521-56890-6. Retrieved 2010-02-09.
• Breitenberger, Ernst (1963). "Analogues of the normal distribution on the circle and the sphere". Biometrika. 50 (1/2): 81–88. doi:10.2307/2333749. JSTOR 2333749.

• Circular Values Math and Statistics with C++11, A C++11 infrastructure for circular values (angles, time-of-day, etc.) mathematics and statistics