Wrapped normal distribution

The probability density function of the wrapped normal distribution is : f_\text{WN}(\theta;\mu,\sigma)=\frac{1}{\sigma \sqrt{2\pi}} \sum^{\infty}_{k=-\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_\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) . where z=e^{i(\theta-\mu)}\, and q=e^{-\sigma^2}. == Moments ==

In terms of the circular variable z=e^{i\theta} the circular moments of the wrapped normal distribution are the characteristic function of the normal distribution evaluated at integer arguments: : \langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_\text{WN}(\theta;\mu,\sigma)\,d\theta = e^{i n \mu-n^2\sigma^2/2}. where \Gamma is some interval of length 2\pi. The first moment is then the average value of z, also known as the mean resultant, or mean resultant vector: : \langle z \rangle=e^{i\mu-\sigma^2/2} The mean angle is : \theta_\mu=\mathrm{Arg}\langle z \rangle = \mu and the length of the mean resultant is : 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: : s=\ln(R^{-2})^{1/2} = \sigma == Estimation of parameters ==

A series of N measurements zn = e iθn drawn from a wrapped normal distribution may be used to estimate certain parameters of the distribution. The average of the series is defined as : \overline{z}=\frac{1}{N}\sum_{n=1}^N z_n and its expectation value will be just the first moment: : \langle\overline{z}\rangle=e^{i\mu-\sigma^2/2}. \, In other words, is an unbiased estimator of the first moment. If we assume that the mean μ lies in the interval [−π, π), then Arg  will be a (biased) estimator of the mean μ. Viewing the zn as a set of vectors in the complex plane, the 2 statistic is the square of the length of the averaged vector: : \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: : \left\langle \overline{R}^2\right\rangle = \frac{1}{N}+\frac{N-1}{N}\,e^{-\sigma^2}\, In other words, the statistic : 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/Re2) will be a (biased) estimator of σ2 == Entropy ==

The information entropy of the wrapped normal distribution is defined as: : H = -\int_\Gamma f_\text{WN}(\theta;\mu,\sigma)\,\ln(f_\text{WN}(\theta;\mu,\sigma))\,d\theta where \Gamma is any interval of length 2\pi. Defining z=e^{i(\theta-\mu)} and q=e^{-\sigma^2}, the Jacobi triple product representation for the wrapped normal is: : f_\text{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 \phi(q)\, is the Euler function. The logarithm of the density of the wrapped normal distribution may be written: : \ln(f_\text{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: : \ln(1+x)=-\sum_{k=1}^\infty \frac{(-1)^k}{k}\,x^k the logarithmic sums may be written as: : \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: : \ln(f_\text{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 \theta\,. Using the characteristic function representation for the wrapped normal distribution in the left side of the integral: : f_\text{WN}(\theta;\mu,\sigma) =\frac{1}{2\pi}\sum_{n=-\infty}^\infty q^{n^2/2}\,z^n the entropy may be written: : 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: : 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} == See also ==