Wrapped normal distribution

Moments

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:

${\displaystyle ⟨z^n⟩={\int }_{\mathrm{\Gamma }}{e}^{in\theta }{f}_{\text{WN}}(\theta ;\mu ,\sigma )d\theta =e^{in\mu -n^2{\sigma }^2/2}.}$

where ${\displaystyle \mathrm{\Gamma }}$ 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:

${\displaystyle ⟨z⟩=e^{i\mu -{\sigma }^2/2}}$

The mean angle is

${\displaystyle {\theta }_{\mu }=\mathrm{A}\mathrm{r}\mathrm{g}⟨z⟩=\mu }$

and the length of the mean resultant is

${\displaystyle R=|⟨z⟩|=e^{-{\sigma }^2/2}}$

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:

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

Estimation of parameters

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:

${\displaystyle ⟨\overline{z}⟩=e^{i\mu -{\sigma }^2/2}.}$

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:

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

and its expected value is:

${\displaystyle ⟨{\overline{R}}^2⟩=\frac{1}{N}+\frac{N-1}{N}{e}^{-{\sigma }^2}}$

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

Entropy

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

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

where ${\displaystyle \mathrm{\Gamma }}$ 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:

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

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

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

Using the series expansion for the logarithm:

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

the logarithmic sums may be written as:

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

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

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

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:

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

the entropy may be written:

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

which may be integrated to yield:

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