Wrapped distribution

Expression in terms of characteristic functions

A fundamental wrapped distribution is the Dirac comb, which is a wrapped Dirac delta function:

${\displaystyle {\mathrm{\Delta }}_{2\pi }(\theta )=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta (\theta +2\pi k)}$.

Using the delta function, a general wrapped distribution can be written

${\displaystyle p_w(\theta )=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p({\theta }^{\prime })\delta (\theta -{\theta }^{\prime }+2\pi k)d{\theta }^{\prime }}$.

Exchanging the order of summation and integration, any wrapped distribution can be written as the convolution of the unwrapped distribution and a Dirac comb:

${\displaystyle p_w(\theta )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p({\theta }^{\prime }){\mathrm{\Delta }}_{2\pi }(\theta -{\theta }^{\prime })d{\theta }^{\prime }}$.

The Dirac comb may also be expressed as a sum of exponentials, so we may write:

${\displaystyle p_w(\theta )=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p({\theta }^{\prime })\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{in(\theta -{\theta }^{\prime })}d{\theta }^{\prime }}$.

Again exchanging the order of summation and integration:

${\displaystyle p_w(\theta )=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p({\theta }^{\prime })e^{in(\theta -{\theta }^{\prime })}d{\theta }^{\prime }}$.

Using the definition of ${\displaystyle \varphi (s)}$, the characteristic function of ${\displaystyle p(\theta )}$ yields a Laurent series about zero for the wrapped distribution in terms of the characteristic function of the unwrapped distribution:

${\displaystyle p_w(\theta )=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\varphi (n)e^{-in\theta }}$

or

${\displaystyle p_{wz}(z)=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\varphi (n)z^{-n}}$

Analogous to linear distributions, ${\displaystyle \varphi (m)}$ is referred to as the characteristic function of the wrapped distribution (or more accurately, the characteristic sequence). This is an instance of the Poisson summation formula, and it can be seen that the coefficients of the Fourier series for the wrapped distribution are simply the coefficients of the Fourier transform of the unwrapped distribution at integer values.

Moments

The moments of the wrapped distribution ${\displaystyle p_w(z)}$ are defined as:

${\displaystyle ⟨z^m⟩=\oint p_{wz}(z)z^{m}dz}$.

Expressing ${\displaystyle p_w(z)}$ in terms of the characteristic function and exchanging the order of integration and summation yields:

${\displaystyle ⟨z^m⟩=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\varphi (n)\oint z^{m-n}dz}$.

From the residue theorem we have

${\displaystyle \oint z^{m-n}dz=2\pi {\delta }_{m-n}}$

where ${\displaystyle {\delta }_k}$ is the Kronecker delta function. It follows that the moments are simply equal to the characteristic function of the unwrapped distribution for integer arguments:

${\displaystyle ⟨z^m⟩=\varphi (m)}$.

Generation of random variates

If ${\displaystyle X}$ is a random variate drawn from a linear probability distribution ${\displaystyle P}$, then ${\displaystyle Z=e^{iX}}$ is a circular variate distributed according to the wrapped ${\displaystyle P}$ distribution, and ${\displaystyle \theta =\arg (Z)}$ is the angular variate distributed according to the wrapped ${\displaystyle P}$ distribution, with ${\displaystyle -\pi <\theta \le \pi }$.

Entropy

The information entropy of a circular distribution with probability density ${\displaystyle p_w(\theta )}$ is defined as:

${\displaystyle H=-{\int }_{\mathrm{\Gamma }}{p}_w(\theta )\ln (p_w(\theta ))d\theta }$

where ${\displaystyle \mathrm{\Gamma }}$ is any interval of length ${\displaystyle 2\pi }$. If both the probability density and its logarithm can be expressed as a Fourier series (or more generally, any integral transform on the circle), the orthogonal basis of the series can be used to obtain a closed form expression for the entropy.

The moments of the distribution ${\displaystyle \varphi (n)}$ are the Fourier coefficients for the Fourier series expansion of the probability density:

${\displaystyle p_w(\theta )=\frac{1}{2\pi }\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\varphi }_n{e}^{-in\theta }}$.

If the logarithm of the probability density can also be expressed as a Fourier series:

${\displaystyle \ln (p_w(\theta ))=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_m{e}^{im\theta }}$

where

${\displaystyle c_m=\frac{1}{2\pi }{\int }_{\mathrm{\Gamma }}\ln (p_w(\theta ))e^{-im\theta }d\theta }$.

Then, exchanging the order of integration and summation, the entropy may be written as:

${\displaystyle H=-\frac{1}{2\pi }\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_m{\varphi }_n{\int }_{\mathrm{\Gamma }}{e}^{i(m-n)\theta }d\theta }$.

Using the orthogonality of the Fourier basis, the integral may be reduced to:

${\displaystyle H=-\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n{\varphi }_n}$.

For the particular case when the probability density is symmetric about the mean, ${\displaystyle c_{-m}=c_m}$ and the logarithm may be written:

${\displaystyle \ln (p_w(\theta ))=c_0+2\sum _{m=1}^{\mathrm{\infty }}{c}_m\cos (m\theta )}$

and

${\displaystyle c_m=\frac{1}{2\pi }{\int }_{\mathrm{\Gamma }}\ln (p_w(\theta ))\cos (m\theta )d\theta }$

and, since normalization requires that ${\displaystyle {\varphi }_0=1}$, the entropy may be written:

${\displaystyle H=-c_0-2\sum _{n=1}^{\mathrm{\infty }}{c}_n{\varphi }_n}$.