# Wrapped Cauchy distribution

Parameters Probability density function The support is chosen to be [-π,π) Cumulative distribution function The support is chosen to be [-π,π) $\mu$ Real $\gamma>0$ $-\pi\le\theta<\pi$ $\frac{1}{2\pi}\,\frac{\sinh(\gamma)}{\cosh(\gamma)-\cos(\theta-\mu)}$ $\,$ $\mu$ (circular) $1-e^{-\gamma}$ (circular) $\ln(2\pi(1-e^{-2\gamma}))$ (differential) $e^{in\mu-|n|\gamma}$

In probability theory and directional statistics, a wrapped Cauchy distribution is a wrapped probability distribution that results from the "wrapping" of the Cauchy distribution around the unit circle. The Cauchy distribution is sometimes known as a Lorentzian distribution, and the wrapped Cauchy distribution may sometimes be referred to as a wrapped Lorentzian distribution.

The wrapped Cauchy distribution is often found in the field of spectroscopy where it is used to analyze diffraction patterns (e.g. see Fabry–Pérot interferometer)

## Description

The probability density function of the wrapped Cauchy distribution is:[1]

$f_{WC}(\theta;\mu,\gamma)=\sum_{n=-\infty}^\infty \frac{\gamma}{\pi(\gamma^2+(\theta-\mu+2\pi n)^2)}$

where $\gamma$ is the scale factor and $\mu$ is the peak position of the "unwrapped" distribution. Expressing the above pdf in terms of the characteristic function of the Cauchy distribution yields:

$f_{WC}(\theta;\mu,\gamma)=\frac{1}{2\pi}\sum_{n=-\infty}^\infty e^{in(\theta-\mu)-|n|\gamma} =\frac{1}{2\pi}\,\,\frac{\sinh\gamma}{\cosh\gamma-\cos(\theta-\mu)}$

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

$\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WC}(\theta;\mu,\gamma)\,d\theta = e^{i n \mu-|n|\gamma}.$

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:

$\langle z \rangle=e^{i\mu-\gamma}$

The mean angle is

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

and the length of the mean resultant is

$R=|\langle z \rangle| = e^{-\gamma}$

## Estimation of parameters

A series of N measurements $z_n=e^{i\theta_n}$ drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series $\overline{z}$ is defined as

$\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n$

and its expectation value will be just the first moment:

$\langle\overline{z}\rangle=e^{i\mu-\gamma}$

In other words, $\overline{z}$ is an unbiased estimator of the first moment. If we assume that the peak position $\mu$ lies in the interval $[-\pi,\pi)$, then Arg$(\overline{z})$ will be a (biased) estimator of the peak position $\mu$.

Viewing the $z_n$ as a set of vectors in the complex plane, the $\overline{R}^2$ statistic is 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 expectation value is

$\langle \overline{R}^2\rangle=\frac{1}{N}+\frac{N-1}{N}e^{-2\gamma}.$

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 $e^{-2\gamma}$, and $\ln(1/R_e^2)/2$ will be a (biased) estimator of $\gamma$.

## Entropy

The information entropy of the wrapped Cauchy distribution is defined as:[1]

$H = -\int_\Gamma f_{WC}(\theta;\mu,\gamma)\,\ln(f_{WC}(\theta;\mu,\gamma))\,d\theta$

where $\Gamma$ is any interval of length $2\pi$. The logarithm of the density of the wrapped Cauchy distribution may be written as a Fourier series in $\theta\,$:

$\ln(f_{WC}(\theta;\mu,\gamma))=c_0+2\sum_{m=1}^\infty c_m \cos(m\theta)$

where

$c_m=\frac{1}{2\pi}\int_\Gamma \ln\left(\frac{\sinh\gamma}{2\pi(\cosh\gamma-\cos\theta)}\right)\cos(m \theta)\,d\theta$

which yields:

$c_0 = \ln\left(\frac{1-e^{-2\gamma}}{2\pi}\right)$

(c.f. Gradshteyn and Ryzhik [2] 4.224.15) and

$c_m=\frac{e^{-m\gamma}}{m}\qquad \mathrm{for}\,m>0$

(c.f. Gradshteyn and Ryzhik [2] 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:

$f_{WC}(\theta;\mu,\gamma) =\frac{1}{2\pi}\left(1+2\sum_{n=1}^\infty\phi_n\cos(n\theta)\right)$

where $\phi_n= e^{-|n|\gamma}$. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:

$H = -c_0-2\sum_{m=1}^\infty \phi_m c_m = -\ln\left(\frac{1-e^{-2\gamma}}{2\pi}\right)-2\sum_{m=1}^\infty \frac{e^{-2n\gamma}}{n}$

The series is just the Taylor expansion for the logarithm of $(1-e^{-2\gamma})$ so the entropy may be written in closed form as:

$H=\ln(2\pi(1-e^{-2\gamma}))\,$