# Wrapped Cauchy distribution

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:
    $f_{WC}(\theta ;\mu ,\gamma )=\sum _{n=-\infty }^{\infty }{\frac {\gamma }{\pi (\gamma ^{2}+(\theta -\mu +2\pi n)^{2})}}\qquad -\pi <\theta <\pi$
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:
    $f_{WC}(\theta ;\mu ,\gamma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{in(\theta -\mu )-|n|\gamma }={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\mu )}}$
The PDF may also be expressed in terms of the circular variable z = e$^{iθ}$ and the complex parameter ζ = e$^{i(μ+iγ)}$
    $f_{WC}(z;\zeta )={\frac {1}{2\pi }}\,\,{\frac {1-|\zeta |^{2}}{|z-\zeta |^{2}}}$
where, as shown below, ζ = ⟨z⟩.
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:
    $\langle z^{n}\rangle =\int _{\Gamma }e^{in\theta }\,f_{WC}(\theta ;\mu ,\gamma )\,d\theta =e^{in\mu -|n|\gamma }.$
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
    $R=|\langle z\rangle |=e^{-\gamma }$
yielding a circular variance of 1 − R.

## 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:
    $H=-\int _{\Gamma }f_{WC}(\theta ;\mu ,\gamma )\,\ln(f_{WC}(\theta ;\mu ,\gamma ))\,d\theta$
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 4.224.15) and
    $c_{m}={\frac {e^{-m\gamma }}{m}}\qquad \mathrm {for} \,m>0$
(c.f. Gradshteyn and Ryzhik 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:
    $H=\ln(2\pi (1-e^{-2\gamma }))\,$

## Circular Cauchy distribution

If X is Cauchy distributed with median μ and scale parameter γ, then the complex variable
    $Z={\frac {X-i}{X+i}}$
has unit modulus and is distributed on the unit circle with density:
    $f_{CC}(\theta ,\mu ,\gamma )={\frac {1}{2\pi }}{\frac {1-|\zeta |^{2}}{|e^{i\theta }-\zeta |^{2}}}$
where
    $\zeta ={\frac {\psi -i}{\psi +i}}$
and ψ expresses the two parameters of the associated linear Cauchy distribution for x as a complex number:
    $\psi =\mu +i\gamma \,$
It can be seen that the circular Cauchy distribution has the same functional form as the wrapped Cauchy distribution in z and ζ (i.e. f$_{WC}$(z,ζ)). The circular Cauchy distribution is a reparameterized wrapped Cauchy distribution:
    $f_{CC}(\theta ,m,\gamma )=f_{WC}\left(e^{i\theta },\,{\frac {m+i\gamma -i}{m+i\gamma +i}}\right)$
The distribution $f_{CC}(\theta ;\mu ,\gamma )$ is called the circular Cauchy distribution (also the complex Cauchy distribution) with parameters μ and γ. (See also McCullagh's parametrization of the Cauchy distributions and Poisson kernel for related concepts.)
The circular Cauchy distribution expressed in complex form has finite moments of all orders
    $\operatorname {E} [Z^{n}]=\zeta ^{n},\quad \operatorname {E} [{\bar {Z}}^{n}]={\bar {\zeta }}^{n}$
for integer n ≥ 1. For |φ| < 1, the transformation
    $U(z,\phi )={\frac {z-\phi }{1-{\bar {\phi }}z}}$
is holomorphic on the unit disk, and the transformed variable U(Z, φ) is distributed as complex Cauchy with parameter U(ζ, φ).
Given a sample z$_1$, ..., z$_n$ of size n > 2, the maximum-likelihood equation
    $n^{-1}U\left(z,{\hat {\zeta }}\right)=n^{-1}\sum U\left(z_{j},{\hat {\zeta }}\right)=0$
can be solved by a simple fixed-point iteration:
    $\zeta ^{(r+1)}=U\left(n^{-1}U(z,\zeta ^{(r)}),\,-\zeta ^{(r)}\right)\,$
starting with ζ$^{(0)}$ = 0. The sequence of likelihood values is non-decreasing, and the solution is unique for samples containing at least three distinct values.
The maximum-likelihood estimate for the median (${\hat {\mu }}$) and scale parameter (${\hat {\gamma }}$) of a real Cauchy sample is obtained by the inverse transformation:
    ${\hat {\mu }}\pm i{\hat {\gamma }}=i{\frac {1+{\hat {\zeta }}}{1-{\hat {\zeta }}}}.$
For n ≤ 4, closed-form expressions are known for ${\hat {\zeta }}$. The density of the maximum-likelihood estimator at t in the unit disk is necessarily of the form:
    ${\frac {1}{4\pi }}{\frac {p_{n}(\chi (t,\zeta ))}{(1-|t|^{2})^{2}}},$
where
    $\chi (t,\zeta )={\frac {|t-\zeta |^{2}}{4(1-|t|^{2})(1-|\zeta |^{2})}}$.
Formulae for p$_3$ and p$_4$ are available.