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Milds # 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$\gamma >0$ $-\pi \leq \theta <\pi$ ${\frac {1}{2\pi }}\,{\frac {\sinh(\gamma )}{\cosh(\gamma )-\cos(\theta -\mu )}}$ $\,$ $\mu$ (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:

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

$f_{WC}(\theta ;\mu ,\gamma )={\frac {1}{2\pi }}\sum _{n=-\infty }^{\infty }e^{in(\theta -\mu )-|n|\gamma }={\frac {1}{2\pi }}\,\,{\frac {\sinh \gamma }{\cosh \gamma -\cos(\theta -\mu )}}$ The PDF may also be expressed in terms of the circular variable z = e i θ and the complex parameter ζ =e i(μ + i γ)

$f_{WC}(z;\zeta )={\frac {1}{2\pi }}\,\,{\frac {1-|\zeta |^{2}}{|z-\zeta |^{2}}}$ 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. fWC(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 z1, ..., zn 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 p3 and p4 are available.