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| {{Probability distribution|
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| name =Wrapped Cauchy|
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| type =density|
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| pdf_image =[[Image:WrappedCauchyPDF.png|325px|Plot of the wrapped Cauchy PDF, <math>\mu=0</math>]]<br /><small>The support is chosen to be [-π,π)</small>|
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| cdf_image =[[Image:WrappedCauchyCDF.png|325px|Plot of the wrapped Cauchy CDF <math>\mu=0</math>]]<br /><small>The support is chosen to be [-π,π)</small>|
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| parameters =<math>\mu</math> Real<br /><math>\gamma>0</math>|
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| support =<math>-\pi\le\theta<\pi</math>|
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| pdf =<math>\frac{1}{2\pi}\,\frac{\sinh(\gamma)}{\cosh(\gamma)-\cos(\theta-\mu)}</math>|
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| cdf =<math>\,</math>|
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| mean =<math>\mu</math> (circular)|
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| median =|
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| mode =|
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| variance =<math>1-e^{-\gamma}</math> (circular)|
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| skewness =|
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| kurtosis =|
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| entropy =<math>\ln(2\pi(1-e^{-2\gamma}))</math> (differential)|
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| mgf =|
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| cf =<math>e^{in\mu-|n|\gamma}</math>|
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| }}
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| In [[probability theory]] and [[directional statistics]], a '''wrapped Cauchy distribution''' is a [[wrapped distribution|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.
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| 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]])
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| == Description ==
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| The [[probability density function]] of the wrapped [[Cauchy distribution]] is:<ref name="Mardia99">{{cite book |title=Directional Statistics |last=Mardia |first=Kantilal |authorlink=Kantilal Mardia |coauthors=Jupp, Peter E. |year=1999|publisher=Wiley |location= |isbn=978-0-471-95333-3 |url=http://www.amazon.com/Directional-Statistics-Kanti-V-Mardia/dp/0471953334/ref=sr_1_1?s=books&ie=UTF8&qid=1311003484&sr=1-1#reader_0471953334 |accessdate=2011-07-19}}</ref>
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| :<math>
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| f_{WC}(\theta;\mu,\gamma)=\sum_{n=-\infty}^\infty \frac{\gamma}{\pi(\gamma^2+(\theta-\mu+2\pi n)^2)}
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| </math> | |
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| where <math>\gamma</math> is the scale factor and <math>\mu</math> is the peak position of the "unwrapped" distribution. [[Wrapped distribution|Expressing]] the above pdf in terms of the [[characteristic function (probability theory)|characteristic function]] of the Cauchy distribution yields:
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| :<math>
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| 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)}
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| </math>
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| In terms of the circular variable <math>z=e^{i\theta}</math> the circular moments of the wrapped Cauchy distribution are the characteristic function of the Cauchy distribution evaluated at integer arguments:
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| :<math>\langle z^n\rangle=\int_\Gamma e^{in\theta}\,f_{WC}(\theta;\mu,\gamma)\,d\theta = e^{i n \mu-|n|\gamma}.</math>
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| where <math>\Gamma\,</math> is some interval of length <math>2\pi</math>. The first moment is then the average value of ''z'', also known as the mean resultant, or mean resultant vector:
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| :<math>
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| \langle z \rangle=e^{i\mu-\gamma}
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| </math>
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| The mean angle is
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| :<math>
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| \langle \theta \rangle=\mathrm{Arg}\langle z \rangle = \mu
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| </math>
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| and the length of the mean resultant is
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|
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| :<math>
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| R=|\langle z \rangle| = e^{-\gamma}
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| </math>
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| == Estimation of parameters ==
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| A series of ''N'' measurements <math>z_n=e^{i\theta_n}</math> drawn from a wrapped Cauchy distribution may be used to estimate certain parameters of the distribution. The average of the series <math>\overline{z}</math> is defined as
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| :<math>\overline{z}=\frac{1}{N}\sum_{n=1}^N z_n</math>
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| and its expectation value will be just the first moment:
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| :<math>\langle\overline{z}\rangle=e^{i\mu-\gamma}</math>
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| In other words, <math>\overline{z}</math> is an unbiased estimator of the first moment. If we assume that the peak position <math>\mu</math> lies in the interval <math>[-\pi,\pi)</math>, then Arg<math>(\overline{z})</math> will be a (biased) estimator of the peak position <math>\mu</math>.
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| Viewing the <math>z_n</math> as a set of vectors in the complex plane, the <math>\overline{R}^2</math> statistic is the length of the averaged vector:
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| :<math>\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</math>
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| and its expectation value is
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| :<math>\langle \overline{R}^2\rangle=\frac{1}{N}+\frac{N-1}{N}e^{-2\gamma}.</math>
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| In other words, the statistic | |
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| :<math>R_e^2=\frac{N}{N-1}\left(\overline{R}^2-\frac{1}{N}\right)</math>
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| will be an unbiased estimator of <math>e^{-2\gamma}</math>, and <math>\ln(1/R_e^2)/2</math> will be a (biased) estimator of <math>\gamma</math>.
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| == Entropy ==
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| The [[Entropy (information theory)|information entropy]] of the wrapped Cauchy distribution is defined as:<ref name="Mardia99"/>
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| :<math>H = -\int_\Gamma f_{WC}(\theta;\mu,\gamma)\,\ln(f_{WC}(\theta;\mu,\gamma))\,d\theta</math>
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| where <math>\Gamma</math> is any interval of length <math>2\pi</math>. The logarithm of the density of the wrapped Cauchy distribution may be written as a [[Fourier series]] in <math>\theta\,</math>:
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| :<math>\ln(f_{WC}(\theta;\mu,\gamma))=c_0+2\sum_{m=1}^\infty c_m \cos(m\theta) </math>
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| where
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| :<math>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</math>
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| which yields:
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| :<math>c_0 = \ln\left(\frac{1-e^{-2\gamma}}{2\pi}\right)</math>
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| (c.f. Gradshteyn and Ryzhik <ref name="G&R">{{cite book |title=Table Of Integrals, Series And Products |last=Gradshteyn |first=I. |authorlink= |coauthors=Ryzhik, I. |year=2007 |publisher=Academic Press|isbn=0-12-373637-4 |edition=7 |editor1-first=Alan |editor1-last=Jeffrey|editor2-first=Daniel |editor2-last=Zwillinger}}</ref> 4.224.15) and
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| :<math>c_m=\frac{e^{-m\gamma}}{m}\qquad \mathrm{for}\,m>0</math>
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| (c.f. Gradshteyn and Ryzhik <ref name="G&R"/> 4.397.6). The characteristic function representation for the wrapped Cauchy distribution in the left side of the integral is:
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| :<math>f_{WC}(\theta;\mu,\gamma) =\frac{1}{2\pi}\left(1+2\sum_{n=1}^\infty\phi_n\cos(n\theta)\right)</math>
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| where <math>\phi_n= e^{-|n|\gamma}</math>. 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:
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| :<math>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}</math>
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| The series is just the [[Taylor expansion]] for the logarithm of <math>(1-e^{-2\gamma})</math> so the entropy may be written in [[closed form expression|closed form]] as:
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| :<math>H=\ln(2\pi(1-e^{-2\gamma}))\,</math>
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| == See also ==
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| * [[Wrapped distribution]]
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| * [[Dirac comb]]
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| * [[Wrapped normal distribution]]
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| * [[Circular uniform distribution]]
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| * [[McCullagh's parametrization of the Cauchy distributions]]
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| == References ==
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| <references/>
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| * {{cite book |title=Statistics of Earth Science Data |last=Borradaile |first=Graham |year=2003 |publisher=Springer |isbn=978-3-540-43603-4 |url=http://books.google.com/books?id=R3GpDglVOSEC&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q=&f=false |accessdate=31 Dec 2009}}
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| * {{cite book |title=Statistical Analysis of Circular Data |last=Fisher |first=N. I. |year=1996 |publisher=Cambridge University Press |location= |isbn=978-0-521-56890-6 |url=http://books.google.com/books?id=IIpeevaNH88C&dq=%22circular+variance%22+fisher&source=gbs_navlinks_s |accessdate=2010-02-09}}
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| {{ProbDistributions|directional}}
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| [[Category:Continuous distributions]]
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| [[Category:Directional statistics]]
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| [[Category:Probability distributions]]
| |
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