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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 ==
| | They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. This platform can be customizedaccording to the requirements of the business. The first thing you need to do is to choose the right web hosting plan. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>If you loved this informative article in addition to you wish to be given more information with regards to [http://ad4.fr/wordpress_backup_plugin_8757605 wordpress backup plugin] generously stop by our webpage. Numerous bloggers are utilizing Word - Press and with good reason. php file in the Word - Press root folder and look for this line (line 73 in our example):. Next you'll go by way of to your simple Word - Press site. So, we have to add our social media sharing buttons in website. If your site does well you can get paid professional designer to create a unique Word - Press theme. <br><br>You will know which of your Word - Press blog posts are attracting more unique visitors which in turn will help you develop better products and services for your customers. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Press CTRL and the numbers one to six to choose your option. |
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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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They're always ready to help, and they're always making changes to the site to make sure you won't have troubles in the first place. The next step is to visit your Word - Press blog dashboard. Wordpress Content management systems, being customer friendly, can be used extensively to write and manage websites and blogs. If you are using videos on your site then this is the plugin to use. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed.
Choosing what kind of links you'll be using is a ctitical aspect of any linkwheel strategy, especially since there are several different types of links that are assessed by search engines. You do not catch a user's attention through big and large pictures that usually takes a millennium to load up. Well Managed Administration The Word - Press can easily absorb the high numbers of traffic by controlling the server load to make sure that the site works properly. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. But in case you want some theme or plugin in sync with your business needs, it is advisable that you must seek some professional help.
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