<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Cauchy-Lorentz Distribution</title> <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> <link rel="home" href="../../../../index.html" title="Math Toolkit"> <link rel="up" href="../dists.html" title="Distributions"> <link rel="prev" href="binomial_dist.html" title="Binomial Distribution"> <link rel="next" href="chi_squared_dist.html" title="Chi Squared Distribution"> </head> <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> <table cellpadding="2" width="100%"><tr> <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../../../boost.png"></td> <td align="center"><a href="../../../../../../../../../index.html">Home</a></td> <td align="center"><a href="../../../../../../../../../libs/libraries.htm">Libraries</a></td> 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Cauchy-Lorentz Distribution</a> </h5></div></div></div> <p> </p> <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">cauchy</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> <p> </p> <pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> <span class="keyword">class</span> <span class="identifier">cauchy_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="identifier">cauchy</span><span class="special">;</span> <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> <span class="keyword">class</span> <span class="identifier">cauchy_distribution</span> <span class="special">{</span> <span class="keyword">public</span><span class="special">:</span> <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> <span class="identifier">cauchy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> <span class="special">};</span> </pre> <p> The <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz distribution</a> is named after Augustin Cauchy and Hendrik Lorentz. It is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous probability distribution</a> with <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability distribution function PDF</a> given by: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/cauchy_ref1.png"></span> </p> <p> The location parameter x<sub>0</sub> is the location of the peak of the distribution (the mode of the distribution), while the scale parameter γ specifies half the width of the PDF at half the maximum height. If the location is zero, and the scale 1, then the result is a standard Cauchy distribution. </p> <p> The distribution is important in physics as it is the solution to the differential equation describing forced resonance, while in spectroscopy it is the description of the line shape of spectral lines. </p> <p> The following graph shows how the distributions moves as the location parameter changes: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/cauchy_pdf1.png" align="middle"></span> </p> <p> While the following graph shows how the shape (scale) parameter alters the distribution: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/cauchy_pdf2.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.member_functions"></a><h5> <a name="id1027102"></a> <a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">cauchy_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span> </pre> <p> Constructs a Cauchy distribution, with location parameter <span class="emphasis"><em>location</em></span> and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take their default values (location = 0, scale = 1) then the result is a Standard Cauchy Distribution. </p> <p> Requires scale > 0, otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the location parameter of the distribution. </p> <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> </pre> <p> Returns the scale parameter of the distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.non_member_accessors"></a><h5> <a name="id1027250"></a> <a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.non_member_accessors">Non-member Accessors</a> </h5> <p> All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor functions</a> that are generic to all distributions are supported: <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. </p> <p> Note however that the Cauchy distribution does not have a mean, standard deviation, etc. See <a class="link" href="../../../policy/pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically undefined function</a> to control whether these should fail to compile with a BOOST_STATIC_ASSERTION_FAILURE, which is the default. </p> <p> Alternately, the functions <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a> and <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a> will all return a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> if called. </p> <p> The domain of the random variable is [-[max_value], +[min_value]]. </p> <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.accuracy"></a><h5> <a name="id1027392"></a> <a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.accuracy">Accuracy</a> </h5> <p> The Cauchy distribution is implemented in terms of the standard library <code class="computeroutput"><span class="identifier">tan</span></code> and <code class="computeroutput"><span class="identifier">atan</span></code> functions, and as such should have very low error rates. </p> <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.implementation"></a><h5> <a name="id1027426"></a> <a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.implementation">Implementation</a> </h5> <p> In the following table x<sub>0 </sub> is the location parameter of the distribution, γ is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. </p> <div class="informaltable"><table class="table"> <colgroup> <col> <col> </colgroup> <thead><tr> <th> <p> Function </p> </th> <th> <p> Implementation Notes </p> </th> </tr></thead> <tbody> <tr> <td> <p> pdf </p> </td> <td> <p> Using the relation: pdf = 1 / (π * γ * (1 + ((x - x<sub>0 </sub>) / γ)<sup>2</sup>) </p> </td> </tr> <tr> <td> <p> cdf and its complement </p> </td> <td> <p> The cdf is normally given by: </p> <p> p = 0.5 + atan(x)/π </p> <p> But that suffers from cancellation error as x -> -∞. So recall that for <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> <span class="number">0</span></code>: </p> <p> atan(x) = -π/2 - atan(1/x) </p> <p> Substituting into the above we get: </p> <p> p = -atan(1/x) ; x < 0 </p> <p> So the procedure is to calculate the cdf for -fabs(x) using the above formula. Note that to factor in the location and scale parameters you must substitute (x - x<sub>0 </sub>) / γ for x in the above. </p> <p> This procedure yields the smaller of <span class="emphasis"><em>p</em></span> and <span class="emphasis"><em>q</em></span>, so the result may need subtracting from 1 depending on whether we want the complement or not, and whether <span class="emphasis"><em>x</em></span> is less than x<sub>0 </sub> or not. </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> The same procedure is used irrespective of whether we're starting from the probability or it's complement. First the argument <span class="emphasis"><em>p</em></span> is reduced to the range [-0.5, 0.5], then the relation </p> <p> x = x<sub>0 </sub> ± γ / tan(π * p) </p> <p> is used to obtain the result. Whether we're adding or subtracting from x<sub>0 </sub> is determined by whether we're starting from the complement or not. </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> The location parameter. </p> </td> </tr> </tbody> </table></div> <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.references"></a><h5> <a name="id1027653"></a> <a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.references">References</a> </h5> <div class="itemizedlist"><ul type="disc"> <li> <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz distribution</a> </li> <li> <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm" target="_top">NIST Exploratory Data Analysis</a> </li> <li> <a href="http://mathworld.wolfram.com/CauchyDistribution.html" target="_top">Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource.</a> </li> </ul></div> </div> <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> <td align="left"></td> <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009 John Maddock, Paul A. Bristow, Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and Thijs van den Berg<p> Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) </p> </div></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="binomial_dist.html"><img src="../../../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../../index.html"><img src="../../../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>