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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist"></a><a class="link" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy-Lorentz |
| Distribution</a> |
| </h4></div></div></div> |
| <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> |
| <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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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.svg"></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.svg" 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.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.member_functions"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.non_member_accessors"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. |
| </p> |
| <p> |
| Note however that the Cauchy distribution does not have a mean, standard |
| deviation, etc. See <a class="link" href="../../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_toolkit.dist_ref.nmp.mean">mean</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> |
| and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a> |
| will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> |
| if called. |
| </p> |
| <p> |
| The domain of the random variable is [-[max_value], +[min_value]]. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.accuracy"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.implementation"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.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 its 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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.cauchy_dist.h4"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.cauchy_dist.references"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist_ref.dists.cauchy_dist.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> |
| <li class="listitem"> |
| <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz |
| distribution</a> |
| </li> |
| <li class="listitem"> |
| <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm" target="_top">NIST |
| Exploratory Data Analysis</a> |
| </li> |
| <li class="listitem"> |
| <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 © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>) |
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