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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.cauchy_dist"></a><a class="link" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution"> 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="id1018998"></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="id1019146"></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="id1019288"></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="id1019322"></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="id1019549"></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> |
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| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 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>) |
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