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<html>
<head>
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<title>Cauchy-Lorentz Distribution</title>
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<div class="section math_toolkit_dist_dist_ref_dists_cauchy_dist">
<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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.h0"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.member_functions"></a></span><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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.h1"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.non_member_accessors"></a></span><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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.h2"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.accuracy"></a></span><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>
<h5>
<a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.h3"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.implementation"></a></span><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 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.dist_ref.dists.cauchy_dist.h4"></a>
<span><a name="math_toolkit.dist.dist_ref.dists.cauchy_dist.references"></a></span><a class="link" href="cauchy_dist.html#math_toolkit.dist.dist_ref.dists.cauchy_dist.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" 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 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>
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