1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Cauchy-Lorentz Distribution</title> 5<link rel="stylesheet" href="../../../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../../../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../dists.html" title="Distributions"> 9<link rel="prev" href="binomial_dist.html" title="Binomial Distribution"> 10<link rel="next" href="chi_squared_dist.html" title="Chi Squared Distribution"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<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> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h4 class="title"> 27<a name="math_toolkit.dist_ref.dists.cauchy_dist"></a><a class="link" href="cauchy_dist.html" title="Cauchy-Lorentz Distribution">Cauchy-Lorentz 28 Distribution</a> 29</h4></div></div></div> 30<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> 31<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> 32 <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 21. 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> 33<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span><span class="special">;</span> 34 35<span class="keyword">typedef</span> <span class="identifier">cauchy_distribution</span><span class="special"><></span> <span class="identifier">cauchy</span><span class="special">;</span> 36 37<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 21. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">></span> 38<span class="keyword">class</span> <span class="identifier">cauchy_distribution</span> 39<span class="special">{</span> 40<span class="keyword">public</span><span class="special">:</span> 41 <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> 42 <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> 43 44 <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> 45 46 <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 47 <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> 48<span class="special">};</span> 49</pre> 50<p> 51 The <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz 52 distribution</a> is named after Augustin Cauchy and Hendrik Lorentz. 53 It is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous 54 probability distribution</a> with <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">probability 55 distribution function PDF</a> given by: 56 </p> 57<div class="blockquote"><blockquote class="blockquote"><p> 58 <span class="inlinemediaobject"><img src="../../../../equations/cauchy_ref1.svg"></span> 59 60 </p></blockquote></div> 61<p> 62 The location parameter <span class="emphasis"><em>x<sub>0</sub></em></span> is the location of the peak 63 of the distribution (the mode of the distribution), while the scale parameter 64 γ specifies half the width of the PDF at half the maximum height. If the 65 location is zero, and the scale 1, then the result is a standard Cauchy 66 distribution. 67 </p> 68<p> 69 The distribution is important in physics as it is the solution to the differential 70 equation describing forced resonance, while in spectroscopy it is the description 71 of the line shape of spectral lines. 72 </p> 73<p> 74 The following graph shows how the distributions moves as the location parameter 75 changes: 76 </p> 77<div class="blockquote"><blockquote class="blockquote"><p> 78 <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf1.svg" align="middle"></span> 79 80 </p></blockquote></div> 81<p> 82 While the following graph shows how the shape (scale) parameter alters 83 the distribution: 84 </p> 85<div class="blockquote"><blockquote class="blockquote"><p> 86 <span class="inlinemediaobject"><img src="../../../../graphs/cauchy_pdf2.svg" align="middle"></span> 87 88 </p></blockquote></div> 89<h5> 90<a name="math_toolkit.dist_ref.dists.cauchy_dist.h0"></a> 91 <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 92 Functions</a> 93 </h5> 94<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> 95</pre> 96<p> 97 Constructs a Cauchy distribution, with location parameter <span class="emphasis"><em>location</em></span> 98 and scale parameter <span class="emphasis"><em>scale</em></span>. When these parameters take 99 their default values (location = 0, scale = 1) then the result is a Standard 100 Cauchy Distribution. 101 </p> 102<p> 103 Requires scale > 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>. 104 </p> 105<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> 106</pre> 107<p> 108 Returns the location parameter of the distribution. 109 </p> 110<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> 111</pre> 112<p> 113 Returns the scale parameter of the distribution. 114 </p> 115<h5> 116<a name="math_toolkit.dist_ref.dists.cauchy_dist.h1"></a> 117 <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 118 Accessors</a> 119 </h5> 120<p> 121 All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor 122 functions</a> that are generic to all distributions are supported: 123 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, 124 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, 125 <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>, 126 <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>, 127 <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>, 128 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 129 <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>, 130 <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>. 131 </p> 132<p> 133 Note however that the Cauchy distribution does not have a mean, standard 134 deviation, etc. See <a class="link" href="../../pol_ref/assert_undefined.html" title="Mathematically Undefined Function Policies">mathematically 135 undefined function</a> to control whether these should fail to compile 136 with a BOOST_STATIC_ASSERTION_FAILURE, which is the default. 137 </p> 138<p> 139 Alternately, the functions <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, 140 <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, 141 <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> 142 and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a> 143 will all return a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a> 144 if called. 145 </p> 146<p> 147 The domain of the random variable is [-[max_value], +[min_value]]. 148 </p> 149<h5> 150<a name="math_toolkit.dist_ref.dists.cauchy_dist.h2"></a> 151 <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> 152 </h5> 153<p> 154 The Cauchy distribution is implemented in terms of the standard library 155 <code class="computeroutput"><span class="identifier">tan</span></code> and <code class="computeroutput"><span class="identifier">atan</span></code> 156 functions, and as such should have very low error rates. 157 </p> 158<h5> 159<a name="math_toolkit.dist_ref.dists.cauchy_dist.h3"></a> 160 <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> 161 </h5> 162<p> 163 In the following table x<sub>0 </sub> is the location parameter of the distribution, 164 γ is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, 165 <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q = 1-p</em></span>. 166 </p> 167<div class="informaltable"><table class="table"> 168<colgroup> 169<col> 170<col> 171</colgroup> 172<thead><tr> 173<th> 174 <p> 175 Function 176 </p> 177 </th> 178<th> 179 <p> 180 Implementation Notes 181 </p> 182 </th> 183</tr></thead> 184<tbody> 185<tr> 186<td> 187 <p> 188 pdf 189 </p> 190 </td> 191<td> 192 <p> 193 Using the relation: <span class="emphasis"><em>pdf = 1 / (π * γ * (1 + ((x - x<sub>0 </sub>) 194 / γ)<sup>2</sup>) </em></span> 195 </p> 196 </td> 197</tr> 198<tr> 199<td> 200 <p> 201 cdf and its complement 202 </p> 203 </td> 204<td> 205 <p> 206 The cdf is normally given by: 207 </p> 208 <div class="blockquote"><blockquote class="blockquote"><p> 209 <span class="serif_italic">p = 0.5 + atan(x)/π</span> 210 </p></blockquote></div> 211 <p> 212 But that suffers from cancellation error as x -> -∞. So recall 213 that for <code class="computeroutput"><span class="identifier">x</span> <span class="special"><</span> 214 <span class="number">0</span></code>: 215 </p> 216 <div class="blockquote"><blockquote class="blockquote"><p> 217 <span class="serif_italic">atan(x) = -π/2 - atan(1/x)</span> 218 </p></blockquote></div> 219 <p> 220 Substituting into the above we get: 221 </p> 222 <div class="blockquote"><blockquote class="blockquote"><p> 223 <span class="serif_italic">p = -atan(1/x) ; x < 0</span> 224 </p></blockquote></div> 225 <p> 226 So the procedure is to calculate the cdf for -fabs(x) using the 227 above formula. Note that to factor in the location and scale 228 parameters you must substitute (x - x<sub>0 </sub>) / γ for x in the above. 229 </p> 230 <p> 231 This procedure yields the smaller of <span class="emphasis"><em>p</em></span> and 232 <span class="emphasis"><em>q</em></span>, so the result may need subtracting from 233 1 depending on whether we want the complement or not, and whether 234 <span class="emphasis"><em>x</em></span> is less than x<sub>0 </sub> or not. 235 </p> 236 </td> 237</tr> 238<tr> 239<td> 240 <p> 241 quantile 242 </p> 243 </td> 244<td> 245 <p> 246 The same procedure is used irrespective of whether we're starting 247 from the probability or its complement. First the argument <span class="emphasis"><em>p</em></span> 248 is reduced to the range [-0.5, 0.5], then the relation 249 </p> 250 <div class="blockquote"><blockquote class="blockquote"><p> 251 <span class="serif_italic">x = x<sub>0 </sub> ± γ / tan(π * p)</span> 252 </p></blockquote></div> 253 <p> 254 is used to obtain the result. Whether we're adding or subtracting 255 from x<sub>0 </sub> is determined by whether we're starting from the complement 256 or not. 257 </p> 258 </td> 259</tr> 260<tr> 261<td> 262 <p> 263 mode 264 </p> 265 </td> 266<td> 267 <p> 268 The location parameter. 269 </p> 270 </td> 271</tr> 272</tbody> 273</table></div> 274<h5> 275<a name="math_toolkit.dist_ref.dists.cauchy_dist.h4"></a> 276 <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> 277 </h5> 278<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "> 279<li class="listitem"> 280 <a href="http://en.wikipedia.org/wiki/Cauchy_distribution" target="_top">Cauchy-Lorentz 281 distribution</a> 282 </li> 283<li class="listitem"> 284 <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm" target="_top">NIST 285 Exploratory Data Analysis</a> 286 </li> 287<li class="listitem"> 288 <a href="http://mathworld.wolfram.com/CauchyDistribution.html" target="_top">Weisstein, 289 Eric W. "Cauchy Distribution." From MathWorld--A Wolfram 290 Web Resource.</a> 291 </li> 292</ul></div> 293</div> 294<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 295<td align="left"></td> 296<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 297 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 298 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 299 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 300 Daryle Walker and Xiaogang Zhang<p> 301 Distributed under the Boost Software License, Version 1.0. (See accompanying 302 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>) 303 </p> 304</div></td> 305</tr></table> 306<hr> 307<div class="spirit-nav"> 308<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> 309</div> 310</body> 311</html> 312
