<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Laplace 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="hypergeometric_dist.html" title="Hypergeometric Distribution"> <link rel="next" href="logistic_dist.html" title="Logistic 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> <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> <td align="center"><a href="../../../../../../../../../more/index.htm">More</a></td> </tr></table> <hr> <div class="spirit-nav"> <a accesskey="p" href="hypergeometric_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="logistic_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> <div class="section" lang="en"> <div class="titlepage"><div><div><h5 class="title"> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution"> Laplace 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">laplace</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span></pre> <p> </p> <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</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">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">laplace_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special"><></span> <span class="identifier">laplace</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">laplace_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="comment">// Construct: </span> <span class="identifier">laplace_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="comment">// Accessors: </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> <span class="special">}}</span> <span class="comment">// namespaces </span></pre> <p> Laplace distribution is the distribution of differences between two independent variates with identical exponential distributions (Abramowitz and Stegun 1972, p. 930). It is also called the double exponential distribution. </p> <p> For location parameter μ and scale parameter σ it is defined by the probability density function: </p> <p> <span class="inlinemediaobject"><img src="../../../../../equations/laplace_pdf.png"></span> </p> <p> The location and scale parameters are equivalent to the mean and standard deviation of the normal or Gaussian distribution. </p> <p> The following graph illustrates the effect of the parameters μ and σ on the PDF. Note that the range of the random variable remains [-∞,+∞] irrespective of the value of the location parameter: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/laplace_pdf.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.member_functions"></a><h5> <a name="id1037744"></a> <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">laplace_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 laplace distribution with location <span class="emphasis"><em>location</em></span> and scale <span class="emphasis"><em>scale</em></span>. </p> <p> The location parameter is the same as the mean of the random variate. </p> <p> The scale parameter is proportional to the standard deviation of the random variate. </p> <p> Requires that the scale parameter is greater than zero, 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 <span class="emphasis"><em>location</em></span> parameter of this 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 <span class="emphasis"><em>scale</em></span> parameter of this distribution. </p> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.non_member_accessors"></a><h5> <a name="id1037906"></a> <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_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> The domain of the random variable is [-∞,+∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.accuracy"></a><h5> <a name="id1038003"></a> <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.accuracy">Accuracy</a> </h5> <p> The laplace distribution is implemented in terms of the standard library log and exp functions and as such should have very small errors. </p> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.implementation"></a><h5> <a name="id1038022"></a> <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.implementation">Implementation</a> </h5> <p> In the following table μ is the location parameter of the distribution, σ 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 its complement <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 = e<sup>-abs(x-μ) / σ</sup> / (2 * σ) </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relations: </p> <p> x < μ : p = e<sup>(x-μ)/σ </sup> / σ </p> <p> x >= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relation: </p> <p> -x < μ : q = e<sup>(-x-μ)/σ </sup> / σ </p> <p> -x >= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> Using the relations: </p> <p> p < 0.5 : x = μ + σ * log(2*p) </p> <p> p >= 0.5 : x = μ - σ * log(2-2*p) </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> Using the relation: </p> <p> q > 0.5: x = μ + σ*log(2-2*q) </p> <p> q <=0.5: x = μ - σ*log( 2*q ) </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> μ </p> </td> </tr> <tr> <td> <p> variance </p> </td> <td> <p> 2 * σ<sup>2</sup> </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> μ </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> 0 </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> 6 </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> 3 </p> </td> </tr> </tbody> </table></div> <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.references"></a><h5> <a name="id1038344"></a> <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.references">References</a> </h5> <div class="itemizedlist"><ul type="disc"> <li> <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein, Eric W. "Laplace Distribution."</a> From MathWorld--A Wolfram Web Resource. </li> <li> <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace Distribution</a> </li> <li> M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, p. 930. </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="hypergeometric_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="logistic_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>