<html> <head> <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> <title>Extreme Value 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="exp_dist.html" title="Exponential Distribution"> <link rel="next" href="f_dist.html" title="F 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 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</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">extreme</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">extreme_value_distribution</span><span class="special">;</span> <span class="keyword">typedef</span> <span class="identifier">extreme_value_distribution</span><span class="special"><></span> <span class="identifier">extreme_value</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">extreme_value_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="identifier">extreme_value_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">scale</span><span class="special">()</span><span class="keyword">const</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="special">};</span> </pre> <p> There are various <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">extreme value distributions</a> : this implementation represents the maximum case, and is variously known as a Fisher-Tippett distribution, a log-Weibull distribution or a Gumbel distribution. </p> <p> Extreme value theory is important for assessing risk for highly unusual events, such as 100-year floods. </p> <p> More information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm" target="_top">NIST</a>, <a href="http://en.wikipedia.org/wiki/Extreme_value_distribution" target="_top">Wikipedia</a>, <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">Mathworld</a>, and <a href="http://en.wikipedia.org/wiki/Extreme_value_theory" target="_top">Extreme value theory</a> websites. </p> <p> The relationship of the types of extreme value distributions, of which this is but one, is discussed by <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme Value Distributions, Theory and Applications Samuel Kotz & Saralees Nadarajah</a>. </p> <p> The distribution has a PDF given by: </p> <p> f(x) = (1/scale) e<sup>-(x-location)/scale</sup> e<sup>-e<sup>-(x-location)/scale</sup></sup> </p> <p> Which in the standard case (scale = 1, location = 0) reduces to: </p> <p> f(x) = e<sup>-x</sup>e<sup>-e<sup>-x</sup></sup> </p> <p> The following graph illustrates how the PDF varies with the location parameter: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/extreme_value_pdf1.png" align="middle"></span> </p> <p> And this graph illustrates how the PDF varies with the shape parameter: </p> <p> <span class="inlinemediaobject"><img src="../../../../../graphs/extreme_value_pdf2.png" align="middle"></span> </p> <a name="math_toolkit.dist.dist_ref.dists.extreme_dist.member_functions"></a><h5> <a name="id1030810"></a> <a class="link" href="extreme_dist.html#math_toolkit.dist.dist_ref.dists.extreme_dist.member_functions">Member Functions</a> </h5> <pre class="programlisting"><span class="identifier">extreme_value_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 an Extreme Value distribution with the specified location and scale parameters. </p> <p> Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">></span> <span class="number">0</span></code>, 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.extreme_dist.non_member_accessors"></a><h5> <a name="id1030966"></a> <a class="link" href="extreme_dist.html#math_toolkit.dist.dist_ref.dists.extreme_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 parameter is [-∞, +∞]. </p> <a name="math_toolkit.dist.dist_ref.dists.extreme_dist.accuracy"></a><h5> <a name="id1031064"></a> <a class="link" href="extreme_dist.html#math_toolkit.dist.dist_ref.dists.extreme_dist.accuracy">Accuracy</a> </h5> <p> The extreme value distribution is implemented in terms of the standard library <code class="computeroutput"><span class="identifier">exp</span></code> and <code class="computeroutput"><span class="identifier">log</span></code> functions and as such should have very low error rates. </p> <a name="math_toolkit.dist.dist_ref.dists.extreme_dist.implementation"></a><h5> <a name="id1031097"></a> <a class="link" href="extreme_dist.html#math_toolkit.dist.dist_ref.dists.extreme_dist.implementation">Implementation</a> </h5> <p> In the following table: <span class="emphasis"><em>a</em></span> is the location parameter, <span class="emphasis"><em>b</em></span> is the 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 = exp((a-x)/b) * exp(-exp((a-x)/b)) / b </p> </td> </tr> <tr> <td> <p> cdf </p> </td> <td> <p> Using the relation: p = exp(-exp((a-x)/b)) </p> </td> </tr> <tr> <td> <p> cdf complement </p> </td> <td> <p> Using the relation: q = -expm1(-exp((a-x)/b)) </p> </td> </tr> <tr> <td> <p> quantile </p> </td> <td> <p> Using the relation: a - log(-log(p)) * b </p> </td> </tr> <tr> <td> <p> quantile from the complement </p> </td> <td> <p> Using the relation: a - log(-log1p(-q)) * b </p> </td> </tr> <tr> <td> <p> mean </p> </td> <td> <p> a + <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant" target="_top">Euler-Mascheroni-constant</a> * b </p> </td> </tr> <tr> <td> <p> standard deviation </p> </td> <td> <p> pi * b / sqrt(6) </p> </td> </tr> <tr> <td> <p> mode </p> </td> <td> <p> The same as the location parameter <span class="emphasis"><em>a</em></span>. </p> </td> </tr> <tr> <td> <p> skewness </p> </td> <td> <p> 12 * sqrt(6) * zeta(3) / pi<sup>3</sup> </p> </td> </tr> <tr> <td> <p> kurtosis </p> </td> <td> <p> 27 / 5 </p> </td> </tr> <tr> <td> <p> kurtosis excess </p> </td> <td> <p> kurtosis - 3 or 12 / 5 </p> </td> </tr> </tbody> </table></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="exp_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="f_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> </div> </body> </html>