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| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.extreme_dist"></a><a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme Value |
| Distribution</a> |
| </h4></div></div></div> |
| <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> |
| <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="Chapter 14. 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> |
| <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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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.svg" 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.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.extreme_dist.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.member_functions"></a></span><a class="link" href="extreme_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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_ref.dists.extreme_dist.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.non_member_accessors"></a></span><a class="link" href="extreme_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>, |
| <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>, |
| <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>, |
| <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>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</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>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>, |
| <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>. |
| </p> |
| <p> |
| The domain of the random parameter is [-∞, +∞]. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.extreme_dist.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.accuracy"></a></span><a class="link" href="extreme_dist.html#math_toolkit.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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.extreme_dist.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.implementation"></a></span><a class="link" href="extreme_dist.html#math_toolkit.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 © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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> |
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