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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">&lt;</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">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</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 22. 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&lt;&gt;</a> <span class="special">&gt;</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">&lt;&gt;</span> <span class="identifier">extreme_value</span><span class="special">;</span>
<span class="keyword">template</span> <span class="special">&lt;</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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&amp;gbpv=0" target="_top">Extreme
Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
Nadarajah</a>.
</p>
<p>
The distribution has a PDF given by:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x) = (1/scale) e<sup>-(x-location)/scale</sup> e<sup>-e<sup>-(x-location)/scale</sup></sup></span>
</p></blockquote></div>
<p>
which in the standard case (scale = 1, location = 0) reduces to:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="serif_italic">f(x) = e<sup>-x</sup>e<sup>-e<sup>-x</sup></sup></span>
</p></blockquote></div>
<p>
The following graph illustrates how the PDF varies with the location parameter:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.svg" align="middle"></span>
</p></blockquote></div>
<p>
And this graph illustrates how the PDF varies with the shape parameter:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.svg" align="middle"></span>
</p></blockquote></div>
<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">&gt;</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>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>)
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