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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.extreme_dist"></a><a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme Value
  28         Distribution</a>
  29 </h4></div></div></div>
  30 <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>
  31 <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>
  32           <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter&#160;20.&#160;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>
  33 <span class="keyword">class</span> <span class="identifier">extreme_value_distribution</span><span class="special">;</span>
  34
  35 <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>
  36
  37 <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&#160;20.&#160;Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
  38 <span class="keyword">class</span> <span class="identifier">extreme_value_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
  43    <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>
  44
  45    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  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="special">};</span>
  48 </pre>
  49 <p>
  50           There are various <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">extreme
  51           value distributions</a> : this implementation represents the maximum
  52           case, and is variously known as a Fisher-Tippett distribution, a log-Weibull
  53           distribution or a Gumbel distribution.
  54         </p>
  55 <p>
  56           Extreme value theory is important for assessing risk for highly unusual
  57           events, such as 100-year floods.
  58         </p>
  59 <p>
  60           More information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm" target="_top">NIST</a>,
  61           <a href="http://en.wikipedia.org/wiki/Extreme_value_distribution" target="_top">Wikipedia</a>,
  62           <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">Mathworld</a>,
  63           and <a href="http://en.wikipedia.org/wiki/Extreme_value_theory" target="_top">Extreme
  64           value theory</a> websites.
  65         </p>
  66 <p>
  67           The relationship of the types of extreme value distributions, of which
  68           this is but one, is discussed by <a href="http://www.worldscibooks.com/mathematics/p191.html" target="_top">Extreme
  69           Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
  70           Nadarajah</a>.
  71         </p>
  72 <p>
  73           The distribution has a PDF given by:
  74         </p>
  75 <div class="blockquote"><blockquote class="blockquote"><p>
  76             <span class="serif_italic">f(x) = (1/scale) e<sup>-(x-location)/scale</sup> e<sup>-e<sup>-(x-location)/scale</sup></sup></span>
  77           </p></blockquote></div>
  78 <p>
  79           which in the standard case (scale = 1, location = 0) reduces to:
  80         </p>
  81 <div class="blockquote"><blockquote class="blockquote"><p>
  82             <span class="serif_italic">f(x) = e<sup>-x</sup>e<sup>-e<sup>-x</sup></sup></span>
  83           </p></blockquote></div>
  84 <p>
  85           The following graph illustrates how the PDF varies with the location parameter:
  86         </p>
  87 <div class="blockquote"><blockquote class="blockquote"><p>
  88             <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.svg" align="middle"></span>
  89
  90           </p></blockquote></div>
  91 <p>
  92           And this graph illustrates how the PDF varies with the shape parameter:
  93         </p>
  94 <div class="blockquote"><blockquote class="blockquote"><p>
  95             <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.svg" align="middle"></span>
  96
  97           </p></blockquote></div>
  98 <h5>
  99 <a name="math_toolkit.dist_ref.dists.extreme_dist.h0"></a>
 100           <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
 101           Functions</a>
 102         </h5>
 103 <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>
 104 </pre>
 105 <p>
 106           Constructs an Extreme Value distribution with the specified location and
 107           scale parameters.
 108         </p>
 109 <p>
 110           Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">&gt;</span>
 111           <span class="number">0</span></code>, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 112         </p>
 113 <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>
 114 </pre>
 115 <p>
 116           Returns the location parameter of the distribution.
 117         </p>
 118 <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>
 119 </pre>
 120 <p>
 121           Returns the scale parameter of the distribution.
 122         </p>
 123 <h5>
 124 <a name="math_toolkit.dist_ref.dists.extreme_dist.h1"></a>
 125           <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
 126           Accessors</a>
 127         </h5>
 128 <p>
 129           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 130           functions</a> that are generic to all distributions are supported:
 131           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 132           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 133           <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>,
 134           <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>,
 135           <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>,
 136           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 137           <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>,
 138           <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>.
 139         </p>
 140 <p>
 141           The domain of the random parameter is [-&#8734;, +&#8734;].
 142         </p>
 143 <h5>
 144 <a name="math_toolkit.dist_ref.dists.extreme_dist.h2"></a>
 145           <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>
 146         </h5>
 147 <p>
 148           The extreme value distribution is implemented in terms of the standard
 149           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
 150           very low error rates.
 151         </p>
 152 <h5>
 153 <a name="math_toolkit.dist_ref.dists.extreme_dist.h3"></a>
 154           <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>
 155         </h5>
 156 <p>
 157           In the following table: <span class="emphasis"><em>a</em></span> is the location parameter,
 158           <span class="emphasis"><em>b</em></span> is the scale parameter, <span class="emphasis"><em>x</em></span> is
 159           the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 160           = 1-p</em></span>.
 161         </p>
 162 <div class="informaltable"><table class="table">
 163 <colgroup>
 164 <col>
 165 <col>
 166 </colgroup>
 167 <thead><tr>
 168 <th>
 169                   <p>
 170                     Function
 171                   </p>
 172                 </th>
 173 <th>
 174                   <p>
 175                     Implementation Notes
 176                   </p>
 177                 </th>
 178 </tr></thead>
 179 <tbody>
 180 <tr>
 181 <td>
 182                   <p>
 183                     pdf
 184                   </p>
 185                 </td>
 186 <td>
 187                   <p>
 188                     Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) /
 189                     b
 190                   </p>
 191                 </td>
 192 </tr>
 193 <tr>
 194 <td>
 195                   <p>
 196                     cdf
 197                   </p>
 198                 </td>
 199 <td>
 200                   <p>
 201                     Using the relation: p = exp(-exp((a-x)/b))
 202                   </p>
 203                 </td>
 204 </tr>
 205 <tr>
 206 <td>
 207                   <p>
 208                     cdf complement
 209                   </p>
 210                 </td>
 211 <td>
 212                   <p>
 213                     Using the relation: q = -expm1(-exp((a-x)/b))
 214                   </p>
 215                 </td>
 216 </tr>
 217 <tr>
 218 <td>
 219                   <p>
 220                     quantile
 221                   </p>
 222                 </td>
 223 <td>
 224                   <p>
 225                     Using the relation: a - log(-log(p)) * b
 226                   </p>
 227                 </td>
 228 </tr>
 229 <tr>
 230 <td>
 231                   <p>
 232                     quantile from the complement
 233                   </p>
 234                 </td>
 235 <td>
 236                   <p>
 237                     Using the relation: a - log(-log1p(-q)) * b
 238                   </p>
 239                 </td>
 240 </tr>
 241 <tr>
 242 <td>
 243                   <p>
 244                     mean
 245                   </p>
 246                 </td>
 247 <td>
 248                   <p>
 249                     a + <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant" target="_top">Euler-Mascheroni-constant</a>
 250                     * b
 251                   </p>
 252                 </td>
 253 </tr>
 254 <tr>
 255 <td>
 256                   <p>
 257                     standard deviation
 258                   </p>
 259                 </td>
 260 <td>
 261                   <p>
 262                     pi * b / sqrt(6)
 263                   </p>
 264                 </td>
 265 </tr>
 266 <tr>
 267 <td>
 268                   <p>
 269                     mode
 270                   </p>
 271                 </td>
 272 <td>
 273                   <p>
 274                     The same as the location parameter <span class="emphasis"><em>a</em></span>.
 275                   </p>
 276                 </td>
 277 </tr>
 278 <tr>
 279 <td>
 280                   <p>
 281                     skewness
 282                   </p>
 283                 </td>
 284 <td>
 285                   <p>
 286                     12 * sqrt(6) * zeta(3) / pi<sup>3</sup>
 287                   </p>
 288                 </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                   <p>
 293                     kurtosis
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     27 / 5
 299                   </p>
 300                 </td>
 301 </tr>
 302 <tr>
 303 <td>
 304                   <p>
 305                     kurtosis excess
 306                   </p>
 307                 </td>
 308 <td>
 309                   <p>
 310                     kurtosis - 3 or 12 / 5
 311                   </p>
 312                 </td>
 313 </tr>
 314 </tbody>
 315 </table></div>
 316 </div>
 317 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 318 <td align="left"></td>
 319 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
 320       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
 321       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
 322       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
 323       Daryle Walker and Xiaogang Zhang<p>
 324         Distributed under the Boost Software License, Version 1.0. (See accompanying
 325         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>)
 326       </p>
 327 </div></td>
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