libs/math/doc/html/math_toolkit/dist_ref/dists/extreme_dist.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Extreme Value Distribution</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.2">
   8 <link rel="up" href="../dists.html" title="Distributions">
   9 <link rel="prev" href="exp_dist.html" title="Exponential Distribution">
  10 <link rel="next" href="f_dist.html" title="F Distribution">
  11 </head>
  12 <body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
  13 <table cellpadding="2" width="100%"><tr>
  14 <td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
  15 <td align="center"><a href="../../../../../../../index.html">Home</a></td>
  16 <td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
  17 <td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
  18 <td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
  19 <td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
  20 </tr></table>
  21 <hr>
  22 <div class="spirit-nav">
  23 <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>
  24 </div>
  25 <div class="section">
  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;15.&#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;15.&#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 <p>
  76           f(x) = (1/scale) e<sup>-(x-location)/scale</sup> e<sup>-e<sup>-(x-location)/scale</sup></sup>
  77         </p>
  78 <p>
  79           Which in the standard case (scale = 1, location = 0) reduces to:
  80         </p>
  81 <p>
  82           f(x) = e<sup>-x</sup>e<sup>-e<sup>-x</sup></sup>
  83         </p>
  84 <p>
  85           The following graph illustrates how the PDF varies with the location parameter:
  86         </p>
  87 <p>
  88           <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.svg" align="middle"></span>
  89         </p>
  90 <p>
  91           And this graph illustrates how the PDF varies with the shape parameter:
  92         </p>
  93 <p>
  94           <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.svg" align="middle"></span>
  95         </p>
  96 <h5>
  97 <a name="math_toolkit.dist_ref.dists.extreme_dist.h0"></a>
  98           <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
  99           Functions</a>
 100         </h5>
 101 <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>
 102 </pre>
 103 <p>
 104           Constructs an Extreme Value distribution with the specified location and
 105           scale parameters.
 106         </p>
 107 <p>
 108           Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">&gt;</span>
 109           <span class="number">0</span></code>, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 110         </p>
 111 <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>
 112 </pre>
 113 <p>
 114           Returns the location parameter of the distribution.
 115         </p>
 116 <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>
 117 </pre>
 118 <p>
 119           Returns the scale parameter of the distribution.
 120         </p>
 121 <h5>
 122 <a name="math_toolkit.dist_ref.dists.extreme_dist.h1"></a>
 123           <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
 124           Accessors</a>
 125         </h5>
 126 <p>
 127           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 128           functions</a> that are generic to all distributions are supported:
 129           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 130           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 131           <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>,
 132           <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>,
 133           <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>,
 134           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 135           <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>,
 136           <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>.
 137         </p>
 138 <p>
 139           The domain of the random parameter is [-&#8734;, +&#8734;].
 140         </p>
 141 <h5>
 142 <a name="math_toolkit.dist_ref.dists.extreme_dist.h2"></a>
 143           <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>
 144         </h5>
 145 <p>
 146           The extreme value distribution is implemented in terms of the standard
 147           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
 148           very low error rates.
 149         </p>
 150 <h5>
 151 <a name="math_toolkit.dist_ref.dists.extreme_dist.h3"></a>
 152           <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>
 153         </h5>
 154 <p>
 155           In the following table: <span class="emphasis"><em>a</em></span> is the location parameter,
 156           <span class="emphasis"><em>b</em></span> is the scale parameter, <span class="emphasis"><em>x</em></span> is
 157           the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 158           = 1-p</em></span>.
 159         </p>
 160 <div class="informaltable"><table class="table">
 161 <colgroup>
 162 <col>
 163 <col>
 164 </colgroup>
 165 <thead><tr>
 166 <th>
 167                   <p>
 168                     Function
 169                   </p>
 170                 </th>
 171 <th>
 172                   <p>
 173                     Implementation Notes
 174                   </p>
 175                 </th>
 176 </tr></thead>
 177 <tbody>
 178 <tr>
 179 <td>
 180                   <p>
 181                     pdf
 182                   </p>
 183                 </td>
 184 <td>
 185                   <p>
 186                     Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) /
 187                     b
 188                   </p>
 189                 </td>
 190 </tr>
 191 <tr>
 192 <td>
 193                   <p>
 194                     cdf
 195                   </p>
 196                 </td>
 197 <td>
 198                   <p>
 199                     Using the relation: p = exp(-exp((a-x)/b))
 200                   </p>
 201                 </td>
 202 </tr>
 203 <tr>
 204 <td>
 205                   <p>
 206                     cdf complement
 207                   </p>
 208                 </td>
 209 <td>
 210                   <p>
 211                     Using the relation: q = -expm1(-exp((a-x)/b))
 212                   </p>
 213                 </td>
 214 </tr>
 215 <tr>
 216 <td>
 217                   <p>
 218                     quantile
 219                   </p>
 220                 </td>
 221 <td>
 222                   <p>
 223                     Using the relation: a - log(-log(p)) * b
 224                   </p>
 225                 </td>
 226 </tr>
 227 <tr>
 228 <td>
 229                   <p>
 230                     quantile from the complement
 231                   </p>
 232                 </td>
 233 <td>
 234                   <p>
 235                     Using the relation: a - log(-log1p(-q)) * b
 236                   </p>
 237                 </td>
 238 </tr>
 239 <tr>
 240 <td>
 241                   <p>
 242                     mean
 243                   </p>
 244                 </td>
 245 <td>
 246                   <p>
 247                     a + <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant" target="_top">Euler-Mascheroni-constant</a>
 248                     * b
 249                   </p>
 250                 </td>
 251 </tr>
 252 <tr>
 253 <td>
 254                   <p>
 255                     standard deviation
 256                   </p>
 257                 </td>
 258 <td>
 259                   <p>
 260                     pi * b / sqrt(6)
 261                   </p>
 262                 </td>
 263 </tr>
 264 <tr>
 265 <td>
 266                   <p>
 267                     mode
 268                   </p>
 269                 </td>
 270 <td>
 271                   <p>
 272                     The same as the location parameter <span class="emphasis"><em>a</em></span>.
 273                   </p>
 274                 </td>
 275 </tr>
 276 <tr>
 277 <td>
 278                   <p>
 279                     skewness
 280                   </p>
 281                 </td>
 282 <td>
 283                   <p>
 284                     12 * sqrt(6) * zeta(3) / pi<sup>3</sup>
 285                   </p>
 286                 </td>
 287 </tr>
 288 <tr>
 289 <td>
 290                   <p>
 291                     kurtosis
 292                   </p>
 293                 </td>
 294 <td>
 295                   <p>
 296                     27 / 5
 297                   </p>
 298                 </td>
 299 </tr>
 300 <tr>
 301 <td>
 302                   <p>
 303                     kurtosis excess
 304                   </p>
 305                 </td>
 306 <td>
 307                   <p>
 308                     kurtosis - 3 or 12 / 5
 309                   </p>
 310                 </td>
 311 </tr>
 312 </tbody>
 313 </table></div>
 314 </div>
 315 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 316 <td align="left"></td>
 317 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 318       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 319       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 320       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 321         Distributed under the Boost Software License, Version 1.0. (See accompanying
 322         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>)
 323       </p>
 324 </div></td>
 325 </tr></table>
 326 <hr>
 327 <div class="spirit-nav">
 328 <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>
 329 </div>
 330 </body>
 331 </html>