ceph/src/boost/libs/math/doc/html/math_toolkit/dist_ref/dists/lognormal_dist.html

   1 <html>
   2 <head>
   3 <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
   4 <title>Log Normal Distribution</title>
   5 <link rel="stylesheet" href="../../../math.css" type="text/css">
   6 <meta name="generator" content="DocBook XSL Stylesheets V1.77.1">
   7 <link rel="home" href="../../../index.html" title="Math Toolkit 2.5.1">
   8 <link rel="up" href="../dists.html" title="Distributions">
   9 <link rel="prev" href="logistic_dist.html" title="Logistic Distribution">
  10 <link rel="next" href="negative_binomial_dist.html" title="Negative Binomial 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="logistic_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="negative_binomial_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.lognormal_dist"></a><a class="link" href="lognormal_dist.html" title="Log Normal Distribution">Log Normal
  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">lognormal</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
  31 <pre class="programlisting"><span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span><span class="special">{</span>
  32
  33 <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>
  34           <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>
  35 <span class="keyword">class</span> <span class="identifier">lognormal_distribution</span><span class="special">;</span>
  36
  37 <span class="keyword">typedef</span> <span class="identifier">lognormal_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">lognormal</span><span class="special">;</span>
  38
  39 <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>
  40 <span class="keyword">class</span> <span class="identifier">lognormal_distribution</span>
  41 <span class="special">{</span>
  42 <span class="keyword">public</span><span class="special">:</span>
  43    <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
  44    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  45    <span class="comment">// Construct:</span>
  46    <span class="identifier">lognormal_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>
  47    <span class="comment">// Accessors:</span>
  48    <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  49    <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
  50 <span class="special">};</span>
  51
  52 <span class="special">}}</span> <span class="comment">// namespaces</span>
  53 </pre>
  54 <p>
  55           The lognormal distribution is the distribution that arises when the logarithm
  56           of the random variable is normally distributed. A lognormal distribution
  57           results when the variable is the product of a large number of independent,
  58           identically-distributed variables.
  59         </p>
  60 <p>
  61           For location and scale parameters <span class="emphasis"><em>m</em></span> and <span class="emphasis"><em>s</em></span>
  62           it is defined by the probability density function:
  63         </p>
  64 <p>
  65           <span class="inlinemediaobject"><img src="../../../../equations/lognormal_ref.svg"></span>
  66         </p>
  67 <p>
  68           The location and scale parameters are equivalent to the mean and standard
  69           deviation of the logarithm of the random variable.
  70         </p>
  71 <p>
  72           The following graph illustrates the effect of the location parameter on
  73           the PDF, note that the range of the random variable remains [0,+&#8734;] irrespective
  74           of the value of the location parameter:
  75         </p>
  76 <p>
  77           <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf1.svg" align="middle"></span>
  78         </p>
  79 <p>
  80           The next graph illustrates the effect of the scale parameter on the PDF:
  81         </p>
  82 <p>
  83           <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf2.svg" align="middle"></span>
  84         </p>
  85 <h5>
  86 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h0"></a>
  87           <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.member_functions"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.member_functions">Member
  88           Functions</a>
  89         </h5>
  90 <pre class="programlisting"><span class="identifier">lognormal_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>
  91 </pre>
  92 <p>
  93           Constructs a lognormal distribution with location <span class="emphasis"><em>location</em></span>
  94           and scale <span class="emphasis"><em>scale</em></span>.
  95         </p>
  96 <p>
  97           The location parameter is the same as the mean of the logarithm of the
  98           random variate.
  99         </p>
 100 <p>
 101           The scale parameter is the same as the standard deviation of the logarithm
 102           of the random variate.
 103         </p>
 104 <p>
 105           Requires that the scale parameter is greater than zero, otherwise calls
 106           <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
 107         </p>
 108 <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>
 109 </pre>
 110 <p>
 111           Returns the <span class="emphasis"><em>location</em></span> parameter of this distribution.
 112         </p>
 113 <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>
 114 </pre>
 115 <p>
 116           Returns the <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 117         </p>
 118 <h5>
 119 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h1"></a>
 120           <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.non_member_accessors"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.non_member_accessors">Non-member
 121           Accessors</a>
 122         </h5>
 123 <p>
 124           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 125           functions</a> that are generic to all distributions are supported:
 126           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 127           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 128           <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>,
 129           <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>,
 130           <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>,
 131           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 132           <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>,
 133           <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>.
 134         </p>
 135 <p>
 136           The domain of the random variable is [0,+&#8734;].
 137         </p>
 138 <h5>
 139 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h2"></a>
 140           <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.accuracy"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.accuracy">Accuracy</a>
 141         </h5>
 142 <p>
 143           The lognormal distribution is implemented in terms of the standard library
 144           log and exp functions, plus the <a class="link" href="../../sf_erf/error_function.html" title="Error Functions">error
 145           function</a>, and as such should have very low error rates.
 146         </p>
 147 <h5>
 148 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h3"></a>
 149           <span class="phrase"><a name="math_toolkit.dist_ref.dists.lognormal_dist.implementation"></a></span><a class="link" href="lognormal_dist.html#math_toolkit.dist_ref.dists.lognormal_dist.implementation">Implementation</a>
 150         </h5>
 151 <p>
 152           In the following table <span class="emphasis"><em>m</em></span> is the location parameter
 153           of the distribution, <span class="emphasis"><em>s</em></span> is its scale parameter, <span class="emphasis"><em>x</em></span>
 154           is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
 155           = 1-p</em></span>.
 156         </p>
 157 <div class="informaltable"><table class="table">
 158 <colgroup>
 159 <col>
 160 <col>
 161 </colgroup>
 162 <thead><tr>
 163 <th>
 164                   <p>
 165                     Function
 166                   </p>
 167                 </th>
 168 <th>
 169                   <p>
 170                     Implementation Notes
 171                   </p>
 172                 </th>
 173 </tr></thead>
 174 <tbody>
 175 <tr>
 176 <td>
 177                   <p>
 178                     pdf
 179                   </p>
 180                 </td>
 181 <td>
 182                   <p>
 183                     Using the relation: pdf = e<sup>-(ln(x) - m)<sup>2 </sup> / 2s<sup>2 </sup> </sup> / (x * s * sqrt(2pi))
 184                   </p>
 185                 </td>
 186 </tr>
 187 <tr>
 188 <td>
 189                   <p>
 190                     cdf
 191                   </p>
 192                 </td>
 193 <td>
 194                   <p>
 195                     Using the relation: p = cdf(normal_distribtion&lt;RealType&gt;(m,
 196                     s), log(x))
 197                   </p>
 198                 </td>
 199 </tr>
 200 <tr>
 201 <td>
 202                   <p>
 203                     cdf complement
 204                   </p>
 205                 </td>
 206 <td>
 207                   <p>
 208                     Using the relation: q = cdf(complement(normal_distribtion&lt;RealType&gt;(m,
 209                     s), log(x)))
 210                   </p>
 211                 </td>
 212 </tr>
 213 <tr>
 214 <td>
 215                   <p>
 216                     quantile
 217                   </p>
 218                 </td>
 219 <td>
 220                   <p>
 221                     Using the relation: x = exp(quantile(normal_distribtion&lt;RealType&gt;(m,
 222                     s), p))
 223                   </p>
 224                 </td>
 225 </tr>
 226 <tr>
 227 <td>
 228                   <p>
 229                     quantile from the complement
 230                   </p>
 231                 </td>
 232 <td>
 233                   <p>
 234                     Using the relation: x = exp(quantile(complement(normal_distribtion&lt;RealType&gt;(m,
 235                     s), q)))
 236                   </p>
 237                 </td>
 238 </tr>
 239 <tr>
 240 <td>
 241                   <p>
 242                     mean
 243                   </p>
 244                 </td>
 245 <td>
 246                   <p>
 247                     e<sup>m + s<sup>2 </sup> / 2 </sup>
 248                   </p>
 249                 </td>
 250 </tr>
 251 <tr>
 252 <td>
 253                   <p>
 254                     variance
 255                   </p>
 256                 </td>
 257 <td>
 258                   <p>
 259                     (e<sup>s<sup>2</sup> </sup> - 1) * e<sup>2m + s<sup>2 </sup> </sup>
 260                   </p>
 261                 </td>
 262 </tr>
 263 <tr>
 264 <td>
 265                   <p>
 266                     mode
 267                   </p>
 268                 </td>
 269 <td>
 270                   <p>
 271                     e<sup>m - s<sup>2 </sup> </sup>
 272                   </p>
 273                 </td>
 274 </tr>
 275 <tr>
 276 <td>
 277                   <p>
 278                     skewness
 279                   </p>
 280                 </td>
 281 <td>
 282                   <p>
 283                     sqrt(e<sup>s<sup>2</sup> </sup> - 1) * (2 + e<sup>s<sup>2</sup> </sup>)
 284                   </p>
 285                 </td>
 286 </tr>
 287 <tr>
 288 <td>
 289                   <p>
 290                     kurtosis
 291                   </p>
 292                 </td>
 293 <td>
 294                   <p>
 295                     e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 3
 296                   </p>
 297                 </td>
 298 </tr>
 299 <tr>
 300 <td>
 301                   <p>
 302                     kurtosis excess
 303                   </p>
 304                 </td>
 305 <td>
 306                   <p>
 307                     e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 6
 308                   </p>
 309                 </td>
 310 </tr>
 311 </tbody>
 312 </table></div>
 313 </div>
 314 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 315 <td align="left"></td>
 316 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 317       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 318       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 319       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 320         Distributed under the Boost Software License, Version 1.0. (See accompanying
 321         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>)
 322       </p>
 323 </div></td>
 324 </tr></table>
 325 <hr>
 326 <div class="spirit-nav">
 327 <a accesskey="p" href="logistic_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="negative_binomial_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
 328 </div>
 329 </body>
 330 </html>