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  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;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>
  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;20.&#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 <div class="blockquote"><blockquote class="blockquote"><p>
  65             <span class="inlinemediaobject"><img src="../../../../equations/lognormal_ref.svg"></span>
  66
  67           </p></blockquote></div>
  68 <p>
  69           The location and scale parameters are equivalent to the mean and standard
  70           deviation of the logarithm of the random variable.
  71         </p>
  72 <p>
  73           The following graph illustrates the effect of the location parameter on
  74           the PDF, note that the range of the random variable remains [0,+&#8734;] irrespective
  75           of the value of the location parameter:
  76         </p>
  77 <div class="blockquote"><blockquote class="blockquote"><p>
  78             <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf1.svg" align="middle"></span>
  79
  80           </p></blockquote></div>
  81 <p>
  82           The next graph illustrates the effect of the scale parameter on the PDF:
  83         </p>
  84 <div class="blockquote"><blockquote class="blockquote"><p>
  85             <span class="inlinemediaobject"><img src="../../../../graphs/lognormal_pdf2.svg" align="middle"></span>
  86
  87           </p></blockquote></div>
  88 <h5>
  89 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h0"></a>
  90           <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
  91           Functions</a>
  92         </h5>
  93 <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>
  94 </pre>
  95 <p>
  96           Constructs a lognormal distribution with location <span class="emphasis"><em>location</em></span>
  97           and scale <span class="emphasis"><em>scale</em></span>.
  98         </p>
  99 <p>
 100           The location parameter is the same as the mean of the logarithm of the
 101           random variate.
 102         </p>
 103 <p>
 104           The scale parameter is the same as the standard deviation of the logarithm
 105           of the random variate.
 106         </p>
 107 <p>
 108           Requires that the scale parameter is greater than zero, otherwise calls
 109           <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 <span class="emphasis"><em>location</em></span> parameter of this 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 <span class="emphasis"><em>scale</em></span> parameter of this distribution.
 120         </p>
 121 <h5>
 122 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h1"></a>
 123           <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
 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 variable is [0,+&#8734;].
 140         </p>
 141 <h5>
 142 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h2"></a>
 143           <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>
 144         </h5>
 145 <p>
 146           The lognormal distribution is implemented in terms of the standard library
 147           log and exp functions, plus the <a class="link" href="../../sf_erf/error_function.html" title="Error Function erf and complement erfc">error
 148           function</a>, and as such should have very low error rates.
 149         </p>
 150 <h5>
 151 <a name="math_toolkit.dist_ref.dists.lognormal_dist.h3"></a>
 152           <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>
 153         </h5>
 154 <p>
 155           In the following table <span class="emphasis"><em>m</em></span> is the location parameter
 156           of the distribution, <span class="emphasis"><em>s</em></span> is its scale parameter, <span class="emphasis"><em>x</em></span>
 157           is 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 = e<sup>-(ln(x) - m)<sup>2 </sup> / 2s<sup>2 </sup> </sup> / (x * s * sqrt(2pi))
 187                   </p>
 188                 </td>
 189 </tr>
 190 <tr>
 191 <td>
 192                   <p>
 193                     cdf
 194                   </p>
 195                 </td>
 196 <td>
 197                   <p>
 198                     Using the relation: p = cdf(normal_distribtion&lt;RealType&gt;(m,
 199                     s), log(x))
 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 = cdf(complement(normal_distribtion&lt;RealType&gt;(m,
 212                     s), log(x)))
 213                   </p>
 214                 </td>
 215 </tr>
 216 <tr>
 217 <td>
 218                   <p>
 219                     quantile
 220                   </p>
 221                 </td>
 222 <td>
 223                   <p>
 224                     Using the relation: x = exp(quantile(normal_distribtion&lt;RealType&gt;(m,
 225                     s), p))
 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: x = exp(quantile(complement(normal_distribtion&lt;RealType&gt;(m,
 238                     s), q)))
 239                   </p>
 240                 </td>
 241 </tr>
 242 <tr>
 243 <td>
 244                   <p>
 245                     mean
 246                   </p>
 247                 </td>
 248 <td>
 249                   <p>
 250                     e<sup>m + s<sup>2 </sup> / 2 </sup>
 251                   </p>
 252                 </td>
 253 </tr>
 254 <tr>
 255 <td>
 256                   <p>
 257                     variance
 258                   </p>
 259                 </td>
 260 <td>
 261                   <p>
 262                     (e<sup>s<sup>2</sup> </sup> - 1) * e<sup>2m + s<sup>2 </sup> </sup>
 263                   </p>
 264                 </td>
 265 </tr>
 266 <tr>
 267 <td>
 268                   <p>
 269                     mode
 270                   </p>
 271                 </td>
 272 <td>
 273                   <p>
 274                     e<sup>m - s<sup>2 </sup> </sup>
 275                   </p>
 276                 </td>
 277 </tr>
 278 <tr>
 279 <td>
 280                   <p>
 281                     skewness
 282                   </p>
 283                 </td>
 284 <td>
 285                   <p>
 286                     sqrt(e<sup>s<sup>2</sup> </sup> - 1) * (2 + e<sup>s<sup>2</sup> </sup>)
 287                   </p>
 288                 </td>
 289 </tr>
 290 <tr>
 291 <td>
 292                   <p>
 293                     kurtosis
 294                   </p>
 295                 </td>
 296 <td>
 297                   <p>
 298                     e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 3
 299                   </p>
 300                 </td>
 301 </tr>
 302 <tr>
 303 <td>
 304                   <p>
 305                     kurtosis excess
 306                   </p>
 307                 </td>
 308 <td>
 309                   <p>
 310                     e<sup>4s<sup>2</sup> </sup> + 2e<sup>3s<sup>2</sup> </sup> + 3e<sup>2s<sup>2</sup> </sup> - 6
 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>
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