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  26 <div class="titlepage"><div><div><h4 class="title">
  27 <a name="math_toolkit.dist_ref.dists.logistic_dist"></a><a class="link" href="logistic_dist.html" title="Logistic Distribution">Logistic
  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">logistic</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">logistic_distribution</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> <span class="identifier">Policy</span><span class="special">&gt;</span>
  38 <span class="keyword">class</span> <span class="identifier">logistic_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    <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>
  43    <span class="comment">// Construct:</span>
  44    <span class="identifier">logistic_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>
  45    <span class="comment">// Accessors:</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> <span class="comment">// location.</span>
  47    <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="comment">// scale.</span>
  48
  49 <span class="special">};</span>
  50
  51 <span class="keyword">typedef</span> <span class="identifier">logistic_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">logistic</span><span class="special">;</span>
  52
  53 <span class="special">}}</span> <span class="comment">// namespaces</span>
  54 </pre>
  55 <p>
  56           The logistic distribution is a continous probability distribution. It has
  57           two parameters - location and scale. The cumulative distribution function
  58           of the logistic distribution appears in logistic regression and feedforward
  59           neural networks. Among other applications, United State Chess Federation
  60           and FIDE use it to calculate chess ratings.
  61         </p>
  62 <p>
  63           The following graph shows how the distribution changes as the parameters
  64           change:
  65         </p>
  66 <p>
  67           <span class="inlinemediaobject"><img src="../../../../graphs/logistic_pdf.svg" align="middle"></span>
  68         </p>
  69 <h5>
  70 <a name="math_toolkit.dist_ref.dists.logistic_dist.h0"></a>
  71           <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.member_functions"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.member_functions">Member
  72           Functions</a>
  73         </h5>
  74 <pre class="programlisting"><span class="identifier">logistic_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">u</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">s</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
  75 </pre>
  76 <p>
  77           Constructs a logistic distribution with location <span class="emphasis"><em>u</em></span>
  78           and scale <span class="emphasis"><em>s</em></span>.
  79         </p>
  80 <p>
  81           Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">&gt;</span>
  82           <span class="number">0</span></code>, otherwise a <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>
  83           is raised.
  84         </p>
  85 <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>
  86 </pre>
  87 <p>
  88           Returns the location of this distribution.
  89         </p>
  90 <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>
  91 </pre>
  92 <p>
  93           Returns the scale of this distribution.
  94         </p>
  95 <h5>
  96 <a name="math_toolkit.dist_ref.dists.logistic_dist.h1"></a>
  97           <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.non_member_accessors"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.non_member_accessors">Non-member
  98           Accessors</a>
  99         </h5>
 100 <p>
 101           All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
 102           functions</a> that are generic to all distributions are supported:
 103           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
 104           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
 105           <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>,
 106           <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>,
 107           <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>,
 108           <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
 109           <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>,
 110           <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>.
 111         </p>
 112 <p>
 113           The domain of the random variable is [-[max_value], +[min_value]]. However,
 114           the pdf and cdf support inputs of +&#8734; and -&#8734;
 115 as special cases if RealType permits.
 116         </p>
 117 <p>
 118           At <code class="computeroutput"><span class="identifier">p</span><span class="special">=</span><span class="number">1</span></code> and <code class="computeroutput"><span class="identifier">p</span><span class="special">=</span><span class="number">0</span></code>, the quantile
 119           function returns the result of +<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
 120           and -<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>,
 121           while the complement quantile function returns the result of -<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
 122           and +<a class="link" href="../../error_handling.html#math_toolkit.error_handling.overflow_error">overflow_error</a>
 123           respectively.
 124         </p>
 125 <h5>
 126 <a name="math_toolkit.dist_ref.dists.logistic_dist.h2"></a>
 127           <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.accuracy"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.accuracy">Accuracy</a>
 128         </h5>
 129 <p>
 130           The logistic distribution is implemented in terms of the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span></code> and the <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">log</span></code>
 131           functions, so its accuracy is related to the accurate implementations of
 132           those functions on a given platform. When calculating the quantile with
 133           a non-zero <span class="emphasis"><em>position</em></span> parameter catastrophic cancellation
 134           errors can occur: in such cases, only a low <span class="emphasis"><em>absolute error</em></span>
 135           can be guaranteed.
 136         </p>
 137 <h5>
 138 <a name="math_toolkit.dist_ref.dists.logistic_dist.h3"></a>
 139           <span class="phrase"><a name="math_toolkit.dist_ref.dists.logistic_dist.implementation"></a></span><a class="link" href="logistic_dist.html#math_toolkit.dist_ref.dists.logistic_dist.implementation">Implementation</a>
 140         </h5>
 141 <div class="informaltable"><table class="table">
 142 <colgroup>
 143 <col>
 144 <col>
 145 </colgroup>
 146 <thead><tr>
 147 <th>
 148                   <p>
 149                     Function
 150                   </p>
 151                 </th>
 152 <th>
 153                   <p>
 154                     Implementation Notes
 155                   </p>
 156                 </th>
 157 </tr></thead>
 158 <tbody>
 159 <tr>
 160 <td>
 161                   <p>
 162                     pdf
 163                   </p>
 164                 </td>
 165 <td>
 166                   <p>
 167                     Using the relation: pdf = e<sup>-(x-u)/s</sup> / (s*(1+e<sup>-(x-u)/s</sup>)<sup>2</sup>)
 168                   </p>
 169                 </td>
 170 </tr>
 171 <tr>
 172 <td>
 173                   <p>
 174                     cdf
 175                   </p>
 176                 </td>
 177 <td>
 178                   <p>
 179                     Using the relation: p = 1/(1+e<sup>-(x-u)/s</sup>)
 180                   </p>
 181                 </td>
 182 </tr>
 183 <tr>
 184 <td>
 185                   <p>
 186                     cdf complement
 187                   </p>
 188                 </td>
 189 <td>
 190                   <p>
 191                     Using the relation: q = 1/(1+e<sup>(x-u)/s</sup>)
 192                   </p>
 193                 </td>
 194 </tr>
 195 <tr>
 196 <td>
 197                   <p>
 198                     quantile
 199                   </p>
 200                 </td>
 201 <td>
 202                   <p>
 203                     Using the relation: x = u - s*log(1/p-1)
 204                   </p>
 205                 </td>
 206 </tr>
 207 <tr>
 208 <td>
 209                   <p>
 210                     quantile from the complement
 211                   </p>
 212                 </td>
 213 <td>
 214                   <p>
 215                     Using the relation: x = u + s*log(p/1-p)
 216                   </p>
 217                 </td>
 218 </tr>
 219 <tr>
 220 <td>
 221                   <p>
 222                     mean
 223                   </p>
 224                 </td>
 225 <td>
 226                   <p>
 227                     u
 228                   </p>
 229                 </td>
 230 </tr>
 231 <tr>
 232 <td>
 233                   <p>
 234                     mode
 235                   </p>
 236                 </td>
 237 <td>
 238                   <p>
 239                     The same as the mean.
 240                   </p>
 241                 </td>
 242 </tr>
 243 <tr>
 244 <td>
 245                   <p>
 246                     skewness
 247                   </p>
 248                 </td>
 249 <td>
 250                   <p>
 251                     0
 252                   </p>
 253                 </td>
 254 </tr>
 255 <tr>
 256 <td>
 257                   <p>
 258                     kurtosis excess
 259                   </p>
 260                 </td>
 261 <td>
 262                   <p>
 263                     6/5
 264                   </p>
 265                 </td>
 266 </tr>
 267 <tr>
 268 <td>
 269                   <p>
 270                     variance
 271                   </p>
 272                 </td>
 273 <td>
 274                   <p>
 275                     (&#960;*s)<sup>2</sup> / 3
 276                   </p>
 277                 </td>
 278 </tr>
 279 </tbody>
 280 </table></div>
 281 </div>
 282 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
 283 <td align="left"></td>
 284 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2010, 2012-2014 Nikhar Agrawal,
 285       Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
 286       Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
 287       Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
 288         Distributed under the Boost Software License, Version 1.0. (See accompanying
 289         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>)
 290       </p>
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