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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.logistic_dist"></a><a class="link" href="logistic_dist.html" title="Logistic Distribution"> |
| Logistic Distribution</a> |
| </h5></div></div></div> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</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">></span></pre> |
| <p> |
| </p> |
| <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> |
| |
| <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span> |
| <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a> <span class="special">=</span> <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> <span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">logistic_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">template</span> <span class="special"><</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">></span> |
| <span class="keyword">class</span> <span class="identifier">logistic_distribution</span> |
| <span class="special">{</span> |
| <span class="keyword">public</span><span class="special">:</span> |
| <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span> |
| <span class="keyword">typedef</span> <span class="identifier">Policy</span> <span class="identifier">policy_type</span><span class="special">;</span> |
| <span class="comment">// Construct: |
| </span> <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> |
| <span class="comment">// Accessors: |
| </span> <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> <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> |
| <span class="special">};</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">logistic_distribution</span><span class="special"><></span> <span class="identifier">logistic</span><span class="special">;</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| The logistic distribution is a continous probability distribution. It |
| has two parameters - location and scale. The cumulative distribution |
| function of the logistic distribution appears in logistic regression |
| and feedforward neural networks. Among other applications, United State |
| Chess Federation and FIDE use it to calculate chess ratings. |
| </p> |
| <p> |
| The following graph shows how the distribution changes as the parameters |
| change: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/logistic_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.logistic_dist.member_functions"></a><h5> |
| <a name="id1035669"></a> |
| <a class="link" href="logistic_dist.html#math_toolkit.dist.dist_ref.dists.logistic_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <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> |
| </pre> |
| <p> |
| Constructs a logistic distribution with location <span class="emphasis"><em>u</em></span> |
| and scale <span class="emphasis"><em>s</em></span>. |
| </p> |
| <p> |
| Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">></span> |
| <span class="number">0</span></code>, otherwise a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> |
| is raised. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the location of this distribution. |
| </p> |
| <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> |
| </pre> |
| <p> |
| Returns the scale of this distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.logistic_dist.non_member_accessors"></a><h5> |
| <a name="id1035835"></a> |
| <a class="link" href="logistic_dist.html#math_toolkit.dist.dist_ref.dists.logistic_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> that are generic to all distributions are supported: |
| <a class="link" href="../nmp.html#math.dist.cdf">Cumulative Distribution Function</a>, |
| <a class="link" href="../nmp.html#math.dist.pdf">Probability Density Function</a>, <a class="link" href="../nmp.html#math.dist.quantile">Quantile</a>, <a class="link" href="../nmp.html#math.dist.hazard">Hazard |
| Function</a>, <a class="link" href="../nmp.html#math.dist.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math.dist.mean">mean</a>, <a class="link" href="../nmp.html#math.dist.median">median</a>, |
| <a class="link" href="../nmp.html#math.dist.mode">mode</a>, <a class="link" href="../nmp.html#math.dist.variance">variance</a>, |
| <a class="link" href="../nmp.html#math.dist.sd">standard deviation</a>, <a class="link" href="../nmp.html#math.dist.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math.dist.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>, |
| <a class="link" href="../nmp.html#math.dist.range">range</a> and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| The domain of the random variable is [-[max_value], +[min_value]]. However, |
| the pdf and cdf support inputs of +∞ and -∞ |
| as special cases if RealType |
| permits. |
| </p> |
| <p> |
| 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 function returns the result of +<a class="link" href="../../../main_overview/error_handling.html#overflow_error">overflow_error</a> |
| and -<a class="link" href="../../../main_overview/error_handling.html#overflow_error">overflow_error</a>, while the |
| complement quantile function returns the result of -<a class="link" href="../../../main_overview/error_handling.html#overflow_error">overflow_error</a> |
| and +<a class="link" href="../../../main_overview/error_handling.html#overflow_error">overflow_error</a> respectively. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.logistic_dist.accuracy"></a><h5> |
| <a name="id1035985"></a> |
| <a class="link" href="logistic_dist.html#math_toolkit.dist.dist_ref.dists.logistic_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| 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> |
| functions, so its accuracy is related to the accurate implementations |
| of those functions on a given platform. When calculating the quantile |
| with a non-zero <span class="emphasis"><em>position</em></span> parameter catastrophic |
| cancellation errors can occur: in such cases, only a low <span class="emphasis"><em>absolute |
| error</em></span> can be guarenteed. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.logistic_dist.implementation"></a><h5> |
| <a name="id1036040"></a> |
| <a class="link" href="logistic_dist.html#math_toolkit.dist.dist_ref.dists.logistic_dist.implementation">Implementation</a> |
| </h5> |
| <div class="informaltable"><table class="table"> |
| <colgroup> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Function |
| </p> |
| </th> |
| <th> |
| <p> |
| Implementation Notes |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| pdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: pdf = e<sup>-(x-u)/s</sup> / (s*(1+e<sup>-(x-u)/s</sup>)<sup>2</sup>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: p = 1/(1+e<sup>-(x-u)/s</sup>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: q = 1/(1+e<sup>(x-u)/s</sup>) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = u - s*log(1/p-1) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = u + s*log(p/1-p) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| u |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| The same as the mean. |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 6/5 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| (π*s)<sup>2</sup> / 3 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> |
| <td align="left"></td> |
| <td align="right"><div class="copyright-footer">Copyright © 2006 , 2007, 2008, 2009, 2010 John Maddock, Paul A. Bristow, |
| Hubert Holin, Xiaogang Zhang, Bruno Lalande, Johan Råde, Gautam Sewani and |
| Thijs van den Berg<p> |
| Distributed under the Boost Software License, Version 1.0. (See accompanying |
| 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>) |
| </p> |
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