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| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution"> Laplace |
| 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">laplace</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">laplace_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">laplace_distribution</span><span class="special"><></span> <span class="identifier">laplace</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> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">laplace_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">laplace_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="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="special">};</span> |
| |
| <span class="special">}}</span> <span class="comment">// namespaces |
| </span></pre> |
| <p> |
| Laplace distribution is the distribution of differences between two independent |
| variates with identical exponential distributions (Abramowitz and Stegun |
| 1972, p. 930). It is also called the double exponential distribution. |
| </p> |
| <p> |
| For location parameter μ ​ and scale parameter σ ​ it is defined by the probability |
| density function: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/laplace_pdf.png"></span> |
| </p> |
| <p> |
| The location and scale parameters are equivalent to the mean and standard |
| deviation of the normal or Gaussian distribution. |
| </p> |
| <p> |
| The following graph illustrates the effect of the parameters μ ​ and σ ​ on the |
| PDF. Note that the range of the random variable remains [-∞,+∞] irrespective |
| of the value of the location parameter: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/laplace_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.member_functions"></a><h5> |
| <a name="id1032269"></a> |
| <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">laplace_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> |
| </pre> |
| <p> |
| Constructs a laplace distribution with location <span class="emphasis"><em>location</em></span> |
| and scale <span class="emphasis"><em>scale</em></span>. |
| </p> |
| <p> |
| The location parameter is the same as the mean of the random variate. |
| </p> |
| <p> |
| The scale parameter is proportional to the standard deviation of the |
| random variate. |
| </p> |
| <p> |
| Requires that the scale parameter is greater than zero, otherwise calls |
| <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>. |
| </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 <span class="emphasis"><em>location</em></span> parameter 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 <span class="emphasis"><em>scale</em></span> parameter of this distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.non_member_accessors"></a><h5> |
| <a name="id1032431"></a> |
| <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_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 [-∞,+∞]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.accuracy"></a><h5> |
| <a name="id1032528"></a> |
| <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The laplace distribution is implemented in terms of the standard library |
| log and exp functions and as such should have very small errors. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.implementation"></a><h5> |
| <a name="id1032548"></a> |
| <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table μ is the location parameter of the distribution, |
| σ is its scale parameter, <span class="emphasis"><em>x</em></span> is the random variate, |
| <span class="emphasis"><em>p</em></span> is the probability and its complement <span class="emphasis"><em>q |
| = 1-p</em></span>. |
| </p> |
| <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>-abs(x-μ) / σ</sup> / (2 * σ) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relations: |
| </p> |
| <p> |
| x < μ : p = e<sup>(x-μ)/σ </sup> / σ |
| </p> |
| <p> |
| x >= μ : p = 1 - e<sup>(μ-x)/σ </sup> / σ |
| |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| cdf complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| -x < μ : q = e<sup>(-x-μ)/σ </sup> / σ |
| </p> |
| <p> |
| -x >= μ : q = 1 - e<sup>(μ+x)/σ </sup> / σ |
| |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relations: |
| </p> |
| <p> |
| p < 0.5 : x = μ + σ * log(2*p) |
| </p> |
| <p> |
| p >= 0.5 : x = μ - σ * log(2-2*p) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: |
| </p> |
| <p> |
| q > 0.5: x = μ + σ*log(2-2*q) |
| </p> |
| <p> |
| q <=0.5: x = μ - σ*log( 2*q ) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| μ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| 2 * σ<sup>2</sup> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| μ |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| skewness |
| </p> |
| </td> |
| <td> |
| <p> |
| 0 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis |
| </p> |
| </td> |
| <td> |
| <p> |
| 6 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis excess |
| </p> |
| </td> |
| <td> |
| <p> |
| 3 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.laplace_dist.references"></a><h5> |
| <a name="id1032869"></a> |
| <a class="link" href="laplace_dist.html#math_toolkit.dist.dist_ref.dists.laplace_dist.references">References</a> |
| </h5> |
| <div class="itemizedlist"><ul type="disc"> |
| <li> |
| <a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein, |
| Eric W. "Laplace Distribution."</a> From MathWorld--A |
| Wolfram Web Resource. |
| </li> |
| <li> |
| <a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace |
| Distribution</a> |
| </li> |
| <li> |
| M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, |
| 1972, p. 930. |
| </li> |
| </ul></div> |
| </div> |
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| <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> |
| </div></td> |
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