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<title>Laplace Distribution</title>
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<div class="section">
<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.laplace_dist"></a><a class="link" href="laplace_dist.html" title="Laplace Distribution">Laplace Distribution</a>
</h4></div></div></div>
<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>
<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="Chapter 14. 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<></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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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 domain 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>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h0"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.member_functions"></a></span><a class="link" href="laplace_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h1"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.non_member_accessors"></a></span><a class="link" href="laplace_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
<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>,
<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>,
<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>,
<a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
<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>,
<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>.
</p>
<p>
The domain of the random variable is [-∞,+∞].
</p>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h2"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.accuracy"></a></span><a class="link" href="laplace_dist.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h3"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.implementation"></a></span><a class="link" href="laplace_dist.html#math_toolkit.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>
<h5>
<a name="math_toolkit.dist_ref.dists.laplace_dist.h4"></a>
<span class="phrase"><a name="math_toolkit.dist_ref.dists.laplace_dist.references"></a></span><a class="link" href="laplace_dist.html#math_toolkit.dist_ref.dists.laplace_dist.references">References</a>
</h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
<a href="http://mathworld.wolfram.com/LaplaceDistribution.html" target="_top">Weisstein,
Eric W. "Laplace Distribution."</a> From MathWorld--A
Wolfram Web Resource.
</li>
<li class="listitem">
<a href="http://en.wikipedia.org/wiki/Laplace_distribution" target="_top">Laplace
Distribution</a>
</li>
<li class="listitem">
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,
1972, p. 930.
</li>
</ul></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 © 2006-2010, 2012-2014 Nikhar Agrawal,
Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta,
Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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>
</tr></table>
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