boost/math
2
0
Fork 0
mirror of https://github.com/boostorg/math.git synced 2026-01-30 20:12:09 +00:00
Code Issues Projects Releases Wiki Activity
Files
develop
math/doc/html/math_toolkit/dist_ref/dists/laplace_dist.html
jzmaddock 4c2cdc2cf6 Regenerate docs.
2024-08-06 13:18:32 +01:00

362 lines
17 KiB
HTML
Raw Permalink Blame History

This file contains invisible Unicode characters
This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Laplace Distribution</title>
<link rel="stylesheet" href="../../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
<link rel="home" href="../../../index.html" title="Math Toolkit 4.2.1">
<link rel="up" href="../dists.html" title="Distributions">
<link rel="prev" href="kolmogorov_smirnov_dist.html" title="Kolmogorov-Smirnov Distribution">
<link rel="next" href="logistic_dist.html" title="Logistic Distribution">
<meta name="viewport" content="width=device-width, initial-scale=1">
</head>
<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF">
<table cellpadding="2" width="100%"><tr>
<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../../../boost.png"></td>
<td align="center"><a href="../../../../../../../index.html">Home</a></td>
<td align="center"><a href="../../../../../../../libs/libraries.htm">Libraries</a></td>
<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td>
<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td>
<td align="center"><a href="../../../../../../../more/index.htm">More</a></td>
</tr></table>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="logistic_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
<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">&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">laplace</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</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">&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>
<span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. 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>
<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">&lt;&gt;</span> <span class="identifier">laplace</span><span class="special">;</span>
<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 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</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 <span class="emphasis"><em>μ</em></span> and scale parameter <span class="emphasis"><em>σ</em></span>,
it is defined by the probability density function:
</p>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../equations/laplace_pdf.svg"></span>
</p></blockquote></div>
<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>
<div class="blockquote"><blockquote class="blockquote"><p>
<span class="inlinemediaobject"><img src="../../../../graphs/laplace_pdf.svg" align="middle"></span>
</p></blockquote></div>
<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 &lt; μ : p = e<sup>(x-μ)/σ </sup> / σ
</p>
<p>
x &gt;= 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 &lt; μ : q = e<sup>(-x-μ)/σ </sup> / σ
</p>
<p>
-x &gt;= 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 &lt; 0.5 : x = μ + σ * log(2*p)
</p>
<p>
p &gt;= 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 &gt; 0.5: x = μ + σ*log(2-2*q)
</p>
<p>
q &lt;=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>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, 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>
<hr>
<div class="spirit-nav">
<a accesskey="p" href="kolmogorov_smirnov_dist.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../dists.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="logistic_dist.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
</div>
</body>
</html>
Reference in New Issue View Git Blame Copy Permalink