diff options
Diffstat (limited to 'libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html')
| -rw-r--r-- | libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html | 375 |
1 files changed, 375 insertions, 0 deletions
diff --git a/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html b/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html new file mode 100644 index 0000000000..50c2c5e0f9 --- /dev/null +++ b/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/gamma_dist.html @@ -0,0 +1,375 @@ +<html> +<head> +<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> +<title>Gamma (and Erlang) Distribution</title> +<link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> +<meta name="generator" content="DocBook XSL Stylesheets V1.76.1"> +<link rel="home" href="../../../../index.html" title="Math Toolkit"> +<link rel="up" href="../dists.html" title="Distributions"> +<link rel="prev" href="f_dist.html" title="F Distribution"> +<link rel="next" href="geometric_dist.html" title="Geometric Distribution"> +</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="f_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="geometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +<div class="section math_toolkit_dist_dist_ref_dists_gamma_dist"> +<div class="titlepage"><div><div><h5 class="title"> +<a name="math_toolkit.dist.dist_ref.dists.gamma_dist"></a><a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">Gamma + (and Erlang) 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">gamma</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">gamma_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="identifier">gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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="identifier">RealType</span> <span class="identifier">shape</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> + The gamma distribution is a continuous probability distribution. When + the shape parameter is an integer then it is known as the Erlang Distribution. + It is also closely related to the Poisson and Chi Squared Distributions. + </p> +<p> + When the shape parameter has an integer value, the distribution is the + <a href="http://en.wikipedia.org/wiki/Erlang_distribution" target="_top">Erlang + distribution</a>. Since this can be produced by ensuring that the + shape parameter has an integer value > 0, the Erlang distribution + is not separately implemented. + </p> +<div class="note"><table border="0" summary="Note"> +<tr> +<td rowspan="2" align="center" valign="top" width="25"><img alt="[Note]" src="../../../../../../../../../doc/src/images/note.png"></td> +<th align="left">Note</th> +</tr> +<tr><td align="left" valign="top"> +<p> + To avoid potential confusion with the gamma functions, this distribution + does not provide the typedef: + </p> +<p> +</p> +<pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">gamma_distribution</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">gamma</span><span class="special">;</span></pre> +<p> + </p> +<p> + Instead if you want a double precision gamma distribution you can write + </p> +<p> +</p> +<pre class="programlisting"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gamma_distribution</span><span class="special"><></span> <span class="identifier">my_gamma</span><span class="special">(</span><span class="number">1</span><span class="special">,</span> <span class="number">1</span><span class="special">);</span></pre> +<p> + </p> +</td></tr> +</table></div> +<p> + For shape parameter <span class="emphasis"><em>k</em></span> and scale parameter θ   it is + defined by the probability density function: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../../equations/gamma_dist_ref1.png"></span> + </p> +<p> + Sometimes an alternative formulation is used: given parameters α  = k and + β  = 1 / θ, then the distribution can be defined by the PDF: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../../equations/gamma_dist_ref2.png"></span> + </p> +<p> + In this form the inverse scale parameter is called a <span class="emphasis"><em>rate parameter</em></span>. + </p> +<p> + Both forms are in common usage: this library uses the first definition + throughout. Therefore to construct a Gamma Distribution from a <span class="emphasis"><em>rate + parameter</em></span>, you should pass the reciprocal of the rate as the + scale parameter. + </p> +<p> + The following two graphs illustrate how the PDF of the gamma distribution + varies as the parameters vary: + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../../graphs/gamma1_pdf.png" align="middle"></span> + </p> +<p> + <span class="inlinemediaobject"><img src="../../../../../graphs/gamma2_pdf.png" align="middle"></span> + </p> +<p> + The <span class="bold"><strong>Erlang Distribution</strong></span> is the same + as the Gamma, but with the shape parameter an integer. It is often expressed + using a <span class="emphasis"><em>rate</em></span> rather than a <span class="emphasis"><em>scale</em></span> + as the second parameter (remember that the rate is the reciprocal of + the scale). + </p> +<p> + Internally the functions used to implement the Gamma Distribution are + already optimised for small-integer arguments, so in general there should + be no great loss of performance from using a Gamma Distribution rather + than a dedicated Erlang Distribution. + </p> +<h5> +<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h0"></a> + <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions"></a></span><a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions">Member + Functions</a> + </h5> +<pre class="programlisting"><span class="identifier">gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</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 gamma distribution with shape <span class="emphasis"><em>shape</em></span> + and scale <span class="emphasis"><em>scale</em></span>. + </p> +<p> + Requires that the shape and scale parameters are 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">shape</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> +</pre> +<p> + Returns the <span class="emphasis"><em>shape</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.dist_ref.dists.gamma_dist.h1"></a> + <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.non_member_accessors"></a></span><a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_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 [0,+∞]. + </p> +<h5> +<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h2"></a> + <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy"></a></span><a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy">Accuracy</a> + </h5> +<p> + The lognormal distribution is implemented in terms of the incomplete + gamma functions <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a> + and <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a> + and their inverses <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a> + and <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>: + refer to the accuracy data for those functions for more information. + </p> +<h5> +<a name="math_toolkit.dist.dist_ref.dists.gamma_dist.h3"></a> + <span><a name="math_toolkit.dist.dist_ref.dists.gamma_dist.implementation"></a></span><a class="link" href="gamma_dist.html#math_toolkit.dist.dist_ref.dists.gamma_dist.implementation">Implementation</a> + </h5> +<p> + In the following table <span class="emphasis"><em>k</em></span> is the shape parameter + of the distribution, θ   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 <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 = <a class="link" href="../../../special/sf_gamma/gamma_derivatives.html" title="Derivative of the Incomplete Gamma Function">gamma_p_derivative</a>(k, + x / θ) / θ + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf + </p> + </td> +<td> + <p> + Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(k, + x / θ) + </p> + </td> +</tr> +<tr> +<td> + <p> + cdf complement + </p> + </td> +<td> + <p> + Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(k, + x / θ) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile + </p> + </td> +<td> + <p> + Using the relation: x = θ  * <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(k, + p) + </p> + </td> +</tr> +<tr> +<td> + <p> + quantile from the complement + </p> + </td> +<td> + <p> + Using the relation: x = θ  * <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(k, + p) + </p> + </td> +</tr> +<tr> +<td> + <p> + mean + </p> + </td> +<td> + <p> + kθ + </p> + </td> +</tr> +<tr> +<td> + <p> + variance + </p> + </td> +<td> + <p> + kθ<sup>2</sup> + </p> + </td> +</tr> +<tr> +<td> + <p> + mode + </p> + </td> +<td> + <p> + (k-1)θ   for <span class="emphasis"><em>k>1</em></span> otherwise a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> + </p> + </td> +</tr> +<tr> +<td> + <p> + skewness + </p> + </td> +<td> + <p> + 2 / sqrt(k) + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis + </p> + </td> +<td> + <p> + 3 + 6 / k + </p> + </td> +</tr> +<tr> +<td> + <p> + kurtosis excess + </p> + </td> +<td> + <p> + 6 / k + </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 © 2006-2010 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> +</tr></table> +<hr> +<div class="spirit-nav"> +<a accesskey="p" href="f_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="geometric_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> +</div> +</body> +</html> |