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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.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_distibution</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 use |
| </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></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> |
| <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.member_functions"></a><h5> |
| <a name="id1026774"></a> |
| <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> |
| <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.non_member_accessors"></a><h5> |
| <a name="id1026919"></a> |
| <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> |
| <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.accuracy"></a><h5> |
| <a name="id1027017"></a> |
| <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> |
| <a name="math_toolkit.dist.dist_ref.dists.gamma_dist.implementation"></a><h5> |
| <a name="id1027055"></a> |
| <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 it's 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 , 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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