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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gamma_dist"></a><a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution"> |
| Inverse Gamma 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">inverse_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">inverse_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">inverse_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 inverse_gamma distribution is a continuous probability distribution |
| of the reciprocal of a variable distributed according to the gamma distribution. |
| </p> |
| <p> |
| The inverse_gamma distribution is used in Bayesian statistics. |
| </p> |
| <p> |
| See <a href="http://en.wikipedia.org/wiki/Inverse-gamma_distribution" target="_top">inverse |
| gamma distribution</a>. |
| </p> |
| <p> |
| <a href="http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html" target="_top">R |
| inverse gamma distribution functions</a>. |
| </p> |
| <p> |
| <a href="http://reference.wolfram.com/mathematica/ref/InverseGammaDistribution.html" target="_top">Wolfram |
| inverse gamma distribution</a>. |
| </p> |
| <p> |
| See also <a class="link" href="gamma_dist.html" title="Gamma (and Erlang) Distribution">Gamma |
| Distribution</a>. |
| </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> |
| In spite of potential confusion with the inverse gamma function, this |
| distribution <span class="bold"><strong>does</strong></span> provide the typedef: |
| </p> |
| <p> |
| |
| </p> |
| <pre class="programlisting"><span class="keyword">typedef</span> <span class="identifier">inverse_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> |
| If you want a <code class="computeroutput"><span class="keyword">double</span></code> 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">inverse_gamma_distribution</span><span class="special"><></span></pre> |
| <p> |
| </p> |
| <p> |
| or you can write <code class="computeroutput"><span class="identifier">inverse_gamma</span> |
| <span class="identifier">my_ig</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="number">3</span><span class="special">);</span></code> |
| </p> |
| </td></tr> |
| </table></div> |
| <p> |
| For shape parameter α and scale parameter β, it is defined by the probability |
| density function (PDF): |
| </p> |
| <p> |
|   f(x;α, β) = β<sup>α</sup> * (1/x) <sup>α+1</sup> exp(-β/x) / Γ(α) |
| </p> |
| <p> |
| and cumulative density function (CDF) |
| </p> |
| <p> |
|   F(x;α, β) = Γ(α, β/x) / Γ(α) |
| </p> |
| <p> |
| The following graphs illustrate how the PDF and CDF of the inverse gamma |
| distribution varies as the parameters vary: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/inverse_gamma_pdf.png" align="middle"></span> |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/inverse_gamma_cdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.member_functions"></a><h5> |
| <a name="id1029656"></a> |
| <a class="link" href="inverse_gamma_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">inverse_gamma_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">shape</span> <span class="special">=</span> <span class="number">1</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 an inverse gamma distribution with shape α and scale β. |
| </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 α shape parameter of this inverse gamma 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 β scale parameter of this inverse gamma distribution. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.non_member_accessors"></a><h5> |
| <a name="id1029792"></a> |
| <a class="link" href="inverse_gamma_dist.html#math_toolkit.dist.dist_ref.dists.inverse_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 variate is [0,+∞]. |
| </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> |
| Unlike some definitions, this implementation supports a random variate |
| equal to zero as a special case, returning zero for pdf and cdf. |
| </p></td></tr> |
| </table></div> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.accuracy"></a><h5> |
| <a name="id1029899"></a> |
| <a class="link" href="inverse_gamma_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The inverse gamma 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. |
| But in general, inverse_gamma results are accurate to a few epsilon, |
| >14 decimal digits accuracy for 64-bit double. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.implementation"></a><h5> |
| <a name="id1029938"></a> |
| <a class="link" href="inverse_gamma_dist.html#math_toolkit.dist.dist_ref.dists.inverse_gamma_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| In the following table α is the shape parameter of the distribution, α ​ 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>(α, |
| β/ x, β) / x * x |
| </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_q</a>(α, |
| β / x) |
| </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_p</a>(α, |
| β / x) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = β ​/ <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_q_inv</a>(α, |
| p) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| quantile from the complement |
| </p> |
| </td> |
| <td> |
| <p> |
| Using the relation: x = α ​/ <a class="link" href="../../../special/sf_gamma/igamma_inv.html" title="Incomplete Gamma Function Inverses">gamma_p_inv</a>(α, |
| q) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mode |
| </p> |
| </td> |
| <td> |
| <p> |
| β / (α + 1) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| median |
| </p> |
| </td> |
| <td> |
| <p> |
| no analytic equation is known, but is evaluated as quantile(0.5) |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| mean |
| </p> |
| </td> |
| <td> |
| <p> |
| β / (α - 1) for α > 1, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| variance |
| </p> |
| </td> |
| <td> |
| <p> |
| (β * β) / ((α - 1) * (α - 1) * (α - 2)) for α >2, else 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> |
| 4 * sqrt (α -2) / (α -3) for α >3, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| kurtosis_excess |
| </p> |
| </td> |
| <td> |
| <p> |
| (30 * α - 66) / ((α-3)*(α - 4)) for α >4, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a> |
| </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>) |
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