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Subject: [Boost-commit] svn:boost r65675 - trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists
From: pbristow_at_[hidden]
Date: 2010-09-29 09:48:11

Author: pbristow
Date: 2010-09-29 09:48:10 EDT (Wed, 29 Sep 2010)
New Revision: 65675
URL: http://svn.boost.org/trac/boost/changeset/65675

Log:
Added two new distributions.
Added:
trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html (contents, props changed)

Added: trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html
==============================================================================
--- (empty file)
+++ trunk/libs/math/doc/sf_and_dist/html/math_toolkit/dist/dist_ref/dists/inverse_chi_squared_dist.html 2010-09-29 09:48:10 EDT (Wed, 29 Sep 2010)
@@ -0,0 +1,471 @@
+<html>
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=US-ASCII">
+<title>Inverse Chi Squared Distribution</title>
+<link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css">
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+</div>
+<div class="section" lang="en">
+<div class="titlepage"><div><div><h5 class="title">
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist"></a><a class="link" href="inverse_chi_squared_dist.html" title="Inverse Chi Squared Distribution">Inverse
+ Chi Squared Distribution</a>
+</h5></div></div></div>
+
+
+
+<pre class="programlisting">#include <boost/math/distributions/inverse_chi_squared.hpp></pre>
+
+ 
+<pre class="programlisting">namespace boost{ namespace math{
+
+template <class RealType = double,
+ class <a class="link" href="../../../policy.html" title="Policies">Policy</a> = <a class="link" href="../../../policy/pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy<></a> >
+class inverse_chi_squared_distribution
+{
+public:
+ typedef RealType value_type;
+ typedef Policy policy_type;
+
+ inverse_chi_squared_distribution(RealType df = 1); // Not explicitly scaled, default 1/df.
+ inverse_chi_squared_distribution(RealType df, RealType scale = 1/df); // Scaled.
+
+ RealType degrees_of_freedom()const; // Default 1.
+ RealType scale()const; // Optional scale [xi] (variance), default 1/degrees_of_freedom.
+};
+
+}} // namespace boost // namespace math
+</pre>
+
+ The inverse chi squared distribution is a continuous probability distribution
+ of the reciprocal of a variable distributed
+ according to the chi squared distribution.
+ 
+
+ The sources below give confusingly different formulae using different
+ symbols for the distribution pdf, but they are all the same, or related
+ by a change of variable, or choice of scale.
+ 
+
+ Two constructors are available to implement both the scaled and (implicitly)
+ unscaled versions.
+ 
+
+ The main version has an explicit scale parameter which implements the
+ <a href="http://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution" target="_top">scaled
+ inverse chi_squared distribution</a>.
+ 
+
+ A second version has an implicit scale = 1/degrees of freedom and gives
+ the 1st definition in the <a href="http://en.wikipedia.org/wiki/Inverse-chi-square_distribution" target="_top">Wikipedia
+ inverse chi_squared distribution</a>. The 2nd Wikipedia inverse chi_squared
+ distribution definition can be implemented by explicitly specifying a
+ scale = 1.
+ 
+
+ Both definitions are also available in Wolfram Mathematica and in The R Project for Statistical Computing
+ (geoR) with default scale = 1/degrees of freedom.
+ 
+
+ See
+ 
+<div class="itemizedlist"><ul type="disc">
+<li>
+ Inverse chi_squared distribution http://en.wikipedia.org/wiki/Inverse-chi-square_distribution
+ </li>
+<li>
+ Scaled inverse chi_squared distributionhttp://en.wikipedia.org/wiki/Scaled-inverse-chi-square_distribution
+ </li>
+<li>
+ R inverse chi_squared distribution functions <a href="http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html" target="_top">R
+ </a>
+ </li>
+<li>
+ Inverse chi_squared distribution functions <a href="http://mathworld.wolfram.com/InverseChi-SquaredDistribution.html" target="_top">Weisstein,
+ Eric W. "Inverse Chi-Squared Distribution." From MathWorld--A
+ Wolfram Web Resource.</a>
+ </li>
+<li>
+ Inverse chi_squared distribution reference <a href="http://reference.wolfram.com/mathematica/ref/InverseChiSquareDistribution.html" target="_top">Weisstein,
+ Eric W. "Inverse Chi-Squared Distribution reference." From
+ Wolfram Mathematica.</a>
+ </li>
+</ul></div>
+
+ The inverse_chi_squared distribution is used in <a href="http://en.wikipedia.org/wiki/Bayesian_statistics" target="_top">Bayesian
+ statistics</a>: the scaled inverse chi-square is conjugate prior
+ for the normal distribution with known mean, model parameter σ² (variance).
+ 
+
+ See <a href="http://en.wikipedia.org/wiki/Conjugate_prior" target="_top">conjugate
+ priors including a table of distributions and their priors.</a>
+ 
+
+ See also <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
+ Gamma Distribution</a> and <a class="link" href="chi_squared_dist.html" title="Chi Squared Distribution">Chi
+ Squared Distribution</a>.
+ 
+
+ The inverse_chi_squared distribution is a psecial case of a inverse_gamma
+ distribution with nu (degrees_of_freedom) shape (α) and scale (β) where
+ 
+
+   α= ν /2 and β = ½.
+ 
+<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">
+
+ This distribution does provide the
+ typedef:
+ 
+
+
+
+<pre class="programlisting">typedef inverse_chi_squared_distribution<double> inverse_chi_squared;</pre>
+
+ 
+
+ If you want a <code class="computeroutput">double</code> precision
+ inverse_chi_squared distribution you can use
+ 
+
+
+
+<pre class="programlisting">boost::math::inverse_chi_squared_distribution<></pre>
+
+ 
+
+ or you can write <code class="computeroutput">inverse_chi_squared
+ my_invchisqr(2, 3);</code>
+ 
+</td></tr>
+</table></div>
+
+ For degrees of freedom parameter ν and scale parameter ξ, it is defined
+ by the probability density function (PDF):
+ 
+
+   f(x;ν, ξ) = 2 -ν/2 e(-1/2x x (-1-ν/2) / Γ(ν/2)
+ 
+
+ and Cumulative Density Function (CDF)
+ 
+
+   F(x;ν, ξ) = Γ( ν /2, νξ/2x) / Γ(ν /2)
+ 
+
+ The following graphs illustrate how the PDF and CDF of the inverse chi_squared
+ distribution varies for a few values of parameters ν and ξ:
+ 
+
+ <img src="../../../../../graphs/inverse_chi_squared_pdf.png" align="middle">
+ 
+
+ <img src="../../../../../graphs/inverse_chi_squared_cdf.png" align="middle">
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.member_functions"></a><h5>
+<a name="id1013289"></a>
+ <a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.member_functions">Member
+ Functions</a>
+ </h5>
+<pre class="programlisting">inverse_chi_squared_distribution(RealType df = 1); // Implicitly scaled 1/df.
+inverse_chi_squared_distribution(RealType df = 1, RealType scale); // Explicitly scaled.
+</pre>
+
+ Constructs an inverse chi_squared distribution with ν degrees of freedom
+ df, and scale scale with default
+ value 1df (1ν.
+ 
+
+ Requires that the degrees of freedom ν parameter is greater than zero,
+ otherwise calls <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>.
+ 
+<pre class="programlisting">RealType degrees_of_freedom()const;
+</pre>
+
+ Returns the degrees_of_freedom ν parameter of this distribution.
+ 
+<pre class="programlisting">RealType scale()const;
+</pre>
+
+ Returns the scale ξ parameter of this distribution.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors"></a><h5>
+<a name="id1013464"></a>
+ <a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.non_member_accessors">Non-member
+ Accessors</a>
+ </h5>
+
+ 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>.
+ 
+
+ The domain of the random variate is [0,+∞].
+ 
+<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">
+ Unlike some definitions, this implementation supports a random variate
+ equal to zero as a special case, returning zero for both pdf and cdf.
+ </td></tr>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.accuracy"></a><h5>
+<a name="id1013569"></a>
+ <a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.accuracy">Accuracy</a>
+ </h5>
+
+ The inverse gamma distribution is implemented in terms of the incomplete
+ gamma functions like the <a class="link" href="inverse_gamma_dist.html" title="Inverse Gamma Distribution">Inverse
+ Gamma Distribution</a> that use <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, gamma (and thus inverse gamma) results are often accurate
+ to a few epsilon, >14 decimal digits accuracy for 64-bit double. unless
+ iteration is involved, as for the estimation of degrees of freedom.
+ 
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.implementation"></a><h5>
+<a name="id1013609"></a>
+ <a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.implementation">Implementation</a>
+ </h5>
+
+ In the following table ν is the degrees of freedom parameter and ξ is the
+ scale parameter of the distribution, x is the random
+ variate, p is the probability and q =
+ 1-p its complement. Parameters α for shape and β for scale are
+ used for the inverse gamma function: α = ν/2 and β = ν * ξ/2.
+ 
+<div class="informaltable"><table class="table">
+<colgroup>
+<col>
+<col>
+</colgroup>
+<thead><tr>
+<th>
+ 
+ Function
+ 
+ </th>
+<th>
+ 
+ Implementation Notes
+ 
+ </th>
+</tr></thead>
+<tbody>
+<tr>
+<td>
+ 
+ pdf
+ 
+ </td>
+<td>
+ 
+ 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
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf
+ 
+ </td>
+<td>
+ 
+ Using the relation: p = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_q</a>(α,
+ β / x)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ cdf complement
+ 
+ </td>
+<td>
+ 
+ Using the relation: q = <a class="link" href="../../../special/sf_gamma/igamma.html" title="Incomplete Gamma Functions">gamma_p</a>(α,
+ β / x)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile
+ 
+ </td>
+<td>
+ 
+ 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)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ quantile from the complement
+ 
+ </td>
+<td>
+ 
+ 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)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mode
+ 
+ </td>
+<td>
+ 
+ ν * ξ / (ν + 2)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ median
+ 
+ </td>
+<td>
+ 
+ no closed form analytic equation is known, but is evaluated
+ as quantile(0.5)
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ mean
+ 
+ </td>
+<td>
+ 
+ 1 / (ν - 1) for ν > 2, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ variance
+ 
+ </td>
+<td>
+ 
+ 2 ν² ξ² / ((ν -2)² (ν -4)) for ν >4, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ skewness
+ 
+ </td>
+<td>
+ 
+ 4 √2 √(ν-4) /(ν-6) for ν >6, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis_excess
+ 
+ </td>
+<td>
+ 
+ 12 * (5ν - 22) / (ν - 6) * (ν - 8) for ν >8, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
+ 
+ </td>
+</tr>
+<tr>
+<td>
+ 
+ kurtosis
+ 
+ </td>
+<td>
+ 
+ 3 + 12 * (5ν - 22) / (ν - 6) * (ν-8) for ν >8, else a <a class="link" href="../../../main_overview/error_handling.html#domain_error">domain_error</a>
+ 
+ </td>
+</tr>
+</tbody>
+</table></div>
+<a name="math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.references"></a><h5>
+<a name="id1013938"></a>
+ <a class="link" href="inverse_chi_squared_dist.html#math_toolkit.dist.dist_ref.dists.inverse_chi_squared_dist.references">References</a>
+ </h5>
+<div class="orderedlist"><ol type="1">
+<li>
+ Bayesian Data Analysis, Andrew Gelman, John B. Carlin, Hal S. Stern,
+ Donald B. Rubin, ISBN-13: 978-1584883883, Chapman & Hall; 2 edition
+ (29 July 2003).
+ </li>
+<li>
+ Bayesian Computation with R, Jim Albert, ISBN-13: 978-0387922973,
+ Springer; 2nd ed. edition (10 Jun 2009)
+ </li>
+</ol></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
+ Distributed under the Boost Software License, Version 1.0. (See accompanying
+ file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
+ 
+</div></td>
+</tr></table>
+<hr>
+<div class="spirit-nav">
+<a accesskey="p" href="gamma_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="inverse_gamma_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a>
+</div>
+</body>
+</html>

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