| <html> |
| <head> |
| <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> |
| <title>Noncentral Beta Distribution</title> |
| <link rel="stylesheet" href="../../../../../../../../../doc/src/boostbook.css" type="text/css"> |
| <meta name="generator" content="DocBook XSL Stylesheets V1.74.0"> |
| <link rel="home" href="../../../../index.html" title="Math Toolkit"> |
| <link rel="up" href="../dists.html" title="Distributions"> |
| <link rel="prev" href="negative_binomial_dist.html" title="Negative Binomial Distribution"> |
| <link rel="next" href="nc_chi_squared_dist.html" title="Noncentral Chi-Squared 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="negative_binomial_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="nc_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| <div class="section" lang="en"> |
| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution"> Noncentral |
| Beta 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">non_central_beta</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">non_central_beta_distribution</span><span class="special">;</span> |
| |
| <span class="keyword">typedef</span> <span class="identifier">non_central_beta_distribution</span><span class="special"><></span> <span class="identifier">non_central_beta</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">class</span> <a class="link" href="../../../policy.html" title="Policies">Policy</a><span class="special">></span> |
| <span class="keyword">class</span> <span class="identifier">non_central_beta_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">// Constructor: |
| </span> <span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> |
| |
| <span class="comment">// Accessor to shape parameters: |
| </span> <span class="identifier">RealType</span> <span class="identifier">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| <span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| |
| <span class="comment">// Accessor to non-centrality parameter lambda: |
| </span> <span class="identifier">RealType</span> <span class="identifier">non_centrality</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 noncentral beta distribution is a generalization of the <a class="link" href="beta_dist.html" title="Beta Distribution">Beta |
| Distribution</a>. |
| </p> |
| <p> |
| It is defined as the ratio X = χ<sub>m</sub><sup>2</sup>(λ) / (χ<sub>m</sub><sup>2</sup>(λ) + χ<sub>n</sub><sup>2</sup>) where χ<sub>m</sub><sup>2</sup>(λ) is a noncentral |
| χ<sup>2</sup> |
| random variable with <span class="emphasis"><em>m</em></span> degrees of freedom, and |
| χ<sub>n</sub><sup>2</sup> |
| is a central χ<sup>2</sup> random variable with <span class="emphasis"><em>n</em></span> degrees of |
| freedom. |
| </p> |
| <p> |
| This gives a PDF that can be expressed as a Poisson mixture of beta distribution |
| PDFs: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_beta_ref1.png"></span> |
| </p> |
| <p> |
| where P(i;λ/2) is the discrete Poisson probablity at <span class="emphasis"><em>i</em></span>, |
| with mean λ/2, and I<sub>x</sub><sup>'</sup>(α, β) is the derivative of the incomplete beta function. |
| This leads to the usual form of the CDF as: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_beta_ref2.png"></span> |
| </p> |
| <p> |
| The following graph illustrates how the distribution changes for different |
| values of λ: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../graphs/nc_beta_pdf.png" align="middle"></span> |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.member_functions"></a><h5> |
| <a name="id1042868"></a> |
| <a class="link" href="nc_beta_dist.html#math_toolkit.dist.dist_ref.dists.nc_beta_dist.member_functions">Member |
| Functions</a> |
| </h5> |
| <pre class="programlisting"><span class="identifier">non_central_beta_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">b</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">);</span> |
| </pre> |
| <p> |
| Constructs a noncentral beta distribution with shape parameters <span class="emphasis"><em>a</em></span> |
| and <span class="emphasis"><em>b</em></span> and non-centrality parameter <span class="emphasis"><em>lambda</em></span>. |
| </p> |
| <p> |
| Requires a > 0, b > 0 and lambda >= 0, 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">alpha</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>a</em></span> from which this object was |
| constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">beta</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>b</em></span> from which this object was |
| constructed. |
| </p> |
| <pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">non_centrality</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span> |
| </pre> |
| <p> |
| Returns the parameter <span class="emphasis"><em>lambda</em></span> from which this object |
| was constructed. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.non_member_accessors"></a><h5> |
| <a name="id1043054"></a> |
| <a class="link" href="nc_beta_dist.html#math_toolkit.dist.dist_ref.dists.nc_beta_dist.non_member_accessors">Non-member |
| Accessors</a> |
| </h5> |
| <p> |
| Most of the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member |
| accessor functions</a> 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.median">median</a>, <a class="link" href="../nmp.html#math.dist.mode">mode</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.range">range</a> |
| and <a class="link" href="../nmp.html#math.dist.support">support</a>. |
| </p> |
| <p> |
| However, the following are not currently implemented: <a class="link" href="../nmp.html#math.dist.mean">mean</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> and <a class="link" href="../nmp.html#math.dist.kurtosis_excess">kurtosis_excess</a>. |
| </p> |
| <p> |
| The domain of the random variable is [0, 1]. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.accuracy"></a><h5> |
| <a name="id1043155"></a> |
| <a class="link" href="nc_beta_dist.html#math_toolkit.dist.dist_ref.dists.nc_beta_dist.accuracy">Accuracy</a> |
| </h5> |
| <p> |
| The following table shows the peak errors (in units of <a href="http://en.wikipedia.org/wiki/Machine_epsilon" target="_top">epsilon</a>) |
| found on various platforms with various floating point types. No comparison |
| to the <a href="http://www.r-project.org/" target="_top">R-2.5.1 Math library</a>, |
| or to the FORTRAN implementations of AS226 or AS310 are given since these |
| appear to only guarantee absolute error: this would causes our test harness |
| to assign an <span class="emphasis"><em>"infinite"</em></span> error to these |
| libraries for some of our test values when measuring <span class="emphasis"><em>relative |
| error</em></span>. Unless otherwise specified any floating-point type |
| that is narrower than the one shown will have <a class="link" href="../../../backgrounders/relative_error.html#zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.errors_in_cdf_of_the_noncentral_beta"></a><p class="title"><b>Table 13. Errors In CDF of the Noncentral Beta</b></p> |
| <div class="table-contents"><table class="table" summary="Errors In CDF of the Noncentral Beta"> |
| <colgroup> |
| <col> |
| <col> |
| <col> |
| <col> |
| </colgroup> |
| <thead><tr> |
| <th> |
| <p> |
| Significand Size |
| </p> |
| </th> |
| <th> |
| <p> |
| Platform and Compiler |
| </p> |
| </th> |
| <th> |
| <p> |
| α, β,λ < 200 |
| </p> |
| </th> |
| <th> |
| <p> |
| α,β,λ > 200 |
| </p> |
| </th> |
| </tr></thead> |
| <tbody> |
| <tr> |
| <td> |
| <p> |
| 53 |
| </p> |
| </td> |
| <td> |
| <p> |
| Win32, Visual C++ 8 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=620 Mean=22 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=8670 Mean=1040 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| RedHat Linux IA32, gcc-4.1.1 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=825 Mean=50 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=2.5x10<sup>4</sup> Mean=4000 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 64 |
| </p> |
| </td> |
| <td> |
| <p> |
| Redhat Linux IA64, gcc-3.4.4 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=825 Mean=30 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=1.7x10<sup>4</sup> Mean=2500 |
| </p> |
| </td> |
| </tr> |
| <tr> |
| <td> |
| <p> |
| 113 |
| </p> |
| </td> |
| <td> |
| <p> |
| HPUX IA64, aCC A.06.06 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=420 Mean=50 |
| </p> |
| </td> |
| <td> |
| <p> |
| Peak=9200 Mean=1200 |
| </p> |
| </td> |
| </tr> |
| </tbody> |
| </table></div> |
| </div> |
| <br class="table-break"><p> |
| Error rates for the PDF, the complement of the CDF and for the quantile |
| functions are broadly similar. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.tests"></a><h5> |
| <a name="id1044572"></a> |
| <a class="link" href="nc_beta_dist.html#math_toolkit.dist.dist_ref.dists.nc_beta_dist.tests">Tests</a> |
| </h5> |
| <p> |
| There are two sets of test data used to verify this implementation: firstly |
| we can compare with a few sample values generated by the <a href="http://www.r-project.org/" target="_top">R |
| library</a>. Secondly, we have tables of test data, computed with |
| this implementation and using interval arithmetic - this data should |
| be accurate to at least 50 decimal digits - and is the used for our accuracy |
| tests. |
| </p> |
| <a name="math_toolkit.dist.dist_ref.dists.nc_beta_dist.implementation"></a><h5> |
| <a name="id1044596"></a> |
| <a class="link" href="nc_beta_dist.html#math_toolkit.dist.dist_ref.dists.nc_beta_dist.implementation">Implementation</a> |
| </h5> |
| <p> |
| The CDF and its complement are evaluated as follows: |
| </p> |
| <p> |
| First we determine which of the two values (the CDF or its complement) |
| is likely to be the smaller, the crossover point is taken to be the mean |
| of the distribution: for this we use the approximation due to: R. Chattamvelli |
| and R. Shanmugam, "Algorithm AS 310: Computing the Non-Central Beta |
| Distribution Function", Applied Statistics, Vol. 46, No. 1. (1997), |
| pp. 146-156. |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_beta_ref3.png"></span> |
| </p> |
| <p> |
| Then either the CDF or its complement is computed using the relations: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_beta_ref4.png"></span> |
| </p> |
| <p> |
| The summation is performed by starting at i = λ/2, and then recursing |
| in both directions, using the usual recurrence relations for the Poisson |
| PDF and incomplete beta functions. This is the "Method 2" described |
| by: |
| </p> |
| <p> |
| Denise Benton and K. Krishnamoorthy, "Computing discrete mixtures |
| of continuous distributions: noncentral chisquare, noncentral t and the |
| distribution of the square of the sample multiple correlation coefficient", |
| Computational Statistics & Data Analysis 43 (2003) 249-267. |
| </p> |
| <p> |
| Specific applications of the above formulae to the noncentral beta distribution |
| can be found in: |
| </p> |
| <p> |
| Russell V. Lenth, "Algorithm AS 226: Computing Noncentral Beta Probabilities", |
| Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244. |
| </p> |
| <p> |
| H. Frick, "Algorithm AS R84: A Remark on Algorithm AS 226: Computing |
| Non-Central Beta Probabilities", Applied Statistics, Vol. 39, No. |
| 2. (1990), pp. 311-312. |
| </p> |
| <p> |
| Ming Long Lam, "Remark AS R95: A Remark on Algorithm AS 226: Computing |
| Non-Central Beta Probabilities", Applied Statistics, Vol. 44, No. |
| 4. (1995), pp. 551-552. |
| </p> |
| <p> |
| Harry O. Posten, "An Effective Algorithm for the Noncentral Beta |
| Distribution Function", The American Statistician, Vol. 47, No. |
| 2. (May, 1993), pp. 129-131. |
| </p> |
| <p> |
| R. Chattamvelli, "A Note on the Noncentral Beta Distribution Function", |
| The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234. |
| </p> |
| <p> |
| Of these, the Posten reference provides the most complete overview, and |
| includes the modification starting iteration at λ/2. |
| </p> |
| <p> |
| The main difference between this implementation and the above references |
| is the direct computation of the complement when most efficient to do |
| so, and the accumulation of the sum to -1 rather than subtracting the |
| result from 1 at the end: this can substantially reduce the number of |
| iterations required when the result is near 1. |
| </p> |
| <p> |
| The PDF is computed using the methodology of Benton and Krishnamoorthy |
| and the relation: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../equations/nc_beta_ref1.png"></span> |
| </p> |
| <p> |
| Quantiles are computed using a specially modified version of <a class="link" href="../../../toolkit/internals1/roots2.html" title="Root Finding Without Derivatives">bracket_and_solve_root</a>, |
| starting the search for the root at the mean of the distribution. (A |
| Cornish-Fisher type expansion was also tried, but while this gets quite |
| close to the root in many cases, when it is wrong it tends to introduce |
| quite pathological behaviour: more investigation in this area is probably |
| warranted). |
| </p> |
| </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> |
| </tr></table> |
| <hr> |
| <div class="spirit-nav"> |
| <a accesskey="p" href="negative_binomial_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="nc_chi_squared_dist.html"><img src="../../../../../../../../../doc/src/images/next.png" alt="Next"></a> |
| </div> |
| </body> |
| </html> |
