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| <div class="section"> |
| <div class="titlepage"><div><div><h4 class="title"> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist"></a><a class="link" href="nc_beta_dist.html" title="Noncentral Beta Distribution">Noncentral |
| Beta Distribution</a> |
| </h4></div></div></div> |
| <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> |
| <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="Chapter 14. Policies: Controlling Precision, Error Handling etc">Policy</a> <span class="special">=</span> <a class="link" href="../../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="Chapter 14. Policies: Controlling Precision, Error Handling etc">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.svg"></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.svg"></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.svg" align="middle"></span> |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h0"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.member_functions"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.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="../../error_handling.html#math_toolkit.error_handling.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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h1"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.non_member_accessors"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.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_toolkit.dist_ref.nmp.cdf">Cumulative |
| Distribution Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability |
| Density Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>. |
| </p> |
| <p> |
| Mean and variance are implemented using hypergeometric pfq functions and |
| relations given in <a href="http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html" target="_top">Wolfram |
| Noncentral Beta Distribution</a>. |
| </p> |
| <p> |
| However, the following are not currently implemented: <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a> and |
| <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>. |
| </p> |
| <p> |
| The domain of the random variable is [0, 1]. |
| </p> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h2"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.accuracy"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.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="../../relative_error.html#math_toolkit.relative_error.zero_error">effectively |
| zero error</a>. |
| </p> |
| <div class="table"> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.errors_in_cdf_of_the_noncentral_"></a><p class="title"><b>Table 5.4. 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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h3"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.tests"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.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> |
| <h5> |
| <a name="math_toolkit.dist_ref.dists.nc_beta_dist.h4"></a> |
| <span class="phrase"><a name="math_toolkit.dist_ref.dists.nc_beta_dist.implementation"></a></span><a class="link" href="nc_beta_dist.html#math_toolkit.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.svg"></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.svg"></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.svg"></span> |
| </p> |
| <p> |
| Quantiles are computed using a specially modified version of <a class="link" href="../../internals1/roots2.html" title="Root Finding Without Derivatives: Bisection, Bracket and TOMS748">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 © 2006-2010, 2012-2014 Nikhar Agrawal, |
| Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert |
| Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, |
| Thijs van den Berg, Daryle Walker and Xiaogang Zhang<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> |
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