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208 lines
7.3 KiB
Plaintext
208 lines
7.3 KiB
Plaintext
[section:nc_beta_dist Noncentral Beta Distribution]
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``#include <boost/math/distributions/non_central_beta.hpp>``
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namespace boost{ namespace math{
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template <class RealType = double,
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class ``__Policy`` = ``__policy_class`` >
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class non_central_beta_distribution;
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typedef non_central_beta_distribution<> non_central_beta;
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template <class RealType, class ``__Policy``>
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class non_central_beta_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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// Constructor:
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BOOST_MATH_GPU_ENABLED non_central_beta_distribution(RealType alpha, RealType beta, RealType lambda);
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// Accessor to shape parameters:
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BOOST_MATH_GPU_ENABLED RealType alpha()const;
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BOOST_MATH_GPU_ENABLED RealType beta()const;
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// Accessor to non-centrality parameter lambda:
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BOOST_MATH_GPU_ENABLED RealType non_centrality()const;
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};
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}} // namespaces
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The noncentral beta distribution is a generalization of the __beta_distrib.
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It is defined as the ratio
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[expression X = [chi][sub m][super 2]([lambda]) \/ ([chi][sub m][super 2]([lambda])
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+ [chi][sub n][super 2])]
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where [role serif_italic [chi][sub m][super 2]([lambda])]
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is a noncentral [role serif_italic [chi][super 2]]
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random variable with /m/ degrees of freedom, and [chi][sub n][super 2]
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is a central [role serif_italic [chi][super 2] ] random variable with /n/ degrees of freedom.
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This gives a PDF that can be expressed as a Poisson mixture
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of beta distribution PDFs:
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[equation nc_beta_ref1]
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where P(i;[lambda]\/2) is the discrete Poisson probability at /i/, with mean
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[lambda]\/2, and I[sub x][super ']([alpha], [beta]) is the derivative of
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the incomplete beta function. This leads to the usual form of the CDF
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as:
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[equation nc_beta_ref2]
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The following graph illustrates how the distribution changes
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for different values of [lambda]:
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[graph nc_beta_pdf]
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[h4 Member Functions]
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BOOST_MATH_GPU_ENABLED non_central_beta_distribution(RealType a, RealType b, RealType lambda);
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Constructs a noncentral beta distribution with shape parameters /a/ and /b/
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and non-centrality parameter /lambda/.
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Requires a > 0, b > 0 and lambda >= 0, otherwise calls __domain_error.
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BOOST_MATH_GPU_ENABLED RealType alpha()const;
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Returns the parameter /a/ from which this object was constructed.
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BOOST_MATH_GPU_ENABLED RealType beta()const;
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Returns the parameter /b/ from which this object was constructed.
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BOOST_MATH_GPU_ENABLED RealType non_centrality()const;
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Returns the parameter /lambda/ from which this object was constructed.
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[h4 Non-member Accessors]
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Most of the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
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are supported: __cdf, __pdf, __quantile, __mean, __variance, __sd,
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__median, __mode, __hazard, __chf, __range and __support.
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For this distribution these functions are marked with `BOOST_MATH_GPU_ENABLED` and can
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be run on both host and device.
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Mean and variance are implemented using hypergeometric pfq functions and relations given in
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[@http://reference.wolfram.com/mathematica/ref/NoncentralBetaDistribution.html Wolfram Noncentral Beta Distribution].
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However, the following are not currently implemented:
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__skewness, __kurtosis and __kurtosis_excess.
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The domain of the random variable is \[0, 1\].
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[h4 Accuracy]
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The following table shows the peak errors
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(in units of [@http://en.wikipedia.org/wiki/Machine_epsilon epsilon])
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found on various platforms with various floating point types.
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The failures in the comparison to the [@http://www.r-project.org/ R Math library],
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seem to be mostly in the corner cases when the probability would be very small.
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Unless otherwise specified any floating-point type that is narrower
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than the one shown will have __zero_error.
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[table_non_central_beta_CDF]
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[table_non_central_beta_CDF_complement]
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Error rates for the PDF, the complement of the CDF and for the quantile
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functions are broadly similar.
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[h4 Tests]
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There are two sets of test data used to verify this implementation:
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firstly we can compare with a few sample values generated by the
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[@http://www.r-project.org/ R library].
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Secondly, we have tables of test data, computed with this
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implementation and using interval arithmetic - this data should
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be accurate to at least 50 decimal digits - and is the used for
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our accuracy tests.
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[h4 Implementation]
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The CDF and its complement are evaluated as follows:
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First we determine which of the two values (the CDF or its
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complement) is likely to be the smaller, the crossover point
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is taken to be the mean of the distribution: for this we use the
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approximation due to: R. Chattamvelli and R. Shanmugam,
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"Algorithm AS 310: Computing the Non-Central Beta Distribution Function",
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Applied Statistics, Vol. 46, No. 1. (1997), pp. 146-156.
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[equation nc_beta_ref3]
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Then either the CDF or its complement is computed using the
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relations:
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[equation nc_beta_ref4]
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The summation is performed by starting at i = [lambda]/2, and then recursing
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in both directions, using the usual recurrence relations for the Poisson
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PDF and incomplete beta functions. This is the "Method 2" described
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by:
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Denise Benton and K. Krishnamoorthy,
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"Computing discrete mixtures of continuous
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distributions: noncentral chisquare, noncentral t
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and the distribution of the square of the sample
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multiple correlation coefficient",
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Computational Statistics & Data Analysis 43 (2003) 249-267.
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Specific applications of the above formulae to the noncentral
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beta distribution can be found in:
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Russell V. Lenth,
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"Algorithm AS 226: Computing Noncentral Beta Probabilities",
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Applied Statistics, Vol. 36, No. 2. (1987), pp. 241-244.
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H. Frick,
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"Algorithm AS R84: A Remark on Algorithm AS 226: Computing Non-Central Beta
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Probabilities", Applied Statistics, Vol. 39, No. 2. (1990), pp. 311-312.
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Ming Long Lam,
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"Remark AS R95: A Remark on Algorithm AS 226: Computing Non-Central Beta
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Probabilities", Applied Statistics, Vol. 44, No. 4. (1995), pp. 551-552.
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Harry O. Posten,
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"An Effective Algorithm for the Noncentral Beta Distribution Function",
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The American Statistician, Vol. 47, No. 2. (May, 1993), pp. 129-131.
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R. Chattamvelli,
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"A Note on the Noncentral Beta Distribution Function",
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The American Statistician, Vol. 49, No. 2. (May, 1995), pp. 231-234.
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Of these, the Posten reference provides the most complete overview,
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and includes the modification starting iteration at [lambda]/2.
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The main difference between this implementation and the above
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references is the direct computation of the complement when most
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efficient to do so, and the accumulation of the sum to -1 rather
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than subtracting the result from 1 at the end: this can substantially
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reduce the number of iterations required when the result is near 1.
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The PDF is computed using the methodology of Benton and Krishnamoorthy
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and the relation:
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[equation nc_beta_ref1]
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Quantiles are computed using a specially modified version of
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__bracket_solve,
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starting the search for the root at the mean of the distribution.
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(A Cornish-Fisher type expansion was also tried, but while this gets
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quite close to the root in many cases, when it is wrong it tends to
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introduce quite pathological behaviour: more investigation in this
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area is probably warranted).
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[endsect] [/section:nc_beta_dist]
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[/ nc_beta.qbk
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Copyright 2008 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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