542 lines
19 KiB
Plaintext
542 lines
19 KiB
Plaintext
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// boost\math\distributions\beta.hpp
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// Copyright John Maddock 2006.
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// Copyright Paul A. Bristow 2006.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// http://en.wikipedia.org/wiki/Beta_distribution
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// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
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// http://mathworld.wolfram.com/BetaDistribution.html
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// The Beta Distribution is a continuous probability distribution.
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// The beta distribution is used to model events which are constrained to take place
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// within an interval defined by maxima and minima,
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// so is used extensively in PERT and other project management systems
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// to describe the time to completion.
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// The cdf of the beta distribution is used as a convenient way
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// of obtaining the sum over a set of binomial outcomes.
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// The beta distribution is also used in Bayesian statistics.
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#ifndef BOOST_MATH_DIST_BETA_HPP
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#define BOOST_MATH_DIST_BETA_HPP
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/special_functions/beta.hpp> // for beta.
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#include <boost/math/distributions/complement.hpp> // complements.
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#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
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#include <boost/math/special_functions/fpclassify.hpp> // isnan.
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#include <boost/math/tools/roots.hpp> // for root finding.
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#if defined (BOOST_MSVC)
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# pragma warning(push)
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# pragma warning(disable: 4702) // unreachable code
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// in domain_error_imp in error_handling
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#endif
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#include <utility>
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namespace boost
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{
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namespace math
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{
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namespace beta_detail
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{
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// Common error checking routines for beta distribution functions:
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template <class RealType, class Policy>
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inline bool check_alpha(const char* function, const RealType& alpha, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(alpha) || (alpha <= 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Alpha argument is %1%, but must be > 0 !", alpha, pol);
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return false;
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}
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return true;
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} // bool check_alpha
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template <class RealType, class Policy>
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inline bool check_beta(const char* function, const RealType& beta, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(beta) || (beta <= 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Beta argument is %1%, but must be > 0 !", beta, pol);
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return false;
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}
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return true;
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} // bool check_beta
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template <class RealType, class Policy>
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inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
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{
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if((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
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return false;
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}
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return true;
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} // bool check_prob
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template <class RealType, class Policy>
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inline bool check_x(const char* function, const RealType& x, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(x) || (x < 0) || (x > 1))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"x argument is %1%, but must be >= 0 and <= 1 !", x, pol);
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return false;
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}
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return true;
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} // bool check_x
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template <class RealType, class Policy>
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inline bool check_dist(const char* function, const RealType& alpha, const RealType& beta, RealType* result, const Policy& pol)
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{ // Check both alpha and beta.
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return check_alpha(function, alpha, result, pol)
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&& check_beta(function, beta, result, pol);
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} // bool check_dist
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template <class RealType, class Policy>
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inline bool check_dist_and_x(const char* function, const RealType& alpha, const RealType& beta, RealType x, RealType* result, const Policy& pol)
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{
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return check_dist(function, alpha, beta, result, pol)
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&& beta_detail::check_x(function, x, result, pol);
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} // bool check_dist_and_x
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template <class RealType, class Policy>
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inline bool check_dist_and_prob(const char* function, const RealType& alpha, const RealType& beta, RealType p, RealType* result, const Policy& pol)
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{
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return check_dist(function, alpha, beta, result, pol)
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&& check_prob(function, p, result, pol);
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} // bool check_dist_and_prob
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template <class RealType, class Policy>
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inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(mean) || (mean <= 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"mean argument is %1%, but must be > 0 !", mean, pol);
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return false;
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}
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return true;
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} // bool check_mean
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template <class RealType, class Policy>
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inline bool check_variance(const char* function, const RealType& variance, RealType* result, const Policy& pol)
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{
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if(!(boost::math::isfinite)(variance) || (variance <= 0))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"variance argument is %1%, but must be > 0 !", variance, pol);
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return false;
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}
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return true;
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} // bool check_variance
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} // namespace beta_detail
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// typedef beta_distribution<double> beta;
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// is deliberately NOT included to avoid a name clash with the beta function.
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// Use beta_distribution<> mybeta(...) to construct type double.
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template <class RealType = double, class Policy = policies::policy<> >
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class 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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beta_distribution(RealType l_alpha = 1, RealType l_beta = 1) : m_alpha(l_alpha), m_beta(l_beta)
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{
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RealType result;
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beta_detail::check_dist(
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"boost::math::beta_distribution<%1%>::beta_distribution",
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m_alpha,
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m_beta,
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&result, Policy());
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} // beta_distribution constructor.
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// Accessor functions:
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RealType alpha() const
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{
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return m_alpha;
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}
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RealType beta() const
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{ // .
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return m_beta;
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}
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// Estimation of the alpha & beta parameters.
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// http://en.wikipedia.org/wiki/Beta_distribution
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// gives formulae in section on parameter estimation.
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// Also NIST EDA page 3 & 4 give the same.
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// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
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// http://www.epi.ucdavis.edu/diagnostictests/betabuster.html
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static RealType find_alpha(
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RealType mean, // Expected value of mean.
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RealType variance) // Expected value of variance.
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{
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static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
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RealType result = 0; // of error checks.
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if(false ==
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(
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beta_detail::check_mean(function, mean, &result, Policy())
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&& beta_detail::check_variance(function, variance, &result, Policy())
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)
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)
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{
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return result;
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}
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return mean * (( (mean * (1 - mean)) / variance)- 1);
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} // RealType find_alpha
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static RealType find_beta(
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RealType mean, // Expected value of mean.
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RealType variance) // Expected value of variance.
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{
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static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
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RealType result = 0; // of error checks.
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if(false ==
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(
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beta_detail::check_mean(function, mean, &result, Policy())
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&&
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beta_detail::check_variance(function, variance, &result, Policy())
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)
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)
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{
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return result;
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}
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return (1 - mean) * (((mean * (1 - mean)) /variance)-1);
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} // RealType find_beta
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// Estimate alpha & beta from either alpha or beta, and x and probability.
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// Uses for these parameter estimators are unclear.
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static RealType find_alpha(
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RealType beta, // from beta.
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RealType x, // x.
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RealType probability) // cdf
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{
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static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
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RealType result = 0; // of error checks.
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if(false ==
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(
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beta_detail::check_prob(function, probability, &result, Policy())
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&&
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beta_detail::check_beta(function, beta, &result, Policy())
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&&
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beta_detail::check_x(function, x, &result, Policy())
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)
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)
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{
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return result;
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}
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return ibeta_inva(beta, x, probability, Policy());
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} // RealType find_alpha(beta, a, probability)
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static RealType find_beta(
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// ibeta_invb(T b, T x, T p); (alpha, x, cdf,)
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RealType alpha, // alpha.
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RealType x, // probability x.
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RealType probability) // probability cdf.
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{
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static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
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RealType result = 0; // of error checks.
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if(false ==
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(
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beta_detail::check_prob(function, probability, &result, Policy())
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&&
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beta_detail::check_alpha(function, alpha, &result, Policy())
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&&
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beta_detail::check_x(function, x, &result, Policy())
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)
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)
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{
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return result;
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}
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return ibeta_invb(alpha, x, probability, Policy());
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} // RealType find_beta(alpha, x, probability)
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private:
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RealType m_alpha; // Two parameters of the beta distribution.
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RealType m_beta;
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}; // template <class RealType, class Policy> class beta_distribution
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const beta_distribution<RealType, Policy>& /* dist */)
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{ // Range of permissible values for random variable x.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
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}
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> support(const beta_distribution<RealType, Policy>& /* dist */)
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{ // Range of supported values for random variable x.
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// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
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return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
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}
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template <class RealType, class Policy>
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inline RealType mean(const beta_distribution<RealType, Policy>& dist)
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{ // Mean of beta distribution = np.
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return dist.alpha() / (dist.alpha() + dist.beta());
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} // mean
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template <class RealType, class Policy>
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inline RealType variance(const beta_distribution<RealType, Policy>& dist)
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{ // Variance of beta distribution = np(1-p).
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RealType a = dist.alpha();
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RealType b = dist.beta();
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return (a * b) / ((a + b ) * (a + b) * (a + b + 1));
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} // variance
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template <class RealType, class Policy>
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inline RealType mode(const beta_distribution<RealType, Policy>& dist)
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{
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static const char* function = "boost::math::mode(beta_distribution<%1%> const&)";
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RealType result;
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if ((dist.alpha() <= 1))
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{
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result = policies::raise_domain_error<RealType>(
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function,
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"mode undefined for alpha = %1%, must be > 1!", dist.alpha(), Policy());
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return result;
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}
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if ((dist.beta() <= 1))
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{
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result = policies::raise_domain_error<RealType>(
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function,
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"mode undefined for beta = %1%, must be > 1!", dist.beta(), Policy());
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return result;
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}
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RealType a = dist.alpha();
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RealType b = dist.beta();
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return (a-1) / (a + b - 2);
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} // mode
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//template <class RealType, class Policy>
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//inline RealType median(const beta_distribution<RealType, Policy>& dist)
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//{ // Median of beta distribution is not defined.
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// return tools::domain_error<RealType>(function, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());
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//} // median
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//But WILL be provided by the derived accessor as quantile(0.5).
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template <class RealType, class Policy>
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inline RealType skewness(const beta_distribution<RealType, Policy>& dist)
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{
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BOOST_MATH_STD_USING // ADL of std functions.
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RealType a = dist.alpha();
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RealType b = dist.beta();
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return (2 * (b-a) * sqrt(a + b + 1)) / ((a + b + 2) * sqrt(a * b));
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} // skewness
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template <class RealType, class Policy>
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inline RealType kurtosis_excess(const beta_distribution<RealType, Policy>& dist)
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{
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RealType a = dist.alpha();
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RealType b = dist.beta();
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RealType a_2 = a * a;
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RealType n = 6 * (a_2 * a - a_2 * (2 * b - 1) + b * b * (b + 1) - 2 * a * b * (b + 2));
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RealType d = a * b * (a + b + 2) * (a + b + 3);
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return n / d;
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} // kurtosis_excess
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template <class RealType, class Policy>
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inline RealType kurtosis(const beta_distribution<RealType, Policy>& dist)
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{
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||
|
|
return 3 + kurtosis_excess(dist);
|
||
|
|
} // kurtosis
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType pdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
|
||
|
|
{ // Probability Density/Mass Function.
|
||
|
|
BOOST_FPU_EXCEPTION_GUARD
|
||
|
|
|
||
|
|
static const char* function = "boost::math::pdf(beta_distribution<%1%> const&, %1%)";
|
||
|
|
|
||
|
|
BOOST_MATH_STD_USING // for ADL of std functions
|
||
|
|
|
||
|
|
RealType a = dist.alpha();
|
||
|
|
RealType b = dist.beta();
|
||
|
|
|
||
|
|
// Argument checks:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == beta_detail::check_dist_and_x(
|
||
|
|
function,
|
||
|
|
a, b, x,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
using boost::math::beta;
|
||
|
|
return ibeta_derivative(a, b, x, Policy());
|
||
|
|
} // pdf
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType cdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
|
||
|
|
{ // Cumulative Distribution Function beta.
|
||
|
|
BOOST_MATH_STD_USING // for ADL of std functions
|
||
|
|
|
||
|
|
static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
|
||
|
|
|
||
|
|
RealType a = dist.alpha();
|
||
|
|
RealType b = dist.beta();
|
||
|
|
|
||
|
|
// Argument checks:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == beta_detail::check_dist_and_x(
|
||
|
|
function,
|
||
|
|
a, b, x,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special cases:
|
||
|
|
if (x == 0)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
else if (x == 1)
|
||
|
|
{
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
return ibeta(a, b, x, Policy());
|
||
|
|
} // beta cdf
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType cdf(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{ // Complemented Cumulative Distribution Function beta.
|
||
|
|
|
||
|
|
BOOST_MATH_STD_USING // for ADL of std functions
|
||
|
|
|
||
|
|
static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
|
||
|
|
|
||
|
|
RealType const& x = c.param;
|
||
|
|
beta_distribution<RealType, Policy> const& dist = c.dist;
|
||
|
|
RealType a = dist.alpha();
|
||
|
|
RealType b = dist.beta();
|
||
|
|
|
||
|
|
// Argument checks:
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == beta_detail::check_dist_and_x(
|
||
|
|
function,
|
||
|
|
a, b, x,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
if (x == 0)
|
||
|
|
{
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
else if (x == 1)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
// Calculate cdf beta using the incomplete beta function.
|
||
|
|
// Use of ibeta here prevents cancellation errors in calculating
|
||
|
|
// 1 - x if x is very small, perhaps smaller than machine epsilon.
|
||
|
|
return ibetac(a, b, x, Policy());
|
||
|
|
} // beta cdf
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const beta_distribution<RealType, Policy>& dist, const RealType& p)
|
||
|
|
{ // Quantile or Percent Point beta function or
|
||
|
|
// Inverse Cumulative probability distribution function CDF.
|
||
|
|
// Return x (0 <= x <= 1),
|
||
|
|
// for a given probability p (0 <= p <= 1).
|
||
|
|
// These functions take a probability as an argument
|
||
|
|
// and return a value such that the probability that a random variable x
|
||
|
|
// will be less than or equal to that value
|
||
|
|
// is whatever probability you supplied as an argument.
|
||
|
|
|
||
|
|
static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
|
||
|
|
|
||
|
|
RealType result = 0; // of argument checks:
|
||
|
|
RealType a = dist.alpha();
|
||
|
|
RealType b = dist.beta();
|
||
|
|
if(false == beta_detail::check_dist_and_prob(
|
||
|
|
function,
|
||
|
|
a, b, p,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special cases:
|
||
|
|
if (p == 0)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
if (p == 1)
|
||
|
|
{
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
return ibeta_inv(a, b, p, static_cast<RealType*>(0), Policy());
|
||
|
|
} // quantile
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{ // Complement Quantile or Percent Point beta function .
|
||
|
|
// Return the number of expected x for a given
|
||
|
|
// complement of the probability q.
|
||
|
|
|
||
|
|
static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
|
||
|
|
|
||
|
|
//
|
||
|
|
// Error checks:
|
||
|
|
RealType q = c.param;
|
||
|
|
const beta_distribution<RealType, Policy>& dist = c.dist;
|
||
|
|
RealType result = 0;
|
||
|
|
RealType a = dist.alpha();
|
||
|
|
RealType b = dist.beta();
|
||
|
|
if(false == beta_detail::check_dist_and_prob(
|
||
|
|
function,
|
||
|
|
a,
|
||
|
|
b,
|
||
|
|
q,
|
||
|
|
&result, Policy()))
|
||
|
|
{
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
// Special cases:
|
||
|
|
if(q == 1)
|
||
|
|
{
|
||
|
|
return 0;
|
||
|
|
}
|
||
|
|
if(q == 0)
|
||
|
|
{
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
|
||
|
|
return ibetac_inv(a, b, q, static_cast<RealType*>(0), Policy());
|
||
|
|
} // Quantile Complement
|
||
|
|
|
||
|
|
} // namespace math
|
||
|
|
} // namespace boost
|
||
|
|
|
||
|
|
// This include must be at the end, *after* the accessors
|
||
|
|
// for this distribution have been defined, in order to
|
||
|
|
// keep compilers that support two-phase lookup happy.
|
||
|
|
#include <boost/math/distributions/detail/derived_accessors.hpp>
|
||
|
|
|
||
|
|
#if defined (BOOST_MSVC)
|
||
|
|
# pragma warning(pop)
|
||
|
|
#endif
|
||
|
|
|
||
|
|
#endif // BOOST_MATH_DIST_BETA_HPP
|
||
|
|
|
||
|
|
|