930 lines
36 KiB
Plaintext
930 lines
36 KiB
Plaintext
// boost\math\distributions\non_central_beta.hpp
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// Copyright John Maddock 2008.
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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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#ifndef BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
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#define BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/special_functions/beta.hpp> // for incomplete gamma. gamma_q
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#include <boost/math/distributions/complement.hpp> // complements
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#include <boost/math/distributions/beta.hpp> // central distribution
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#include <boost/math/distributions/detail/generic_mode.hpp>
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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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#include <boost/math/tools/series.hpp>
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namespace boost
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{
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namespace math
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{
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template <class RealType, class Policy>
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class non_central_beta_distribution;
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namespace detail{
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template <class T, class Policy>
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T non_central_beta_p(T a, T b, T lam, T x, T y, const Policy& pol, T init_val = 0)
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{
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BOOST_MATH_STD_USING
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using namespace boost::math;
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//
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// Variables come first:
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//
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boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
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T errtol = boost::math::policies::get_epsilon<T, Policy>();
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T l2 = lam / 2;
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//
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// k is the starting point for iteration, and is the
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// maximum of the poisson weighting term,
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// note that unlike other similar code, we do not set
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// k to zero, when l2 is small, as forward iteration
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// is unstable:
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//
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int k = itrunc(l2);
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if(k == 0)
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k = 1;
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// Starting Poisson weight:
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T pois = gamma_p_derivative(T(k+1), l2, pol);
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if(pois == 0)
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return init_val;
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// recurance term:
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T xterm;
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// Starting beta term:
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T beta = x < y
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? detail::ibeta_imp(T(a + k), b, x, pol, false, true, &xterm)
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: detail::ibeta_imp(b, T(a + k), y, pol, true, true, &xterm);
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xterm *= y / (a + b + k - 1);
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T poisf(pois), betaf(beta), xtermf(xterm);
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T sum = init_val;
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if((beta == 0) && (xterm == 0))
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return init_val;
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//
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// Backwards recursion first, this is the stable
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// direction for recursion:
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//
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T last_term = 0;
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boost::uintmax_t count = k;
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for(int i = k; i >= 0; --i)
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{
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T term = beta * pois;
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sum += term;
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if(((fabs(term/sum) < errtol) && (last_term >= term)) || (term == 0))
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{
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count = k - i;
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break;
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}
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pois *= i / l2;
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beta += xterm;
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xterm *= (a + i - 1) / (x * (a + b + i - 2));
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last_term = term;
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}
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for(int i = k + 1; ; ++i)
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{
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poisf *= l2 / i;
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xtermf *= (x * (a + b + i - 2)) / (a + i - 1);
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betaf -= xtermf;
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T term = poisf * betaf;
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sum += term;
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if((fabs(term/sum) < errtol) || (term == 0))
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{
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break;
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}
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if(static_cast<boost::uintmax_t>(count + i - k) > max_iter)
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{
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return policies::raise_evaluation_error(
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"cdf(non_central_beta_distribution<%1%>, %1%)",
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"Series did not converge, closest value was %1%", sum, pol);
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}
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}
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return sum;
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}
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template <class T, class Policy>
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T non_central_beta_q(T a, T b, T lam, T x, T y, const Policy& pol, T init_val = 0)
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{
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BOOST_MATH_STD_USING
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using namespace boost::math;
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//
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// Variables come first:
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//
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boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
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T errtol = boost::math::policies::get_epsilon<T, Policy>();
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T l2 = lam / 2;
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//
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// k is the starting point for iteration, and is the
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// maximum of the poisson weighting term:
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//
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int k = itrunc(l2);
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T pois;
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if(k <= 30)
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{
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//
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// Might as well start at 0 since we'll likely have this number of terms anyway:
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//
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if(a + b > 1)
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k = 0;
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else if(k == 0)
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k = 1;
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}
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if(k == 0)
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{
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// Starting Poisson weight:
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pois = exp(-l2);
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}
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else
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{
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// Starting Poisson weight:
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pois = gamma_p_derivative(T(k+1), l2, pol);
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}
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if(pois == 0)
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return init_val;
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// recurance term:
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T xterm;
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// Starting beta term:
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T beta = x < y
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? detail::ibeta_imp(T(a + k), b, x, pol, true, true, &xterm)
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: detail::ibeta_imp(b, T(a + k), y, pol, false, true, &xterm);
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xterm *= y / (a + b + k - 1);
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T poisf(pois), betaf(beta), xtermf(xterm);
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T sum = init_val;
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if((beta == 0) && (xterm == 0))
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return init_val;
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//
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// Forwards recursion first, this is the stable
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// direction for recursion, and the location
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// of the bulk of the sum:
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//
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T last_term = 0;
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boost::uintmax_t count = 0;
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for(int i = k + 1; ; ++i)
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{
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poisf *= l2 / i;
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xtermf *= (x * (a + b + i - 2)) / (a + i - 1);
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betaf += xtermf;
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T term = poisf * betaf;
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sum += term;
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if((fabs(term/sum) < errtol) && (last_term >= term))
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{
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count = i - k;
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break;
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}
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if(static_cast<boost::uintmax_t>(i - k) > max_iter)
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{
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return policies::raise_evaluation_error(
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"cdf(non_central_beta_distribution<%1%>, %1%)",
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"Series did not converge, closest value was %1%", sum, pol);
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}
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last_term = term;
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}
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for(int i = k; i >= 0; --i)
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{
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T term = beta * pois;
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sum += term;
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if(fabs(term/sum) < errtol)
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{
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break;
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}
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if(static_cast<boost::uintmax_t>(count + k - i) > max_iter)
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{
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return policies::raise_evaluation_error(
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"cdf(non_central_beta_distribution<%1%>, %1%)",
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"Series did not converge, closest value was %1%", sum, pol);
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}
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pois *= i / l2;
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beta -= xterm;
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xterm *= (a + i - 1) / (x * (a + b + i - 2));
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}
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return sum;
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}
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template <class RealType, class Policy>
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inline RealType non_central_beta_cdf(RealType x, RealType y, RealType a, RealType b, RealType l, bool invert, const Policy&)
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{
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typedef typename policies::evaluation<RealType, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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BOOST_MATH_STD_USING
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if(x == 0)
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return invert ? 1.0f : 0.0f;
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if(y == 0)
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return invert ? 0.0f : 1.0f;
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value_type result;
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value_type c = a + b + l / 2;
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value_type cross = 1 - (b / c) * (1 + l / (2 * c * c));
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if(l == 0)
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result = cdf(boost::math::beta_distribution<RealType, Policy>(a, b), x);
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else if(x > cross)
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{
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// Complement is the smaller of the two:
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result = detail::non_central_beta_q(
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static_cast<value_type>(a),
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static_cast<value_type>(b),
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static_cast<value_type>(l),
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static_cast<value_type>(x),
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static_cast<value_type>(y),
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forwarding_policy(),
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static_cast<value_type>(invert ? 0 : -1));
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invert = !invert;
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}
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else
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{
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result = detail::non_central_beta_p(
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static_cast<value_type>(a),
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static_cast<value_type>(b),
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static_cast<value_type>(l),
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static_cast<value_type>(x),
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static_cast<value_type>(y),
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forwarding_policy(),
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static_cast<value_type>(invert ? -1 : 0));
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}
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if(invert)
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result = -result;
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return policies::checked_narrowing_cast<RealType, forwarding_policy>(
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result,
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"boost::math::non_central_beta_cdf<%1%>(%1%, %1%, %1%)");
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}
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template <class T, class Policy>
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struct nc_beta_quantile_functor
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{
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nc_beta_quantile_functor(const non_central_beta_distribution<T,Policy>& d, T t, bool c)
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: dist(d), target(t), comp(c) {}
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T operator()(const T& x)
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{
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return comp ?
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T(target - cdf(complement(dist, x)))
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: T(cdf(dist, x) - target);
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}
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private:
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non_central_beta_distribution<T,Policy> dist;
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T target;
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bool comp;
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};
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//
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// This is more or less a copy of bracket_and_solve_root, but
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// modified to search only the interval [0,1] using similar
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// heuristics.
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//
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template <class F, class T, class Tol, class Policy>
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std::pair<T, T> bracket_and_solve_root_01(F f, const T& guess, T factor, bool rising, Tol tol, boost::uintmax_t& max_iter, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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static const char* function = "boost::math::tools::bracket_and_solve_root_01<%1%>";
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//
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// Set up inital brackets:
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//
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T a = guess;
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T b = a;
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T fa = f(a);
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T fb = fa;
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//
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// Set up invocation count:
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//
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boost::uintmax_t count = max_iter - 1;
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if((fa < 0) == (guess < 0 ? !rising : rising))
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{
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//
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// Zero is to the right of b, so walk upwards
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// until we find it:
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//
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while((boost::math::sign)(fb) == (boost::math::sign)(fa))
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{
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if(count == 0)
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{
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b = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", b, pol);
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return std::make_pair(a, b);
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}
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//
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// Heuristic: every 20 iterations we double the growth factor in case the
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// initial guess was *really* bad !
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//
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if((max_iter - count) % 20 == 0)
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factor *= 2;
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//
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// Now go ahead and move are guess by "factor",
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// we do this by reducing 1-guess by factor:
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//
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a = b;
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fa = fb;
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b = 1 - ((1 - b) / factor);
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fb = f(b);
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--count;
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BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
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}
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}
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else
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{
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//
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// Zero is to the left of a, so walk downwards
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// until we find it:
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//
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while((boost::math::sign)(fb) == (boost::math::sign)(fa))
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{
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if(fabs(a) < tools::min_value<T>())
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{
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// Escape route just in case the answer is zero!
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max_iter -= count;
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max_iter += 1;
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return a > 0 ? std::make_pair(T(0), T(a)) : std::make_pair(T(a), T(0));
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}
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if(count == 0)
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{
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a = policies::raise_evaluation_error(function, "Unable to bracket root, last nearest value was %1%", a, pol);
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return std::make_pair(a, b);
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}
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//
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// Heuristic: every 20 iterations we double the growth factor in case the
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// initial guess was *really* bad !
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//
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if((max_iter - count) % 20 == 0)
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factor *= 2;
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//
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// Now go ahead and move are guess by "factor":
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//
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b = a;
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fb = fa;
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a /= factor;
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fa = f(a);
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--count;
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BOOST_MATH_INSTRUMENT_CODE("a = " << a << " b = " << b << " fa = " << fa << " fb = " << fb << " count = " << count);
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}
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}
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max_iter -= count;
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max_iter += 1;
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std::pair<T, T> r = toms748_solve(
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f,
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(a < 0 ? b : a),
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(a < 0 ? a : b),
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(a < 0 ? fb : fa),
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(a < 0 ? fa : fb),
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tol,
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count,
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pol);
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max_iter += count;
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BOOST_MATH_INSTRUMENT_CODE("max_iter = " << max_iter << " count = " << count);
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return r;
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}
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template <class RealType, class Policy>
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RealType nc_beta_quantile(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& p, bool comp)
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{
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static const char* function = "quantile(non_central_beta_distribution<%1%>, %1%)";
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typedef typename policies::evaluation<RealType, Policy>::type value_type;
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typedef typename policies::normalise<
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Policy,
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policies::promote_float<false>,
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policies::promote_double<false>,
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policies::discrete_quantile<>,
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policies::assert_undefined<> >::type forwarding_policy;
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value_type a = dist.alpha();
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value_type b = dist.beta();
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value_type l = dist.non_centrality();
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value_type r;
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if(!beta_detail::check_alpha(
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function,
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a, &r, Policy())
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||
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!beta_detail::check_beta(
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function,
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b, &r, Policy())
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!detail::check_non_centrality(
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function,
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l,
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&r,
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Policy())
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||
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!detail::check_probability(
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function,
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static_cast<value_type>(p),
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&r,
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Policy()))
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return (RealType)r;
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//
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// Special cases first:
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//
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if(p == 0)
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return comp
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? 1.0f
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: 0.0f;
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if(p == 1)
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return !comp
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? 1.0f
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: 0.0f;
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value_type c = a + b + l / 2;
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value_type mean = 1 - (b / c) * (1 + l / (2 * c * c));
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/*
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//
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// Calculate a normal approximation to the quantile,
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// uses mean and variance approximations from:
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// Algorithm AS 310:
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// Computing the Non-Central Beta Distribution Function
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// R. Chattamvelli; R. Shanmugam
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// Applied Statistics, Vol. 46, No. 1. (1997), pp. 146-156.
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//
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// Unfortunately, when this is wrong it tends to be *very*
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// wrong, so it's disabled for now, even though it often
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// gets the initial guess quite close. Probably we could
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// do much better by factoring in the skewness if only
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// we could calculate it....
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//
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value_type delta = l / 2;
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value_type delta2 = delta * delta;
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value_type delta3 = delta * delta2;
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value_type delta4 = delta2 * delta2;
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value_type G = c * (c + 1) + delta;
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value_type alpha = a + b;
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value_type alpha2 = alpha * alpha;
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value_type eta = (2 * alpha + 1) * (2 * alpha + 1) + 1;
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value_type H = 3 * alpha2 + 5 * alpha + 2;
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value_type F = alpha2 * (alpha + 1) + H * delta
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+ (2 * alpha + 4) * delta2 + delta3;
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value_type P = (3 * alpha + 1) * (9 * alpha + 17)
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+ 2 * alpha * (3 * alpha + 2) * (3 * alpha + 4) + 15;
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value_type Q = 54 * alpha2 + 162 * alpha + 130;
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value_type R = 6 * (6 * alpha + 11);
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value_type D = delta
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* (H * H + 2 * P * delta + Q * delta2 + R * delta3 + 9 * delta4);
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value_type variance = (b / G)
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* (1 + delta * (l * l + 3 * l + eta) / (G * G))
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- (b * b / F) * (1 + D / (F * F));
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value_type sd = sqrt(variance);
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value_type guess = comp
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? quantile(complement(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p))
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: quantile(normal_distribution<RealType, Policy>(static_cast<RealType>(mean), static_cast<RealType>(sd)), p);
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if(guess >= 1)
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guess = mean;
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if(guess <= tools::min_value<value_type>())
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guess = mean;
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*/
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value_type guess = mean;
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detail::nc_beta_quantile_functor<value_type, Policy>
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f(non_central_beta_distribution<value_type, Policy>(a, b, l), p, comp);
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tools::eps_tolerance<value_type> tol(policies::digits<RealType, Policy>());
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boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
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std::pair<value_type, value_type> ir
|
|
= bracket_and_solve_root_01(
|
|
f, guess, value_type(2.5), true, tol,
|
|
max_iter, Policy());
|
|
value_type result = ir.first + (ir.second - ir.first) / 2;
|
|
|
|
if(max_iter >= policies::get_max_root_iterations<Policy>())
|
|
{
|
|
return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
|
|
" either there is no answer to quantile of the non central beta distribution"
|
|
" or the answer is infinite. Current best guess is %1%",
|
|
policies::checked_narrowing_cast<RealType, forwarding_policy>(
|
|
result,
|
|
function), Policy());
|
|
}
|
|
return policies::checked_narrowing_cast<RealType, forwarding_policy>(
|
|
result,
|
|
function);
|
|
}
|
|
|
|
template <class T, class Policy>
|
|
T non_central_beta_pdf(T a, T b, T lam, T x, T y, const Policy& pol)
|
|
{
|
|
BOOST_MATH_STD_USING
|
|
//
|
|
// Special cases:
|
|
//
|
|
if((x == 0) || (y == 0))
|
|
return 0;
|
|
//
|
|
// Variables come first:
|
|
//
|
|
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
|
|
T errtol = boost::math::policies::get_epsilon<T, Policy>();
|
|
T l2 = lam / 2;
|
|
//
|
|
// k is the starting point for iteration, and is the
|
|
// maximum of the poisson weighting term:
|
|
//
|
|
int k = itrunc(l2);
|
|
// Starting Poisson weight:
|
|
T pois = gamma_p_derivative(T(k+1), l2, pol);
|
|
// Starting beta term:
|
|
T beta = x < y ?
|
|
ibeta_derivative(a + k, b, x, pol)
|
|
: ibeta_derivative(b, a + k, y, pol);
|
|
T sum = 0;
|
|
T poisf(pois);
|
|
T betaf(beta);
|
|
|
|
//
|
|
// Stable backwards recursion first:
|
|
//
|
|
boost::uintmax_t count = k;
|
|
for(int i = k; i >= 0; --i)
|
|
{
|
|
T term = beta * pois;
|
|
sum += term;
|
|
if((fabs(term/sum) < errtol) || (term == 0))
|
|
{
|
|
count = k - i;
|
|
break;
|
|
}
|
|
pois *= i / l2;
|
|
beta *= (a + i - 1) / (x * (a + i + b - 1));
|
|
}
|
|
for(int i = k + 1; ; ++i)
|
|
{
|
|
poisf *= l2 / i;
|
|
betaf *= x * (a + b + i - 1) / (a + i - 1);
|
|
|
|
T term = poisf * betaf;
|
|
sum += term;
|
|
if((fabs(term/sum) < errtol) || (term == 0))
|
|
{
|
|
break;
|
|
}
|
|
if(static_cast<boost::uintmax_t>(count + i - k) > max_iter)
|
|
{
|
|
return policies::raise_evaluation_error(
|
|
"pdf(non_central_beta_distribution<%1%>, %1%)",
|
|
"Series did not converge, closest value was %1%", sum, pol);
|
|
}
|
|
}
|
|
return sum;
|
|
}
|
|
|
|
template <class RealType, class Policy>
|
|
RealType nc_beta_pdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
|
|
{
|
|
BOOST_MATH_STD_USING
|
|
static const char* function = "pdf(non_central_beta_distribution<%1%>, %1%)";
|
|
typedef typename policies::evaluation<RealType, Policy>::type value_type;
|
|
typedef typename policies::normalise<
|
|
Policy,
|
|
policies::promote_float<false>,
|
|
policies::promote_double<false>,
|
|
policies::discrete_quantile<>,
|
|
policies::assert_undefined<> >::type forwarding_policy;
|
|
|
|
value_type a = dist.alpha();
|
|
value_type b = dist.beta();
|
|
value_type l = dist.non_centrality();
|
|
value_type r;
|
|
if(!beta_detail::check_alpha(
|
|
function,
|
|
a, &r, Policy())
|
|
||
|
|
!beta_detail::check_beta(
|
|
function,
|
|
b, &r, Policy())
|
|
||
|
|
!detail::check_non_centrality(
|
|
function,
|
|
l,
|
|
&r,
|
|
Policy())
|
|
||
|
|
!beta_detail::check_x(
|
|
function,
|
|
static_cast<value_type>(x),
|
|
&r,
|
|
Policy()))
|
|
return (RealType)r;
|
|
|
|
if(l == 0)
|
|
return pdf(boost::math::beta_distribution<RealType, Policy>(dist.alpha(), dist.beta()), x);
|
|
return policies::checked_narrowing_cast<RealType, forwarding_policy>(
|
|
non_central_beta_pdf(a, b, l, static_cast<value_type>(x), value_type(1 - static_cast<value_type>(x)), forwarding_policy()),
|
|
"function");
|
|
}
|
|
|
|
template <class T>
|
|
struct hypergeometric_2F2_sum
|
|
{
|
|
typedef T result_type;
|
|
hypergeometric_2F2_sum(T a1_, T a2_, T b1_, T b2_, T z_) : a1(a1_), a2(a2_), b1(b1_), b2(b2_), z(z_), term(1), k(0) {}
|
|
T operator()()
|
|
{
|
|
T result = term;
|
|
term *= a1 * a2 / (b1 * b2);
|
|
a1 += 1;
|
|
a2 += 1;
|
|
b1 += 1;
|
|
b2 += 1;
|
|
k += 1;
|
|
term /= k;
|
|
term *= z;
|
|
return result;
|
|
}
|
|
T a1, a2, b1, b2, z, term, k;
|
|
};
|
|
|
|
template <class T, class Policy>
|
|
T hypergeometric_2F2(T a1, T a2, T b1, T b2, T z, const Policy& pol)
|
|
{
|
|
typedef typename policies::evaluation<T, Policy>::type value_type;
|
|
|
|
const char* function = "boost::math::detail::hypergeometric_2F2<%1%>(%1%,%1%,%1%,%1%,%1%)";
|
|
|
|
hypergeometric_2F2_sum<value_type> s(a1, a2, b1, b2, z);
|
|
boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>();
|
|
#if BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
|
|
value_type zero = 0;
|
|
value_type result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<value_type, Policy>(), max_iter, zero);
|
|
#else
|
|
value_type result = boost::math::tools::sum_series(s, boost::math::policies::get_epsilon<value_type, Policy>(), max_iter);
|
|
#endif
|
|
policies::check_series_iterations<T>(function, max_iter, pol);
|
|
return policies::checked_narrowing_cast<T, Policy>(result, function);
|
|
}
|
|
|
|
} // namespace detail
|
|
|
|
template <class RealType = double, class Policy = policies::policy<> >
|
|
class non_central_beta_distribution
|
|
{
|
|
public:
|
|
typedef RealType value_type;
|
|
typedef Policy policy_type;
|
|
|
|
non_central_beta_distribution(RealType a_, RealType b_, RealType lambda) : a(a_), b(b_), ncp(lambda)
|
|
{
|
|
const char* function = "boost::math::non_central_beta_distribution<%1%>::non_central_beta_distribution(%1%,%1%)";
|
|
RealType r;
|
|
beta_detail::check_alpha(
|
|
function,
|
|
a, &r, Policy());
|
|
beta_detail::check_beta(
|
|
function,
|
|
b, &r, Policy());
|
|
detail::check_non_centrality(
|
|
function,
|
|
lambda,
|
|
&r,
|
|
Policy());
|
|
} // non_central_beta_distribution constructor.
|
|
|
|
RealType alpha() const
|
|
{ // Private data getter function.
|
|
return a;
|
|
}
|
|
RealType beta() const
|
|
{ // Private data getter function.
|
|
return b;
|
|
}
|
|
RealType non_centrality() const
|
|
{ // Private data getter function.
|
|
return ncp;
|
|
}
|
|
private:
|
|
// Data member, initialized by constructor.
|
|
RealType a; // alpha.
|
|
RealType b; // beta.
|
|
RealType ncp; // non-centrality parameter
|
|
}; // template <class RealType, class Policy> class non_central_beta_distribution
|
|
|
|
typedef non_central_beta_distribution<double> non_central_beta; // Reserved name of type double.
|
|
|
|
// Non-member functions to give properties of the distribution.
|
|
|
|
template <class RealType, class Policy>
|
|
inline const std::pair<RealType, RealType> range(const non_central_beta_distribution<RealType, Policy>& /* dist */)
|
|
{ // Range of permissible values for random variable k.
|
|
using boost::math::tools::max_value;
|
|
return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
|
|
}
|
|
|
|
template <class RealType, class Policy>
|
|
inline const std::pair<RealType, RealType> support(const non_central_beta_distribution<RealType, Policy>& /* dist */)
|
|
{ // Range of supported values for random variable k.
|
|
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
|
|
using boost::math::tools::max_value;
|
|
return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
|
|
}
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType mode(const non_central_beta_distribution<RealType, Policy>& dist)
|
|
{ // mode.
|
|
static const char* function = "mode(non_central_beta_distribution<%1%> const&)";
|
|
|
|
RealType a = dist.alpha();
|
|
RealType b = dist.beta();
|
|
RealType l = dist.non_centrality();
|
|
RealType r;
|
|
if(!beta_detail::check_alpha(
|
|
function,
|
|
a, &r, Policy())
|
|
||
|
|
!beta_detail::check_beta(
|
|
function,
|
|
b, &r, Policy())
|
|
||
|
|
!detail::check_non_centrality(
|
|
function,
|
|
l,
|
|
&r,
|
|
Policy()))
|
|
return (RealType)r;
|
|
RealType c = a + b + l / 2;
|
|
RealType mean = 1 - (b / c) * (1 + l / (2 * c * c));
|
|
return detail::generic_find_mode_01(
|
|
dist,
|
|
mean,
|
|
function);
|
|
}
|
|
|
|
//
|
|
// We don't have the necessary information to implement
|
|
// these at present. These are just disabled for now,
|
|
// prototypes retained so we can fill in the blanks
|
|
// later:
|
|
//
|
|
template <class RealType, class Policy>
|
|
inline RealType mean(const non_central_beta_distribution<RealType, Policy>& dist)
|
|
{
|
|
BOOST_MATH_STD_USING
|
|
RealType a = dist.alpha();
|
|
RealType b = dist.beta();
|
|
RealType d = dist.non_centrality();
|
|
RealType apb = a + b;
|
|
return exp(-d / 2) * a * detail::hypergeometric_2F2<RealType, Policy>(1 + a, apb, a, 1 + apb, d / 2, Policy()) / apb;
|
|
} // mean
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType variance(const non_central_beta_distribution<RealType, Policy>& dist)
|
|
{
|
|
//
|
|
// Relative error of this function may be arbitarily large... absolute
|
|
// error will be small however... that's the best we can do for now.
|
|
//
|
|
BOOST_MATH_STD_USING
|
|
RealType a = dist.alpha();
|
|
RealType b = dist.beta();
|
|
RealType d = dist.non_centrality();
|
|
RealType apb = a + b;
|
|
RealType result = detail::hypergeometric_2F2(RealType(1 + a), apb, a, RealType(1 + apb), RealType(d / 2), Policy());
|
|
result *= result * -exp(-d) * a * a / (apb * apb);
|
|
result += exp(-d / 2) * a * (1 + a) * detail::hypergeometric_2F2(RealType(2 + a), apb, a, RealType(2 + apb), RealType(d / 2), Policy()) / (apb * (1 + apb));
|
|
return result;
|
|
}
|
|
|
|
// RealType standard_deviation(const non_central_beta_distribution<RealType, Policy>& dist)
|
|
// standard_deviation provided by derived accessors.
|
|
template <class RealType, class Policy>
|
|
inline RealType skewness(const non_central_beta_distribution<RealType, Policy>& /*dist*/)
|
|
{ // skewness = sqrt(l).
|
|
const char* function = "boost::math::non_central_beta_distribution<%1%>::skewness()";
|
|
typedef typename Policy::assert_undefined_type assert_type;
|
|
BOOST_STATIC_ASSERT(assert_type::value == 0);
|
|
|
|
return policies::raise_evaluation_error<RealType>(
|
|
function,
|
|
"This function is not yet implemented, the only sensible result is %1%.",
|
|
std::numeric_limits<RealType>::quiet_NaN(), Policy()); // infinity?
|
|
}
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType kurtosis_excess(const non_central_beta_distribution<RealType, Policy>& /*dist*/)
|
|
{
|
|
const char* function = "boost::math::non_central_beta_distribution<%1%>::kurtosis_excess()";
|
|
typedef typename Policy::assert_undefined_type assert_type;
|
|
BOOST_STATIC_ASSERT(assert_type::value == 0);
|
|
|
|
return policies::raise_evaluation_error<RealType>(
|
|
function,
|
|
"This function is not yet implemented, the only sensible result is %1%.",
|
|
std::numeric_limits<RealType>::quiet_NaN(), Policy()); // infinity?
|
|
} // kurtosis_excess
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType kurtosis(const non_central_beta_distribution<RealType, Policy>& dist)
|
|
{
|
|
return kurtosis_excess(dist) + 3;
|
|
}
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType pdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
|
|
{ // Probability Density/Mass Function.
|
|
return detail::nc_beta_pdf(dist, x);
|
|
} // pdf
|
|
|
|
template <class RealType, class Policy>
|
|
RealType cdf(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& x)
|
|
{
|
|
const char* function = "boost::math::non_central_beta_distribution<%1%>::cdf(%1%)";
|
|
RealType a = dist.alpha();
|
|
RealType b = dist.beta();
|
|
RealType l = dist.non_centrality();
|
|
RealType r;
|
|
if(!beta_detail::check_alpha(
|
|
function,
|
|
a, &r, Policy())
|
|
||
|
|
!beta_detail::check_beta(
|
|
function,
|
|
b, &r, Policy())
|
|
||
|
|
!detail::check_non_centrality(
|
|
function,
|
|
l,
|
|
&r,
|
|
Policy())
|
|
||
|
|
!beta_detail::check_x(
|
|
function,
|
|
x,
|
|
&r,
|
|
Policy()))
|
|
return (RealType)r;
|
|
|
|
if(l == 0)
|
|
return cdf(beta_distribution<RealType, Policy>(a, b), x);
|
|
|
|
return detail::non_central_beta_cdf(x, RealType(1 - x), a, b, l, false, Policy());
|
|
} // cdf
|
|
|
|
template <class RealType, class Policy>
|
|
RealType cdf(const complemented2_type<non_central_beta_distribution<RealType, Policy>, RealType>& c)
|
|
{ // Complemented Cumulative Distribution Function
|
|
const char* function = "boost::math::non_central_beta_distribution<%1%>::cdf(%1%)";
|
|
non_central_beta_distribution<RealType, Policy> const& dist = c.dist;
|
|
RealType a = dist.alpha();
|
|
RealType b = dist.beta();
|
|
RealType l = dist.non_centrality();
|
|
RealType x = c.param;
|
|
RealType r;
|
|
if(!beta_detail::check_alpha(
|
|
function,
|
|
a, &r, Policy())
|
|
||
|
|
!beta_detail::check_beta(
|
|
function,
|
|
b, &r, Policy())
|
|
||
|
|
!detail::check_non_centrality(
|
|
function,
|
|
l,
|
|
&r,
|
|
Policy())
|
|
||
|
|
!beta_detail::check_x(
|
|
function,
|
|
x,
|
|
&r,
|
|
Policy()))
|
|
return (RealType)r;
|
|
|
|
if(l == 0)
|
|
return cdf(complement(beta_distribution<RealType, Policy>(a, b), x));
|
|
|
|
return detail::non_central_beta_cdf(x, RealType(1 - x), a, b, l, true, Policy());
|
|
} // ccdf
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType quantile(const non_central_beta_distribution<RealType, Policy>& dist, const RealType& p)
|
|
{ // Quantile (or Percent Point) function.
|
|
return detail::nc_beta_quantile(dist, p, false);
|
|
} // quantile
|
|
|
|
template <class RealType, class Policy>
|
|
inline RealType quantile(const complemented2_type<non_central_beta_distribution<RealType, Policy>, RealType>& c)
|
|
{ // Quantile (or Percent Point) function.
|
|
return detail::nc_beta_quantile(c.dist, c.param, true);
|
|
} // 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>
|
|
|
|
#endif // BOOST_MATH_SPECIAL_NON_CENTRAL_BETA_HPP
|
|
|