635 lines
22 KiB
C++
635 lines
22 KiB
C++
// Copyright 2014 Marco Guazzone (marco.guazzone@gmail.com)
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//
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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//
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// This module implements the Hyper-Exponential distribution.
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//
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// References:
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// - "Queueing Theory in Manufacturing Systems Analysis and Design" by H.T. Papadopolous, C. Heavey and J. Browne (Chapman & Hall/CRC, 1993)
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// - http://reference.wolfram.com/language/ref/HyperexponentialDistribution.html
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// - http://en.wikipedia.org/wiki/Hyperexponential_distribution
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//
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#ifndef BOOST_MATH_DISTRIBUTIONS_HYPEREXPONENTIAL_HPP
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#define BOOST_MATH_DISTRIBUTIONS_HYPEREXPONENTIAL_HPP
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#include <boost/config.hpp>
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#include <boost/math/distributions/complement.hpp>
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#include <boost/math/distributions/detail/common_error_handling.hpp>
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#include <boost/math/distributions/exponential.hpp>
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#include <boost/math/policies/policy.hpp>
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#include <boost/math/special_functions/fpclassify.hpp>
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#include <boost/math/tools/precision.hpp>
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#include <boost/math/tools/roots.hpp>
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#include <boost/range/begin.hpp>
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#include <boost/range/end.hpp>
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#include <boost/range/size.hpp>
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#include <boost/type_traits/has_pre_increment.hpp>
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#include <cstddef>
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#include <iterator>
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#include <limits>
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#include <numeric>
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#include <utility>
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#include <vector>
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#if !defined(BOOST_NO_CXX11_HDR_INITIALIZER_LIST)
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# include <initializer_list>
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#endif
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#ifdef _MSC_VER
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# pragma warning (push)
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# pragma warning(disable:4127) // conditional expression is constant
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# pragma warning(disable:4389) // '==' : signed/unsigned mismatch in test_tools
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#endif // _MSC_VER
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namespace boost { namespace math {
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namespace detail {
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template <typename Dist>
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typename Dist::value_type generic_quantile(const Dist& dist, const typename Dist::value_type& p, const typename Dist::value_type& guess, bool comp, const char* function);
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} // Namespace detail
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template <typename RealT, typename PolicyT>
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class hyperexponential_distribution;
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namespace /*<unnamed>*/ { namespace hyperexp_detail {
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template <typename T>
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void normalize(std::vector<T>& v)
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{
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if(!v.size())
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return; // Our error handlers will get this later
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const T sum = std::accumulate(v.begin(), v.end(), static_cast<T>(0));
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T final_sum = 0;
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const typename std::vector<T>::iterator end = --v.end();
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for (typename std::vector<T>::iterator it = v.begin();
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it != end;
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++it)
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{
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*it /= sum;
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final_sum += *it;
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}
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*end = 1 - final_sum; // avoids round off errors, ensures the probs really do sum to 1.
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}
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template <typename RealT, typename PolicyT>
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bool check_probabilities(char const* function, std::vector<RealT> const& probabilities, RealT* presult, PolicyT const& pol)
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{
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BOOST_MATH_STD_USING
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const std::size_t n = probabilities.size();
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RealT sum = 0;
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for (std::size_t i = 0; i < n; ++i)
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{
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if (probabilities[i] < 0
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|| probabilities[i] > 1
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|| !(boost::math::isfinite)(probabilities[i]))
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{
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*presult = policies::raise_domain_error<RealT>(function,
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"The elements of parameter \"probabilities\" must be >= 0 and <= 1, but at least one of them was: %1%.",
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probabilities[i],
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pol);
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return false;
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}
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sum += probabilities[i];
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}
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//
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// We try to keep phase probabilities correctly normalized in the distribution constructors,
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// however in practice we have to allow for a very slight divergence from a sum of exactly 1:
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//
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if (fabs(sum - 1) > tools::epsilon<RealT>() * 2)
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{
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*presult = policies::raise_domain_error<RealT>(function,
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"The elements of parameter \"probabilities\" must sum to 1, but their sum is: %1%.",
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sum,
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pol);
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return false;
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}
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return true;
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}
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template <typename RealT, typename PolicyT>
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bool check_rates(char const* function, std::vector<RealT> const& rates, RealT* presult, PolicyT const& pol)
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{
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const std::size_t n = rates.size();
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for (std::size_t i = 0; i < n; ++i)
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{
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if (rates[i] <= 0
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|| !(boost::math::isfinite)(rates[i]))
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{
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*presult = policies::raise_domain_error<RealT>(function,
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"The elements of parameter \"rates\" must be > 0, but at least one of them is: %1%.",
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rates[i],
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pol);
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return false;
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}
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}
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return true;
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}
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template <typename RealT, typename PolicyT>
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bool check_dist(char const* function, std::vector<RealT> const& probabilities, std::vector<RealT> const& rates, RealT* presult, PolicyT const& pol)
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{
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BOOST_MATH_STD_USING
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if (probabilities.size() != rates.size())
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{
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*presult = policies::raise_domain_error<RealT>(function,
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"The parameters \"probabilities\" and \"rates\" must have the same length, but their size differ by: %1%.",
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fabs(static_cast<RealT>(probabilities.size())-static_cast<RealT>(rates.size())),
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pol);
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return false;
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}
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return check_probabilities(function, probabilities, presult, pol)
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&& check_rates(function, rates, presult, pol);
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}
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template <typename RealT, typename PolicyT>
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bool check_x(char const* function, RealT x, RealT* presult, PolicyT const& pol)
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{
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if (x < 0 || (boost::math::isnan)(x))
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{
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*presult = policies::raise_domain_error<RealT>(function, "The random variable must be >= 0, but is: %1%.", x, pol);
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return false;
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}
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return true;
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}
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template <typename RealT, typename PolicyT>
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bool check_probability(char const* function, RealT p, RealT* presult, PolicyT const& pol)
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{
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if (p < 0 || p > 1 || (boost::math::isnan)(p))
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{
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*presult = policies::raise_domain_error<RealT>(function, "The probability be >= 0 and <= 1, but is: %1%.", p, pol);
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return false;
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}
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return true;
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}
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template <typename RealT, typename PolicyT>
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RealT quantile_impl(hyperexponential_distribution<RealT, PolicyT> const& dist, RealT const& p, bool comp)
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{
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// Don't have a closed form so try to numerically solve the inverse CDF...
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typedef typename policies::evaluation<RealT, PolicyT>::type value_type;
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typedef typename policies::normalise<PolicyT,
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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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static const char* function = comp ? "boost::math::quantile(const boost::math::complemented2_type<boost::math::hyperexponential_distribution<%1%>, %1%>&)"
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: "boost::math::quantile(const boost::math::hyperexponential_distribution<%1%>&, %1%)";
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RealT result = 0;
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if (!check_probability(function, p, &result, PolicyT()))
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{
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return result;
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}
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const std::size_t n = dist.num_phases();
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const std::vector<RealT> probs = dist.probabilities();
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const std::vector<RealT> rates = dist.rates();
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// A possible (but inaccurate) approximation is given below, where the
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// quantile is given by the weighted sum of exponential quantiles:
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RealT guess = 0;
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if (comp)
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{
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for (std::size_t i = 0; i < n; ++i)
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{
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const exponential_distribution<RealT,PolicyT> exp(rates[i]);
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guess += probs[i]*quantile(complement(exp, p));
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}
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}
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else
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{
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for (std::size_t i = 0; i < n; ++i)
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{
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const exponential_distribution<RealT,PolicyT> exp(rates[i]);
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guess += probs[i]*quantile(exp, p);
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}
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}
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// Fast return in case the Hyper-Exponential is essentially an Exponential
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if (n == 1)
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{
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return guess;
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}
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value_type q;
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q = detail::generic_quantile(hyperexponential_distribution<RealT,forwarding_policy>(probs, rates),
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p,
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guess,
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comp,
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function);
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result = policies::checked_narrowing_cast<RealT,forwarding_policy>(q, function);
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return result;
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}
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}} // Namespace <unnamed>::hyperexp_detail
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template <typename RealT = double, typename PolicyT = policies::policy<> >
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class hyperexponential_distribution
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{
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public: typedef RealT value_type;
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public: typedef PolicyT policy_type;
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public: hyperexponential_distribution()
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: probs_(1, 1),
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rates_(1, 1)
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{
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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// Four arg constructor: no ambiguity here, the arguments must be two pairs of iterators:
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public: template <typename ProbIterT, typename RateIterT>
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hyperexponential_distribution(ProbIterT prob_first, ProbIterT prob_last,
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RateIterT rate_first, RateIterT rate_last)
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: probs_(prob_first, prob_last),
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rates_(rate_first, rate_last)
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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// Two arg constructor from 2 ranges, we SFINAE this out of existance if
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// either argument type is incrementable as in that case the type is
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// probably an iterator:
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public: template <typename ProbRangeT, typename RateRangeT>
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hyperexponential_distribution(ProbRangeT const& prob_range,
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RateRangeT const& rate_range,
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typename boost::disable_if_c<boost::has_pre_increment<ProbRangeT>::value || boost::has_pre_increment<RateRangeT>::value>::type* = 0)
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: probs_(boost::begin(prob_range), boost::end(prob_range)),
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rates_(boost::begin(rate_range), boost::end(rate_range))
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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// Two arg constructor for a pair of iterators: we SFINAE this out of
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// existance if neither argument types are incrementable.
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// Note that we allow different argument types here to allow for
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// construction from an array plus a pointer into that array.
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public: template <typename RateIterT, typename RateIterT2>
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hyperexponential_distribution(RateIterT const& rate_first,
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RateIterT2 const& rate_last,
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typename boost::enable_if_c<boost::has_pre_increment<RateIterT>::value || boost::has_pre_increment<RateIterT2>::value>::type* = 0)
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: probs_(std::distance(rate_first, rate_last), 1), // will be normalized below
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rates_(rate_first, rate_last)
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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#if !defined(BOOST_NO_CXX11_HDR_INITIALIZER_LIST)
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// Initializer list constructor: allows for construction from array literals:
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public: hyperexponential_distribution(std::initializer_list<RealT> l1, std::initializer_list<RealT> l2)
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: probs_(l1.begin(), l1.end()),
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rates_(l2.begin(), l2.end())
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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public: hyperexponential_distribution(std::initializer_list<RealT> l1)
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: probs_(l1.size(), 1),
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rates_(l1.begin(), l1.end())
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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#endif // !defined(BOOST_NO_CXX11_HDR_INITIALIZER_LIST)
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// Single argument constructor: argument must be a range.
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public: template <typename RateRangeT>
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hyperexponential_distribution(RateRangeT const& rate_range)
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: probs_(boost::size(rate_range), 1), // will be normalized below
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rates_(boost::begin(rate_range), boost::end(rate_range))
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{
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hyperexp_detail::normalize(probs_);
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RealT err;
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hyperexp_detail::check_dist("boost::math::hyperexponential_distribution<%1%>::hyperexponential_distribution",
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probs_,
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rates_,
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&err,
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PolicyT());
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}
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public: std::vector<RealT> probabilities() const
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{
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return probs_;
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}
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public: std::vector<RealT> rates() const
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{
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return rates_;
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}
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public: std::size_t num_phases() const
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{
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return rates_.size();
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}
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private: std::vector<RealT> probs_;
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private: std::vector<RealT> rates_;
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}; // class hyperexponential_distribution
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// Convenient type synonym for double.
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typedef hyperexponential_distribution<double> hyperexponential;
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// Range of permissible values for random variable x
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template <typename RealT, typename PolicyT>
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std::pair<RealT,RealT> range(hyperexponential_distribution<RealT,PolicyT> const&)
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{
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if (std::numeric_limits<RealT>::has_infinity)
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{
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return std::make_pair(static_cast<RealT>(0), std::numeric_limits<RealT>::infinity()); // 0 to +inf.
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}
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return std::make_pair(static_cast<RealT>(0), tools::max_value<RealT>()); // 0 to +<max value>
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}
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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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template <typename RealT, typename PolicyT>
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std::pair<RealT,RealT> support(hyperexponential_distribution<RealT,PolicyT> const&)
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{
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return std::make_pair(tools::min_value<RealT>(), tools::max_value<RealT>()); // <min value> to +<max value>.
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}
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template <typename RealT, typename PolicyT>
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RealT pdf(hyperexponential_distribution<RealT, PolicyT> const& dist, RealT const& x)
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{
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BOOST_MATH_STD_USING
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RealT result = 0;
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if (!hyperexp_detail::check_x("boost::math::pdf(const boost::math::hyperexponential_distribution<%1%>&, %1%)", x, &result, PolicyT()))
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{
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return result;
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}
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const std::size_t n = dist.num_phases();
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const std::vector<RealT> probs = dist.probabilities();
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const std::vector<RealT> rates = dist.rates();
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for (std::size_t i = 0; i < n; ++i)
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{
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const exponential_distribution<RealT,PolicyT> exp(rates[i]);
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result += probs[i]*pdf(exp, x);
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//result += probs[i]*rates[i]*exp(-rates[i]*x);
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}
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return result;
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}
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template <typename RealT, typename PolicyT>
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RealT cdf(hyperexponential_distribution<RealT, PolicyT> const& dist, RealT const& x)
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{
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RealT result = 0;
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if (!hyperexp_detail::check_x("boost::math::cdf(const boost::math::hyperexponential_distribution<%1%>&, %1%)", x, &result, PolicyT()))
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{
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return result;
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}
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const std::size_t n = dist.num_phases();
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const std::vector<RealT> probs = dist.probabilities();
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const std::vector<RealT> rates = dist.rates();
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for (std::size_t i = 0; i < n; ++i)
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{
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const exponential_distribution<RealT,PolicyT> exp(rates[i]);
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result += probs[i]*cdf(exp, x);
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}
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return result;
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}
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template <typename RealT, typename PolicyT>
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RealT quantile(hyperexponential_distribution<RealT, PolicyT> const& dist, RealT const& p)
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{
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return hyperexp_detail::quantile_impl(dist, p , false);
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}
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template <typename RealT, typename PolicyT>
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RealT cdf(complemented2_type<hyperexponential_distribution<RealT,PolicyT>, RealT> const& c)
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{
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RealT const& x = c.param;
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hyperexponential_distribution<RealT,PolicyT> const& dist = c.dist;
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RealT result = 0;
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if (!hyperexp_detail::check_x("boost::math::cdf(boost::math::complemented2_type<const boost::math::hyperexponential_distribution<%1%>&, %1%>)", x, &result, PolicyT()))
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{
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return result;
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}
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const std::size_t n = dist.num_phases();
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const std::vector<RealT> probs = dist.probabilities();
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const std::vector<RealT> rates = dist.rates();
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for (std::size_t i = 0; i < n; ++i)
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{
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const exponential_distribution<RealT,PolicyT> exp(rates[i]);
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result += probs[i]*cdf(complement(exp, x));
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}
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return result;
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}
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|
|
|
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template <typename RealT, typename PolicyT>
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RealT quantile(complemented2_type<hyperexponential_distribution<RealT, PolicyT>, RealT> const& c)
|
|
{
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RealT const& p = c.param;
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hyperexponential_distribution<RealT,PolicyT> const& dist = c.dist;
|
|
|
|
return hyperexp_detail::quantile_impl(dist, p , true);
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}
|
|
|
|
template <typename RealT, typename PolicyT>
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|
RealT mean(hyperexponential_distribution<RealT, PolicyT> const& dist)
|
|
{
|
|
RealT result = 0;
|
|
|
|
const std::size_t n = dist.num_phases();
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|
const std::vector<RealT> probs = dist.probabilities();
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const std::vector<RealT> rates = dist.rates();
|
|
|
|
for (std::size_t i = 0; i < n; ++i)
|
|
{
|
|
const exponential_distribution<RealT,PolicyT> exp(rates[i]);
|
|
|
|
result += probs[i]*mean(exp);
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|
}
|
|
|
|
return result;
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|
}
|
|
|
|
template <typename RealT, typename PolicyT>
|
|
RealT variance(hyperexponential_distribution<RealT, PolicyT> const& dist)
|
|
{
|
|
RealT result = 0;
|
|
|
|
const std::size_t n = dist.num_phases();
|
|
const std::vector<RealT> probs = dist.probabilities();
|
|
const std::vector<RealT> rates = dist.rates();
|
|
|
|
for (std::size_t i = 0; i < n; ++i)
|
|
{
|
|
result += probs[i]/(rates[i]*rates[i]);
|
|
}
|
|
|
|
const RealT mean = boost::math::mean(dist);
|
|
|
|
result = 2*result-mean*mean;
|
|
|
|
return result;
|
|
}
|
|
|
|
template <typename RealT, typename PolicyT>
|
|
RealT skewness(hyperexponential_distribution<RealT,PolicyT> const& dist)
|
|
{
|
|
BOOST_MATH_STD_USING
|
|
const std::size_t n = dist.num_phases();
|
|
const std::vector<RealT> probs = dist.probabilities();
|
|
const std::vector<RealT> rates = dist.rates();
|
|
|
|
RealT s1 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i}
|
|
RealT s2 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i^2}
|
|
RealT s3 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i^3}
|
|
for (std::size_t i = 0; i < n; ++i)
|
|
{
|
|
const RealT p = probs[i];
|
|
const RealT r = rates[i];
|
|
const RealT r2 = r*r;
|
|
const RealT r3 = r2*r;
|
|
|
|
s1 += p/r;
|
|
s2 += p/r2;
|
|
s3 += p/r3;
|
|
}
|
|
|
|
const RealT s1s1 = s1*s1;
|
|
|
|
const RealT num = (6*s3 - (3*(2*s2 - s1s1) + s1s1)*s1);
|
|
const RealT den = (2*s2 - s1s1);
|
|
|
|
return num / pow(den, static_cast<RealT>(1.5));
|
|
}
|
|
|
|
template <typename RealT, typename PolicyT>
|
|
RealT kurtosis(hyperexponential_distribution<RealT,PolicyT> const& dist)
|
|
{
|
|
const std::size_t n = dist.num_phases();
|
|
const std::vector<RealT> probs = dist.probabilities();
|
|
const std::vector<RealT> rates = dist.rates();
|
|
|
|
RealT s1 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i}
|
|
RealT s2 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i^2}
|
|
RealT s3 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i^3}
|
|
RealT s4 = 0; // \sum_{i=1}^n \frac{p_i}{\lambda_i^4}
|
|
for (std::size_t i = 0; i < n; ++i)
|
|
{
|
|
const RealT p = probs[i];
|
|
const RealT r = rates[i];
|
|
const RealT r2 = r*r;
|
|
const RealT r3 = r2*r;
|
|
const RealT r4 = r3*r;
|
|
|
|
s1 += p/r;
|
|
s2 += p/r2;
|
|
s3 += p/r3;
|
|
s4 += p/r4;
|
|
}
|
|
|
|
const RealT s1s1 = s1*s1;
|
|
|
|
const RealT num = (24*s4 - 24*s3*s1 + 3*(2*(2*s2 - s1s1) + s1s1)*s1s1);
|
|
const RealT den = (2*s2 - s1s1);
|
|
|
|
return num/(den*den);
|
|
}
|
|
|
|
template <typename RealT, typename PolicyT>
|
|
RealT kurtosis_excess(hyperexponential_distribution<RealT,PolicyT> const& dist)
|
|
{
|
|
return kurtosis(dist) - 3;
|
|
}
|
|
|
|
template <typename RealT, typename PolicyT>
|
|
RealT mode(hyperexponential_distribution<RealT,PolicyT> const& /*dist*/)
|
|
{
|
|
return 0;
|
|
}
|
|
|
|
}} // namespace boost::math
|
|
|
|
#ifdef BOOST_MSVC
|
|
#pragma warning (pop)
|
|
#endif
|
|
// 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>
|
|
#include <boost/math/distributions/detail/generic_quantile.hpp>
|
|
|
|
#endif // BOOST_MATH_DISTRIBUTIONS_HYPEREXPONENTIAL
|