720 lines
26 KiB
Plaintext
720 lines
26 KiB
Plaintext
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// Copyright Benjamin Sobotta 2012
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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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#ifndef BOOST_STATS_SKEW_NORMAL_HPP
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#define BOOST_STATS_SKEW_NORMAL_HPP
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// http://en.wikipedia.org/wiki/Skew_normal_distribution
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// http://azzalini.stat.unipd.it/SN/
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// Also:
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// Azzalini, A. (1985). "A class of distributions which includes the normal ones".
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// Scand. J. Statist. 12: 171-178.
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#include <boost/math/distributions/fwd.hpp> // TODO add skew_normal distribution to fwd.hpp!
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#include <boost/math/special_functions/owens_t.hpp> // Owen's T function
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#include <boost/math/distributions/complement.hpp>
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#include <boost/math/distributions/normal.hpp>
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#include <boost/math/distributions/detail/common_error_handling.hpp>
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/tools/tuple.hpp>
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#include <boost/math/tools/roots.hpp> // Newton-Raphson
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#include <boost/assert.hpp>
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#include <boost/math/distributions/detail/generic_mode.hpp> // pdf max finder.
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#include <utility>
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#include <algorithm> // std::lower_bound, std::distance
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namespace boost{ namespace math{
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namespace detail
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{
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template <class RealType, class Policy>
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inline bool check_skew_normal_shape(
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const char* function,
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RealType shape,
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RealType* result,
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const Policy& pol)
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{
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if(!(boost::math::isfinite)(shape))
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{
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*result =
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policies::raise_domain_error<RealType>(function,
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"Shape parameter is %1%, but must be finite!",
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shape, 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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} // namespace detail
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template <class RealType = double, class Policy = policies::policy<> >
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class skew_normal_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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skew_normal_distribution(RealType l_location = 0, RealType l_scale = 1, RealType l_shape = 0)
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: location_(l_location), scale_(l_scale), shape_(l_shape)
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{ // Default is a 'standard' normal distribution N01. (shape=0 results in the normal distribution with no skew)
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static const char* function = "boost::math::skew_normal_distribution<%1%>::skew_normal_distribution";
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RealType result;
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detail::check_scale(function, l_scale, &result, Policy());
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detail::check_location(function, l_location, &result, Policy());
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detail::check_skew_normal_shape(function, l_shape, &result, Policy());
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}
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RealType location()const
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{
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return location_;
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}
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RealType scale()const
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{
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return scale_;
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}
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RealType shape()const
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{
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return shape_;
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}
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private:
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//
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// Data members:
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//
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RealType location_; // distribution location.
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RealType scale_; // distribution scale.
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RealType shape_; // distribution shape.
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}; // class skew_normal_distribution
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typedef skew_normal_distribution<double> skew_normal;
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const skew_normal_distribution<RealType, Policy>& /*dist*/)
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{ // Range of permissible values for random variable x.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(
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std::numeric_limits<RealType>::has_infinity ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>(),
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std::numeric_limits<RealType>::has_infinity ? std::numeric_limits<RealType>::infinity() : max_value<RealType>()); // - to + max value.
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}
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> support(const skew_normal_distribution<RealType, Policy>& /*dist*/)
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{ // Range of supported values for random variable x.
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// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>()); // - to + max value.
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}
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template <class RealType, class Policy>
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inline RealType pdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
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{
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const RealType scale = dist.scale();
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const RealType location = dist.location();
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const RealType shape = dist.shape();
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static const char* function = "boost::math::pdf(const skew_normal_distribution<%1%>&, %1%)";
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RealType result = 0;
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if(false == detail::check_scale(function, scale, &result, Policy()))
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{
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return result;
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}
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if(false == detail::check_location(function, location, &result, Policy()))
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{
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return result;
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}
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if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
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{
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return result;
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}
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if((boost::math::isinf)(x))
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{
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return 0; // pdf + and - infinity is zero.
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}
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// Below produces MSVC 4127 warnings, so the above used instead.
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//if(std::numeric_limits<RealType>::has_infinity && abs(x) == std::numeric_limits<RealType>::infinity())
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//{ // pdf + and - infinity is zero.
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// return 0;
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//}
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if(false == detail::check_x(function, x, &result, Policy()))
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{
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return result;
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}
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const RealType transformed_x = (x-location)/scale;
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normal_distribution<RealType, Policy> std_normal;
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result = pdf(std_normal, transformed_x) * cdf(std_normal, shape*transformed_x) * 2 / scale;
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return result;
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} // pdf
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template <class RealType, class Policy>
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inline RealType cdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
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{
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const RealType scale = dist.scale();
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const RealType location = dist.location();
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const RealType shape = dist.shape();
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static const char* function = "boost::math::cdf(const skew_normal_distribution<%1%>&, %1%)";
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RealType result = 0;
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if(false == detail::check_scale(function, scale, &result, Policy()))
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{
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return result;
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}
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if(false == detail::check_location(function, location, &result, Policy()))
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{
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return result;
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}
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if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
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{
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return result;
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}
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if((boost::math::isinf)(x))
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{
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if(x < 0) return 0; // -infinity
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return 1; // + infinity
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}
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// These produce MSVC 4127 warnings, so the above used instead.
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//if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
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//{ // cdf +infinity is unity.
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// return 1;
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//}
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//if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
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//{ // cdf -infinity is zero.
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// return 0;
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//}
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if(false == detail::check_x(function, x, &result, Policy()))
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{
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return result;
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}
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const RealType transformed_x = (x-location)/scale;
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normal_distribution<RealType, Policy> std_normal;
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result = cdf(std_normal, transformed_x) - owens_t(transformed_x, shape)*static_cast<RealType>(2);
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return result;
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} // cdf
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template <class RealType, class Policy>
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inline RealType cdf(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
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{
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const RealType scale = c.dist.scale();
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const RealType location = c.dist.location();
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const RealType shape = c.dist.shape();
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const RealType x = c.param;
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static const char* function = "boost::math::cdf(const complement(skew_normal_distribution<%1%>&), %1%)";
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if((boost::math::isinf)(x))
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{
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if(x < 0) return 1; // cdf complement -infinity is unity.
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return 0; // cdf complement +infinity is zero
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}
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// These produce MSVC 4127 warnings, so the above used instead.
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//if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
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//{ // cdf complement +infinity is zero.
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// return 0;
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//}
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//if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
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//{ // cdf complement -infinity is unity.
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// return 1;
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//}
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RealType result = 0;
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if(false == detail::check_scale(function, scale, &result, Policy()))
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return result;
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if(false == detail::check_location(function, location, &result, Policy()))
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return result;
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if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
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return result;
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if(false == detail::check_x(function, x, &result, Policy()))
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return result;
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const RealType transformed_x = (x-location)/scale;
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normal_distribution<RealType, Policy> std_normal;
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result = cdf(complement(std_normal, transformed_x)) + owens_t(transformed_x, shape)*static_cast<RealType>(2);
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return result;
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} // cdf complement
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template <class RealType, class Policy>
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inline RealType location(const skew_normal_distribution<RealType, Policy>& dist)
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{
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return dist.location();
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}
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template <class RealType, class Policy>
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inline RealType scale(const skew_normal_distribution<RealType, Policy>& dist)
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{
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return dist.scale();
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}
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template <class RealType, class Policy>
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inline RealType shape(const skew_normal_distribution<RealType, Policy>& dist)
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{
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return dist.shape();
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}
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template <class RealType, class Policy>
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inline RealType mean(const skew_normal_distribution<RealType, Policy>& dist)
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{
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BOOST_MATH_STD_USING // for ADL of std functions
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using namespace boost::math::constants;
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//const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());
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//return dist.location() + dist.scale() * delta * root_two_div_pi<RealType>();
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return dist.location() + dist.scale() * dist.shape() / sqrt(pi<RealType>()+pi<RealType>()*dist.shape()*dist.shape()) * root_two<RealType>();
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}
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template <class RealType, class Policy>
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inline RealType variance(const skew_normal_distribution<RealType, Policy>& dist)
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{
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using namespace boost::math::constants;
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const RealType delta2 = static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape()));
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//const RealType inv_delta2 = static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape());
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RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()*delta2);
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//RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()/inv_delta2);
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return variance;
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}
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namespace detail
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{
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/*
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TODO No closed expression for mode, so use max of pdf.
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*/
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template <class RealType, class Policy>
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inline RealType mode_fallback(const skew_normal_distribution<RealType, Policy>& dist)
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{ // mode.
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static const char* function = "mode(skew_normal_distribution<%1%> const&)";
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const RealType scale = dist.scale();
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const RealType location = dist.location();
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const RealType shape = dist.shape();
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RealType result;
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if(!detail::check_scale(
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function,
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scale, &result, Policy())
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!detail::check_skew_normal_shape(
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function,
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shape,
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&result,
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Policy()))
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return result;
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if( shape == 0 )
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{
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return location;
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}
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if( shape < 0 )
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{
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skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
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result = mode_fallback(D);
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result = location-scale*result;
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return result;
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}
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BOOST_MATH_STD_USING
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// 21 elements
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static const RealType shapes[] = {
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0.0,
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1.000000000000000e-004,
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2.069138081114790e-004,
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4.281332398719396e-004,
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8.858667904100824e-004,
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1.832980710832436e-003,
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3.792690190732250e-003,
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7.847599703514606e-003,
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1.623776739188722e-002,
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3.359818286283781e-002,
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6.951927961775606e-002,
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1.438449888287663e-001,
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2.976351441631319e-001,
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6.158482110660261e-001,
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1.274274985703135e+000,
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2.636650898730361e+000,
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5.455594781168514e+000,
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1.128837891684688e+001,
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2.335721469090121e+001,
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4.832930238571753e+001,
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1.000000000000000e+002};
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// 21 elements
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static const RealType guess[] = {
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|
0.0,
|
||
|
|
5.000050000525391e-005,
|
||
|
|
1.500015000148736e-004,
|
||
|
|
3.500035000350010e-004,
|
||
|
|
7.500075000752560e-004,
|
||
|
|
1.450014500145258e-003,
|
||
|
|
3.050030500305390e-003,
|
||
|
|
6.250062500624765e-003,
|
||
|
|
1.295012950129504e-002,
|
||
|
|
2.675026750267495e-002,
|
||
|
|
5.525055250552491e-002,
|
||
|
|
1.132511325113255e-001,
|
||
|
|
2.249522495224952e-001,
|
||
|
|
3.992539925399257e-001,
|
||
|
|
5.353553535535358e-001,
|
||
|
|
4.954549545495457e-001,
|
||
|
|
3.524535245352451e-001,
|
||
|
|
2.182521825218249e-001,
|
||
|
|
1.256512565125654e-001,
|
||
|
|
6.945069450694508e-002,
|
||
|
|
3.735037350373460e-002
|
||
|
|
};
|
||
|
|
|
||
|
|
const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);
|
||
|
|
|
||
|
|
typedef typename std::iterator_traits<RealType*>::difference_type diff_type;
|
||
|
|
|
||
|
|
const diff_type d = std::distance(shapes, result_ptr);
|
||
|
|
|
||
|
|
BOOST_ASSERT(d > static_cast<diff_type>(0));
|
||
|
|
|
||
|
|
// refine
|
||
|
|
if(d < static_cast<diff_type>(21)) // shape smaller 100
|
||
|
|
{
|
||
|
|
result = guess[d-static_cast<diff_type>(1)]
|
||
|
|
+ (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
|
||
|
|
* (shape-shapes[d-static_cast<diff_type>(1)]);
|
||
|
|
}
|
||
|
|
else // shape greater 100
|
||
|
|
{
|
||
|
|
result = 1e-4;
|
||
|
|
}
|
||
|
|
|
||
|
|
skew_normal_distribution<RealType, Policy> helper(0, 1, shape);
|
||
|
|
|
||
|
|
result = detail::generic_find_mode_01(helper, result, function);
|
||
|
|
|
||
|
|
result = result*scale + location;
|
||
|
|
|
||
|
|
return result;
|
||
|
|
} // mode_fallback
|
||
|
|
|
||
|
|
|
||
|
|
/*
|
||
|
|
* TODO No closed expression for mode, so use f'(x) = 0
|
||
|
|
*/
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
struct skew_normal_mode_functor
|
||
|
|
{
|
||
|
|
skew_normal_mode_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist)
|
||
|
|
: distribution(dist)
|
||
|
|
{
|
||
|
|
}
|
||
|
|
|
||
|
|
boost::math::tuple<RealType, RealType> operator()(RealType const& x)
|
||
|
|
{
|
||
|
|
normal_distribution<RealType, Policy> std_normal;
|
||
|
|
const RealType shape = distribution.shape();
|
||
|
|
const RealType pdf_x = pdf(distribution, x);
|
||
|
|
const RealType normpdf_x = pdf(std_normal, x);
|
||
|
|
const RealType normpdf_ax = pdf(std_normal, x*shape);
|
||
|
|
RealType fx = static_cast<RealType>(2)*shape*normpdf_ax*normpdf_x - x*pdf_x;
|
||
|
|
RealType dx = static_cast<RealType>(2)*shape*x*normpdf_x*normpdf_ax*(static_cast<RealType>(1) + shape*shape) + pdf_x + x*fx;
|
||
|
|
// return both function evaluation difference f(x) and 1st derivative f'(x).
|
||
|
|
return boost::math::make_tuple(fx, -dx);
|
||
|
|
}
|
||
|
|
private:
|
||
|
|
const boost::math::skew_normal_distribution<RealType, Policy> distribution;
|
||
|
|
};
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType mode(const skew_normal_distribution<RealType, Policy>& dist)
|
||
|
|
{
|
||
|
|
const RealType scale = dist.scale();
|
||
|
|
const RealType location = dist.location();
|
||
|
|
const RealType shape = dist.shape();
|
||
|
|
|
||
|
|
static const char* function = "boost::math::mode(const skew_normal_distribution<%1%>&, %1%)";
|
||
|
|
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == detail::check_scale(function, scale, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_location(function, location, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
|
||
|
|
if( shape == 0 )
|
||
|
|
{
|
||
|
|
return location;
|
||
|
|
}
|
||
|
|
|
||
|
|
if( shape < 0 )
|
||
|
|
{
|
||
|
|
skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
|
||
|
|
result = mode(D);
|
||
|
|
result = location-scale*result;
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
// 21 elements
|
||
|
|
static const RealType shapes[] = {
|
||
|
|
0.0,
|
||
|
|
static_cast<RealType>(1.000000000000000e-004),
|
||
|
|
static_cast<RealType>(2.069138081114790e-004),
|
||
|
|
static_cast<RealType>(4.281332398719396e-004),
|
||
|
|
static_cast<RealType>(8.858667904100824e-004),
|
||
|
|
static_cast<RealType>(1.832980710832436e-003),
|
||
|
|
static_cast<RealType>(3.792690190732250e-003),
|
||
|
|
static_cast<RealType>(7.847599703514606e-003),
|
||
|
|
static_cast<RealType>(1.623776739188722e-002),
|
||
|
|
static_cast<RealType>(3.359818286283781e-002),
|
||
|
|
static_cast<RealType>(6.951927961775606e-002),
|
||
|
|
static_cast<RealType>(1.438449888287663e-001),
|
||
|
|
static_cast<RealType>(2.976351441631319e-001),
|
||
|
|
static_cast<RealType>(6.158482110660261e-001),
|
||
|
|
static_cast<RealType>(1.274274985703135e+000),
|
||
|
|
static_cast<RealType>(2.636650898730361e+000),
|
||
|
|
static_cast<RealType>(5.455594781168514e+000),
|
||
|
|
static_cast<RealType>(1.128837891684688e+001),
|
||
|
|
static_cast<RealType>(2.335721469090121e+001),
|
||
|
|
static_cast<RealType>(4.832930238571753e+001),
|
||
|
|
static_cast<RealType>(1.000000000000000e+002)
|
||
|
|
};
|
||
|
|
|
||
|
|
// 21 elements
|
||
|
|
static const RealType guess[] = {
|
||
|
|
0.0,
|
||
|
|
static_cast<RealType>(5.000050000525391e-005),
|
||
|
|
static_cast<RealType>(1.500015000148736e-004),
|
||
|
|
static_cast<RealType>(3.500035000350010e-004),
|
||
|
|
static_cast<RealType>(7.500075000752560e-004),
|
||
|
|
static_cast<RealType>(1.450014500145258e-003),
|
||
|
|
static_cast<RealType>(3.050030500305390e-003),
|
||
|
|
static_cast<RealType>(6.250062500624765e-003),
|
||
|
|
static_cast<RealType>(1.295012950129504e-002),
|
||
|
|
static_cast<RealType>(2.675026750267495e-002),
|
||
|
|
static_cast<RealType>(5.525055250552491e-002),
|
||
|
|
static_cast<RealType>(1.132511325113255e-001),
|
||
|
|
static_cast<RealType>(2.249522495224952e-001),
|
||
|
|
static_cast<RealType>(3.992539925399257e-001),
|
||
|
|
static_cast<RealType>(5.353553535535358e-001),
|
||
|
|
static_cast<RealType>(4.954549545495457e-001),
|
||
|
|
static_cast<RealType>(3.524535245352451e-001),
|
||
|
|
static_cast<RealType>(2.182521825218249e-001),
|
||
|
|
static_cast<RealType>(1.256512565125654e-001),
|
||
|
|
static_cast<RealType>(6.945069450694508e-002),
|
||
|
|
static_cast<RealType>(3.735037350373460e-002)
|
||
|
|
};
|
||
|
|
|
||
|
|
const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);
|
||
|
|
|
||
|
|
typedef typename std::iterator_traits<RealType*>::difference_type diff_type;
|
||
|
|
|
||
|
|
const diff_type d = std::distance(shapes, result_ptr);
|
||
|
|
|
||
|
|
BOOST_ASSERT(d > static_cast<diff_type>(0));
|
||
|
|
|
||
|
|
// TODO: make the search bounds smarter, depending on the shape parameter
|
||
|
|
RealType search_min = 0; // below zero was caught above
|
||
|
|
RealType search_max = 0.55f; // will never go above 0.55
|
||
|
|
|
||
|
|
// refine
|
||
|
|
if(d < static_cast<diff_type>(21)) // shape smaller 100
|
||
|
|
{
|
||
|
|
// it is safe to assume that d > 0, because shape==0.0 is caught earlier
|
||
|
|
result = guess[d-static_cast<diff_type>(1)]
|
||
|
|
+ (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
|
||
|
|
* (shape-shapes[d-static_cast<diff_type>(1)]);
|
||
|
|
}
|
||
|
|
else // shape greater 100
|
||
|
|
{
|
||
|
|
result = 1e-4f;
|
||
|
|
search_max = guess[19]; // set 19 instead of 20 to have a safety margin because the table may not be exact @ shape=100
|
||
|
|
}
|
||
|
|
|
||
|
|
const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
|
||
|
|
boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
|
||
|
|
|
||
|
|
skew_normal_distribution<RealType, Policy> helper(0, 1, shape);
|
||
|
|
|
||
|
|
result = tools::newton_raphson_iterate(detail::skew_normal_mode_functor<RealType, Policy>(helper), result,
|
||
|
|
search_min, search_max, get_digits, m);
|
||
|
|
|
||
|
|
result = result*scale + location;
|
||
|
|
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType skewness(const skew_normal_distribution<RealType, Policy>& dist)
|
||
|
|
{
|
||
|
|
BOOST_MATH_STD_USING // for ADL of std functions
|
||
|
|
using namespace boost::math::constants;
|
||
|
|
|
||
|
|
static const RealType factor = four_minus_pi<RealType>()/static_cast<RealType>(2);
|
||
|
|
const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());
|
||
|
|
|
||
|
|
return factor * pow(root_two_div_pi<RealType>() * delta, 3) /
|
||
|
|
pow(static_cast<RealType>(1)-two_div_pi<RealType>()*delta*delta, static_cast<RealType>(1.5));
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType kurtosis(const skew_normal_distribution<RealType, Policy>& dist)
|
||
|
|
{
|
||
|
|
return kurtosis_excess(dist)+static_cast<RealType>(3);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType kurtosis_excess(const skew_normal_distribution<RealType, Policy>& dist)
|
||
|
|
{
|
||
|
|
using namespace boost::math::constants;
|
||
|
|
|
||
|
|
static const RealType factor = pi_minus_three<RealType>()*static_cast<RealType>(2);
|
||
|
|
|
||
|
|
const RealType delta2 = static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape()));
|
||
|
|
|
||
|
|
const RealType x = static_cast<RealType>(1)-two_div_pi<RealType>()*delta2;
|
||
|
|
const RealType y = two_div_pi<RealType>() * delta2;
|
||
|
|
|
||
|
|
return factor * y*y / (x*x);
|
||
|
|
}
|
||
|
|
|
||
|
|
namespace detail
|
||
|
|
{
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
struct skew_normal_quantile_functor
|
||
|
|
{
|
||
|
|
skew_normal_quantile_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist, RealType const& p)
|
||
|
|
: distribution(dist), prob(p)
|
||
|
|
{
|
||
|
|
}
|
||
|
|
|
||
|
|
boost::math::tuple<RealType, RealType> operator()(RealType const& x)
|
||
|
|
{
|
||
|
|
RealType c = cdf(distribution, x);
|
||
|
|
RealType fx = c - prob; // Difference cdf - value - to minimize.
|
||
|
|
RealType dx = pdf(distribution, x); // pdf is 1st derivative.
|
||
|
|
// return both function evaluation difference f(x) and 1st derivative f'(x).
|
||
|
|
return boost::math::make_tuple(fx, dx);
|
||
|
|
}
|
||
|
|
private:
|
||
|
|
const boost::math::skew_normal_distribution<RealType, Policy> distribution;
|
||
|
|
RealType prob;
|
||
|
|
};
|
||
|
|
|
||
|
|
} // namespace detail
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const skew_normal_distribution<RealType, Policy>& dist, const RealType& p)
|
||
|
|
{
|
||
|
|
const RealType scale = dist.scale();
|
||
|
|
const RealType location = dist.location();
|
||
|
|
const RealType shape = dist.shape();
|
||
|
|
|
||
|
|
static const char* function = "boost::math::quantile(const skew_normal_distribution<%1%>&, %1%)";
|
||
|
|
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == detail::check_scale(function, scale, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_location(function, location, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_probability(function, p, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
|
||
|
|
// Compute initial guess via Cornish-Fisher expansion.
|
||
|
|
RealType x = -boost::math::erfc_inv(2 * p, Policy()) * constants::root_two<RealType>();
|
||
|
|
|
||
|
|
// Avoid unnecessary computations if there is no skew.
|
||
|
|
if(shape != 0)
|
||
|
|
{
|
||
|
|
const RealType skew = skewness(dist);
|
||
|
|
const RealType exk = kurtosis_excess(dist);
|
||
|
|
|
||
|
|
x = x + (x*x-static_cast<RealType>(1))*skew/static_cast<RealType>(6)
|
||
|
|
+ x*(x*x-static_cast<RealType>(3))*exk/static_cast<RealType>(24)
|
||
|
|
- x*(static_cast<RealType>(2)*x*x-static_cast<RealType>(5))*skew*skew/static_cast<RealType>(36);
|
||
|
|
} // if(shape != 0)
|
||
|
|
|
||
|
|
result = standard_deviation(dist)*x+mean(dist);
|
||
|
|
|
||
|
|
// handle special case of non-skew normal distribution.
|
||
|
|
if(shape == 0)
|
||
|
|
return result;
|
||
|
|
|
||
|
|
// refine the result by numerically searching the root of (p-cdf)
|
||
|
|
|
||
|
|
const RealType search_min = range(dist).first;
|
||
|
|
const RealType search_max = range(dist).second;
|
||
|
|
|
||
|
|
const int get_digits = policies::digits<RealType, Policy>();// get digits from policy,
|
||
|
|
boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
|
||
|
|
|
||
|
|
result = tools::newton_raphson_iterate(detail::skew_normal_quantile_functor<RealType, Policy>(dist, p), result,
|
||
|
|
search_min, search_max, get_digits, m);
|
||
|
|
|
||
|
|
return result;
|
||
|
|
} // quantile
|
||
|
|
|
||
|
|
template <class RealType, class Policy>
|
||
|
|
inline RealType quantile(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
|
||
|
|
{
|
||
|
|
const RealType scale = c.dist.scale();
|
||
|
|
const RealType location = c.dist.location();
|
||
|
|
const RealType shape = c.dist.shape();
|
||
|
|
|
||
|
|
static const char* function = "boost::math::quantile(const complement(skew_normal_distribution<%1%>&), %1%)";
|
||
|
|
RealType result = 0;
|
||
|
|
if(false == detail::check_scale(function, scale, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_location(function, location, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
RealType q = c.param;
|
||
|
|
if(false == detail::check_probability(function, q, &result, Policy()))
|
||
|
|
return result;
|
||
|
|
|
||
|
|
skew_normal_distribution<RealType, Policy> D(-location, scale, -shape);
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result = -quantile(D, q);
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return result;
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} // quantile
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} // namespace math
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} // namespace boost
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// This include must be at the end, *after* the accessors
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// for this distribution have been defined, in order to
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// keep compilers that support two-phase lookup happy.
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#include <boost/math/distributions/detail/derived_accessors.hpp>
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#endif // BOOST_STATS_SKEW_NORMAL_HPP
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