Chris@16: //  Copyright Benjamin Sobotta 2012
Chris@16: 
Chris@16: //  Use, modification and distribution are subject to the
Chris@16: //  Boost Software License, Version 1.0. (See accompanying file
Chris@16: //  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: #ifndef BOOST_STATS_SKEW_NORMAL_HPP
Chris@16: #define BOOST_STATS_SKEW_NORMAL_HPP
Chris@16: 
Chris@16: // http://en.wikipedia.org/wiki/Skew_normal_distribution
Chris@16: // http://azzalini.stat.unipd.it/SN/
Chris@16: // Also:
Chris@16: // Azzalini, A. (1985). "A class of distributions which includes the normal ones".
Chris@16: // Scand. J. Statist. 12: 171-178.
Chris@16: 
Chris@16: #include <boost/math/distributions/fwd.hpp> // TODO add skew_normal distribution to fwd.hpp!
Chris@16: #include <boost/math/special_functions/owens_t.hpp> // Owen's T function
Chris@16: #include <boost/math/distributions/complement.hpp>
Chris@16: #include <boost/math/distributions/normal.hpp>
Chris@16: #include <boost/math/distributions/detail/common_error_handling.hpp>
Chris@16: #include <boost/math/constants/constants.hpp>
Chris@16: #include <boost/math/tools/tuple.hpp>
Chris@16: #include <boost/math/tools/roots.hpp> // Newton-Raphson
Chris@16: #include <boost/assert.hpp>
Chris@16: #include <boost/math/distributions/detail/generic_mode.hpp> // pdf max finder.
Chris@16: 
Chris@16: #include <utility>
Chris@16: #include <algorithm> // std::lower_bound, std::distance
Chris@16: 
Chris@16: namespace boost{ namespace math{
Chris@16: 
Chris@16:   namespace detail
Chris@16:   {
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline bool check_skew_normal_shape(
Chris@16:       const char* function,
Chris@16:       RealType shape,
Chris@16:       RealType* result,
Chris@16:       const Policy& pol)
Chris@16:     {
Chris@16:       if(!(boost::math::isfinite)(shape))
Chris@16:       {
Chris@16:         *result =
Chris@16:           policies::raise_domain_error<RealType>(function,
Chris@16:           "Shape parameter is %1%, but must be finite!",
Chris@16:           shape, pol);
Chris@16:         return false;
Chris@16:       }
Chris@16:       return true;
Chris@16:     }
Chris@16: 
Chris@16:   } // namespace detail
Chris@16: 
Chris@16:   template <class RealType = double, class Policy = policies::policy<> >
Chris@16:   class skew_normal_distribution
Chris@16:   {
Chris@16:   public:
Chris@16:     typedef RealType value_type;
Chris@16:     typedef Policy policy_type;
Chris@16: 
Chris@16:     skew_normal_distribution(RealType l_location = 0, RealType l_scale = 1, RealType l_shape = 0)
Chris@16:       : location_(l_location), scale_(l_scale), shape_(l_shape)
Chris@16:     { // Default is a 'standard' normal distribution N01. (shape=0 results in the normal distribution with no skew)
Chris@16:       static const char* function = "boost::math::skew_normal_distribution<%1%>::skew_normal_distribution";
Chris@16: 
Chris@16:       RealType result;
Chris@16:       detail::check_scale(function, l_scale, &result, Policy());
Chris@16:       detail::check_location(function, l_location, &result, Policy());
Chris@16:       detail::check_skew_normal_shape(function, l_shape, &result, Policy());
Chris@16:     }
Chris@16: 
Chris@16:     RealType location()const
Chris@16:     { 
Chris@16:       return location_;
Chris@16:     }
Chris@16: 
Chris@16:     RealType scale()const
Chris@16:     { 
Chris@16:       return scale_;
Chris@16:     }
Chris@16: 
Chris@16:     RealType shape()const
Chris@16:     { 
Chris@16:       return shape_;
Chris@16:     }
Chris@16: 
Chris@16: 
Chris@16:   private:
Chris@16:     //
Chris@16:     // Data members:
Chris@16:     //
Chris@16:     RealType location_;  // distribution location.
Chris@16:     RealType scale_;    // distribution scale.
Chris@16:     RealType shape_;    // distribution shape.
Chris@16:   }; // class skew_normal_distribution
Chris@16: 
Chris@16:   typedef skew_normal_distribution<double> skew_normal;
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline const std::pair<RealType, RealType> range(const skew_normal_distribution<RealType, Policy>& /*dist*/)
Chris@16:   { // Range of permissible values for random variable x.
Chris@16:     using boost::math::tools::max_value;
Chris@16:     return std::pair<RealType, RealType>(
Chris@16:        std::numeric_limits<RealType>::has_infinity ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>(), 
Chris@16:        std::numeric_limits<RealType>::has_infinity ? std::numeric_limits<RealType>::infinity() : max_value<RealType>()); // - to + max value.
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline const std::pair<RealType, RealType> support(const skew_normal_distribution<RealType, Policy>& /*dist*/)
Chris@16:   { // Range of supported values for random variable x.
Chris@16:     // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
Chris@16: 
Chris@16:     using boost::math::tools::max_value;
Chris@16:     return std::pair<RealType, RealType>(-max_value<RealType>(),  max_value<RealType>()); // - to + max value.
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType pdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:   {
Chris@16:     const RealType scale = dist.scale();
Chris@16:     const RealType location = dist.location();
Chris@16:     const RealType shape = dist.shape();
Chris@16: 
Chris@16:     static const char* function = "boost::math::pdf(const skew_normal_distribution<%1%>&, %1%)";
Chris@16:     if((boost::math::isinf)(x))
Chris@16:     {
Chris@16:       return 0; // pdf + and - infinity is zero.
Chris@16:     }
Chris@16:     // Below produces MSVC 4127 warnings, so the above used instead.
Chris@16:     //if(std::numeric_limits<RealType>::has_infinity && abs(x) == std::numeric_limits<RealType>::infinity())
Chris@16:     //{ // pdf + and - infinity is zero.
Chris@16:     //  return 0;
Chris@16:     //}
Chris@16: 
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if(false == detail::check_x(function, x, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16: 
Chris@16:     const RealType transformed_x = (x-location)/scale;
Chris@16: 
Chris@16:     normal_distribution<RealType, Policy> std_normal;
Chris@16: 
Chris@16:     result = pdf(std_normal, transformed_x) * cdf(std_normal, shape*transformed_x) * 2 / scale;
Chris@16: 
Chris@16:     return result;
Chris@16:   } // pdf
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType cdf(const skew_normal_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:   {
Chris@16:     const RealType scale = dist.scale();
Chris@16:     const RealType location = dist.location();
Chris@16:     const RealType shape = dist.shape();
Chris@16: 
Chris@16:     static const char* function = "boost::math::cdf(const skew_normal_distribution<%1%>&, %1%)";
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16:     if((boost::math::isinf)(x))
Chris@16:     {
Chris@16:       if(x < 0) return 0; // -infinity
Chris@16:       return 1; // + infinity
Chris@16:     }
Chris@16:     // These produce MSVC 4127 warnings, so the above used instead.
Chris@16:     //if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
Chris@16:     //{ // cdf +infinity is unity.
Chris@16:     //  return 1;
Chris@16:     //}
Chris@16:     //if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
Chris@16:     //{ // cdf -infinity is zero.
Chris@16:     //  return 0;
Chris@16:     //}
Chris@16:     if(false == detail::check_x(function, x, &result, Policy()))
Chris@16:     {
Chris@16:       return result;
Chris@16:     }
Chris@16: 
Chris@16:     const RealType transformed_x = (x-location)/scale;
Chris@16: 
Chris@16:     normal_distribution<RealType, Policy> std_normal;
Chris@16: 
Chris@16:     result = cdf(std_normal, transformed_x) - owens_t(transformed_x, shape)*static_cast<RealType>(2);
Chris@16: 
Chris@16:     return result;
Chris@16:   } // cdf
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType cdf(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
Chris@16:   {
Chris@16:     const RealType scale = c.dist.scale();
Chris@16:     const RealType location = c.dist.location();
Chris@16:     const RealType shape = c.dist.shape();
Chris@16:     const RealType x = c.param;
Chris@16: 
Chris@16:     static const char* function = "boost::math::cdf(const complement(skew_normal_distribution<%1%>&), %1%)";
Chris@16: 
Chris@16:     if((boost::math::isinf)(x))
Chris@16:     {
Chris@16:       if(x < 0) return 1; // cdf complement -infinity is unity.
Chris@16:       return 0; // cdf complement +infinity is zero
Chris@16:     }
Chris@16:     // These produce MSVC 4127 warnings, so the above used instead.
Chris@16:     //if(std::numeric_limits<RealType>::has_infinity && x == std::numeric_limits<RealType>::infinity())
Chris@16:     //{ // cdf complement +infinity is zero.
Chris@16:     //  return 0;
Chris@16:     //}
Chris@16:     //if(std::numeric_limits<RealType>::has_infinity && x == -std::numeric_limits<RealType>::infinity())
Chris@16:     //{ // cdf complement -infinity is unity.
Chris@16:     //  return 1;
Chris@16:     //}
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_x(function, x, &result, Policy()))
Chris@16:       return result;
Chris@16: 
Chris@16:     const RealType transformed_x = (x-location)/scale;
Chris@16: 
Chris@16:     normal_distribution<RealType, Policy> std_normal;
Chris@16: 
Chris@16:     result = cdf(complement(std_normal, transformed_x)) + owens_t(transformed_x, shape)*static_cast<RealType>(2);
Chris@16:     return result;
Chris@16:   } // cdf complement
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType location(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     return dist.location();
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType scale(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     return dist.scale();
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType shape(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     return dist.shape();
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType mean(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     BOOST_MATH_STD_USING  // for ADL of std functions
Chris@16: 
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     //const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());
Chris@16: 
Chris@16:     //return dist.location() + dist.scale() * delta * root_two_div_pi<RealType>();
Chris@16: 
Chris@16:     return dist.location() + dist.scale() * dist.shape() / sqrt(pi<RealType>()+pi<RealType>()*dist.shape()*dist.shape()) * root_two<RealType>();
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType variance(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     const RealType delta2 = static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape()));
Chris@16:     //const RealType inv_delta2 = static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape());
Chris@16: 
Chris@16:     RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()*delta2);
Chris@16:     //RealType variance = dist.scale()*dist.scale()*(static_cast<RealType>(1)-two_div_pi<RealType>()/inv_delta2);
Chris@16: 
Chris@16:     return variance;
Chris@16:   }
Chris@16: 
Chris@16:   namespace detail
Chris@16:   {
Chris@16:     /*
Chris@16:       TODO No closed expression for mode, so use max of pdf.
Chris@16:     */
Chris@16:     
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType mode_fallback(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:     { // mode.
Chris@16:         static const char* function = "mode(skew_normal_distribution<%1%> const&)";
Chris@16:         const RealType scale = dist.scale();
Chris@16:         const RealType location = dist.location();
Chris@16:         const RealType shape = dist.shape();
Chris@16:         
Chris@16:         RealType result;
Chris@16:         if(!detail::check_scale(
Chris@16:           function,
Chris@16:           scale, &result, Policy())
Chris@16:           ||
Chris@16:         !detail::check_skew_normal_shape(
Chris@16:           function,
Chris@16:           shape,
Chris@16:           &result,
Chris@16:           Policy()))
Chris@16:         return result;
Chris@16: 
Chris@16:         if( shape == 0 )
Chris@16:         {
Chris@16:           return location;
Chris@16:         }
Chris@16: 
Chris@16:         if( shape < 0 )
Chris@16:         {
Chris@16:           skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
Chris@16:           result = mode_fallback(D);
Chris@16:           result = location-scale*result;
Chris@16:           return result;
Chris@16:         }
Chris@16:         
Chris@16:         BOOST_MATH_STD_USING
Chris@16: 
Chris@16:         // 21 elements
Chris@16:         static const RealType shapes[] = {
Chris@16:           0.0,
Chris@16:           1.000000000000000e-004,
Chris@16:           2.069138081114790e-004,
Chris@16:           4.281332398719396e-004,
Chris@16:           8.858667904100824e-004,
Chris@16:           1.832980710832436e-003,
Chris@16:           3.792690190732250e-003,
Chris@16:           7.847599703514606e-003,
Chris@16:           1.623776739188722e-002,
Chris@16:           3.359818286283781e-002,
Chris@16:           6.951927961775606e-002,
Chris@16:           1.438449888287663e-001,
Chris@16:           2.976351441631319e-001,
Chris@16:           6.158482110660261e-001,
Chris@16:           1.274274985703135e+000,
Chris@16:           2.636650898730361e+000,
Chris@16:           5.455594781168514e+000,
Chris@16:           1.128837891684688e+001,
Chris@16:           2.335721469090121e+001,
Chris@16:           4.832930238571753e+001,
Chris@16:           1.000000000000000e+002};
Chris@16: 
Chris@16:         // 21 elements
Chris@16:         static const RealType guess[] = {
Chris@16:           0.0,
Chris@16:           5.000050000525391e-005,
Chris@16:           1.500015000148736e-004,
Chris@16:           3.500035000350010e-004,
Chris@16:           7.500075000752560e-004,
Chris@16:           1.450014500145258e-003,
Chris@16:           3.050030500305390e-003,
Chris@16:           6.250062500624765e-003,
Chris@16:           1.295012950129504e-002,
Chris@16:           2.675026750267495e-002,
Chris@16:           5.525055250552491e-002,
Chris@16:           1.132511325113255e-001,
Chris@16:           2.249522495224952e-001,
Chris@16:           3.992539925399257e-001,
Chris@16:           5.353553535535358e-001,
Chris@16:           4.954549545495457e-001,
Chris@16:           3.524535245352451e-001,
Chris@16:           2.182521825218249e-001,
Chris@16:           1.256512565125654e-001,
Chris@16:           6.945069450694508e-002,
Chris@16:           3.735037350373460e-002
Chris@16:         };
Chris@16: 
Chris@16:         const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);
Chris@16: 
Chris@16:         typedef typename std::iterator_traits<RealType*>::difference_type diff_type;
Chris@16:         
Chris@16:         const diff_type d = std::distance(shapes, result_ptr);
Chris@16:         
Chris@16:         BOOST_ASSERT(d > static_cast<diff_type>(0));
Chris@16: 
Chris@16:         // refine
Chris@16:         if(d < static_cast<diff_type>(21)) // shape smaller 100
Chris@16:         {
Chris@16:           result = guess[d-static_cast<diff_type>(1)]
Chris@16:             + (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
Chris@16:             * (shape-shapes[d-static_cast<diff_type>(1)]);
Chris@16:         }
Chris@16:         else // shape greater 100
Chris@16:         {
Chris@16:           result = 1e-4;
Chris@16:         }
Chris@16: 
Chris@16:         skew_normal_distribution<RealType, Policy> helper(0, 1, shape);
Chris@16:         
Chris@16:         result = detail::generic_find_mode_01(helper, result, function);
Chris@16:         
Chris@16:         result = result*scale + location;
Chris@16:         
Chris@16:         return result;
Chris@16:     } // mode_fallback
Chris@16:     
Chris@16:     
Chris@16:     /*
Chris@16:      * TODO No closed expression for mode, so use f'(x) = 0
Chris@16:      */
Chris@16:     template <class RealType, class Policy>
Chris@16:     struct skew_normal_mode_functor
Chris@16:     { 
Chris@16:       skew_normal_mode_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist)
Chris@16:         : distribution(dist)
Chris@16:       {
Chris@16:       }
Chris@16: 
Chris@16:       boost::math::tuple<RealType, RealType> operator()(RealType const& x)
Chris@16:       {
Chris@16:         normal_distribution<RealType, Policy> std_normal;
Chris@16:         const RealType shape = distribution.shape();
Chris@16:         const RealType pdf_x = pdf(distribution, x);
Chris@16:         const RealType normpdf_x = pdf(std_normal, x);
Chris@16:         const RealType normpdf_ax = pdf(std_normal, x*shape);
Chris@16:         RealType fx = static_cast<RealType>(2)*shape*normpdf_ax*normpdf_x - x*pdf_x;
Chris@16:         RealType dx = static_cast<RealType>(2)*shape*x*normpdf_x*normpdf_ax*(static_cast<RealType>(1) + shape*shape) + pdf_x + x*fx;
Chris@16:         // return both function evaluation difference f(x) and 1st derivative f'(x).
Chris@16:         return boost::math::make_tuple(fx, -dx);
Chris@16:       }
Chris@16:     private:
Chris@16:       const boost::math::skew_normal_distribution<RealType, Policy> distribution;
Chris@16:     };
Chris@16:     
Chris@16:   } // namespace detail
Chris@16:   
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType mode(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     const RealType scale = dist.scale();
Chris@16:     const RealType location = dist.location();
Chris@16:     const RealType shape = dist.shape();
Chris@16: 
Chris@16:     static const char* function = "boost::math::mode(const skew_normal_distribution<%1%>&, %1%)";
Chris@16: 
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:       return result;
Chris@16: 
Chris@16:     if( shape == 0 )
Chris@16:     {
Chris@16:       return location;
Chris@16:     }
Chris@16: 
Chris@16:     if( shape < 0 )
Chris@16:     {
Chris@16:       skew_normal_distribution<RealType, Policy> D(0, 1, -shape);
Chris@16:       result = mode(D);
Chris@16:       result = location-scale*result;
Chris@16:       return result;
Chris@16:     }
Chris@16: 
Chris@16:     // 21 elements
Chris@16:     static const RealType shapes[] = {
Chris@16:       0.0,
Chris@16:       static_cast<RealType>(1.000000000000000e-004),
Chris@16:       static_cast<RealType>(2.069138081114790e-004),
Chris@16:       static_cast<RealType>(4.281332398719396e-004),
Chris@16:       static_cast<RealType>(8.858667904100824e-004),
Chris@16:       static_cast<RealType>(1.832980710832436e-003),
Chris@16:       static_cast<RealType>(3.792690190732250e-003),
Chris@16:       static_cast<RealType>(7.847599703514606e-003),
Chris@16:       static_cast<RealType>(1.623776739188722e-002),
Chris@16:       static_cast<RealType>(3.359818286283781e-002),
Chris@16:       static_cast<RealType>(6.951927961775606e-002),
Chris@16:       static_cast<RealType>(1.438449888287663e-001),
Chris@16:       static_cast<RealType>(2.976351441631319e-001),
Chris@16:       static_cast<RealType>(6.158482110660261e-001),
Chris@16:       static_cast<RealType>(1.274274985703135e+000),
Chris@16:       static_cast<RealType>(2.636650898730361e+000),
Chris@16:       static_cast<RealType>(5.455594781168514e+000),
Chris@16:       static_cast<RealType>(1.128837891684688e+001),
Chris@16:       static_cast<RealType>(2.335721469090121e+001),
Chris@16:       static_cast<RealType>(4.832930238571753e+001),
Chris@16:       static_cast<RealType>(1.000000000000000e+002)
Chris@16:     };
Chris@16: 
Chris@16:     // 21 elements
Chris@16:     static const RealType guess[] = {
Chris@16:       0.0,
Chris@16:       static_cast<RealType>(5.000050000525391e-005),
Chris@16:       static_cast<RealType>(1.500015000148736e-004),
Chris@16:       static_cast<RealType>(3.500035000350010e-004),
Chris@16:       static_cast<RealType>(7.500075000752560e-004),
Chris@16:       static_cast<RealType>(1.450014500145258e-003),
Chris@16:       static_cast<RealType>(3.050030500305390e-003),
Chris@16:       static_cast<RealType>(6.250062500624765e-003),
Chris@16:       static_cast<RealType>(1.295012950129504e-002),
Chris@16:       static_cast<RealType>(2.675026750267495e-002),
Chris@16:       static_cast<RealType>(5.525055250552491e-002),
Chris@16:       static_cast<RealType>(1.132511325113255e-001),
Chris@16:       static_cast<RealType>(2.249522495224952e-001),
Chris@16:       static_cast<RealType>(3.992539925399257e-001),
Chris@16:       static_cast<RealType>(5.353553535535358e-001),
Chris@16:       static_cast<RealType>(4.954549545495457e-001),
Chris@16:       static_cast<RealType>(3.524535245352451e-001),
Chris@16:       static_cast<RealType>(2.182521825218249e-001),
Chris@16:       static_cast<RealType>(1.256512565125654e-001),
Chris@16:       static_cast<RealType>(6.945069450694508e-002),
Chris@16:       static_cast<RealType>(3.735037350373460e-002)
Chris@16:     };
Chris@16: 
Chris@16:     const RealType* result_ptr = std::lower_bound(shapes, shapes+21, shape);
Chris@16: 
Chris@16:     typedef typename std::iterator_traits<RealType*>::difference_type diff_type;
Chris@16:     
Chris@16:     const diff_type d = std::distance(shapes, result_ptr);
Chris@16:     
Chris@16:     BOOST_ASSERT(d > static_cast<diff_type>(0));
Chris@16: 
Chris@16:     // TODO: make the search bounds smarter, depending on the shape parameter
Chris@16:     RealType search_min = 0; // below zero was caught above
Chris@16:     RealType search_max = 0.55f; // will never go above 0.55
Chris@16: 
Chris@16:     // refine
Chris@16:     if(d < static_cast<diff_type>(21)) // shape smaller 100
Chris@16:     {
Chris@16:       // it is safe to assume that d > 0, because shape==0.0 is caught earlier
Chris@16:       result = guess[d-static_cast<diff_type>(1)]
Chris@16:         + (guess[d]-guess[d-static_cast<diff_type>(1)])/(shapes[d]-shapes[d-static_cast<diff_type>(1)])
Chris@16:         * (shape-shapes[d-static_cast<diff_type>(1)]);
Chris@16:     }
Chris@16:     else // shape greater 100
Chris@16:     {
Chris@16:       result = 1e-4f;
Chris@16:       search_max = guess[19]; // set 19 instead of 20 to have a safety margin because the table may not be exact @ shape=100
Chris@16:     }
Chris@16:     
Chris@16:     const int get_digits = policies::digits<RealType, Policy>();// get digits from policy, 
Chris@16:     boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
Chris@16: 
Chris@16:     skew_normal_distribution<RealType, Policy> helper(0, 1, shape);
Chris@16: 
Chris@16:     result = tools::newton_raphson_iterate(detail::skew_normal_mode_functor<RealType, Policy>(helper), result,
Chris@16:       search_min, search_max, get_digits, m);
Chris@16:     
Chris@16:     result = result*scale + location;
Chris@16: 
Chris@16:     return result;
Chris@16:   }
Chris@16:   
Chris@16: 
Chris@16:   
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType skewness(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     BOOST_MATH_STD_USING  // for ADL of std functions
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     static const RealType factor = four_minus_pi<RealType>()/static_cast<RealType>(2);
Chris@16:     const RealType delta = dist.shape() / sqrt(static_cast<RealType>(1)+dist.shape()*dist.shape());
Chris@16: 
Chris@16:     return factor * pow(root_two_div_pi<RealType>() * delta, 3) /
Chris@16:       pow(static_cast<RealType>(1)-two_div_pi<RealType>()*delta*delta, static_cast<RealType>(1.5));
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType kurtosis(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     return kurtosis_excess(dist)+static_cast<RealType>(3);
Chris@16:   }
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType kurtosis_excess(const skew_normal_distribution<RealType, Policy>& dist)
Chris@16:   {
Chris@16:     using namespace boost::math::constants;
Chris@16: 
Chris@16:     static const RealType factor = pi_minus_three<RealType>()*static_cast<RealType>(2);
Chris@16: 
Chris@16:     const RealType delta2 = static_cast<RealType>(1) / (static_cast<RealType>(1)+static_cast<RealType>(1)/(dist.shape()*dist.shape()));
Chris@16: 
Chris@16:     const RealType x = static_cast<RealType>(1)-two_div_pi<RealType>()*delta2;
Chris@16:     const RealType y = two_div_pi<RealType>() * delta2;
Chris@16: 
Chris@16:     return factor * y*y / (x*x);
Chris@16:   }
Chris@16: 
Chris@16:   namespace detail
Chris@16:   {
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     struct skew_normal_quantile_functor
Chris@16:     { 
Chris@16:       skew_normal_quantile_functor(const boost::math::skew_normal_distribution<RealType, Policy> dist, RealType const& p)
Chris@16:         : distribution(dist), prob(p)
Chris@16:       {
Chris@16:       }
Chris@16: 
Chris@16:       boost::math::tuple<RealType, RealType> operator()(RealType const& x)
Chris@16:       {
Chris@16:         RealType c = cdf(distribution, x);
Chris@16:         RealType fx = c - prob;  // Difference cdf - value - to minimize.
Chris@16:         RealType dx = pdf(distribution, x); // pdf is 1st derivative.
Chris@16:         // return both function evaluation difference f(x) and 1st derivative f'(x).
Chris@16:         return boost::math::make_tuple(fx, dx);
Chris@16:       }
Chris@16:     private:
Chris@16:       const boost::math::skew_normal_distribution<RealType, Policy> distribution;
Chris@16:       RealType prob; 
Chris@16:     };
Chris@16: 
Chris@16:   } // namespace detail
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType quantile(const skew_normal_distribution<RealType, Policy>& dist, const RealType& p)
Chris@16:   {
Chris@16:     const RealType scale = dist.scale();
Chris@16:     const RealType location = dist.location();
Chris@16:     const RealType shape = dist.shape();
Chris@16: 
Chris@16:     static const char* function = "boost::math::quantile(const skew_normal_distribution<%1%>&, %1%)";
Chris@16: 
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_probability(function, p, &result, Policy()))
Chris@16:       return result;
Chris@16: 
Chris@16:     // Compute initial guess via Cornish-Fisher expansion.
Chris@16:     RealType x = -boost::math::erfc_inv(2 * p, Policy()) * constants::root_two<RealType>();
Chris@16: 
Chris@16:     // Avoid unnecessary computations if there is no skew.
Chris@16:     if(shape != 0)
Chris@16:     {
Chris@16:       const RealType skew = skewness(dist);
Chris@16:       const RealType exk = kurtosis_excess(dist);
Chris@16: 
Chris@16:       x = x + (x*x-static_cast<RealType>(1))*skew/static_cast<RealType>(6)
Chris@16:       + x*(x*x-static_cast<RealType>(3))*exk/static_cast<RealType>(24)
Chris@16:       - x*(static_cast<RealType>(2)*x*x-static_cast<RealType>(5))*skew*skew/static_cast<RealType>(36);
Chris@16:     } // if(shape != 0)
Chris@16: 
Chris@16:     result = standard_deviation(dist)*x+mean(dist);
Chris@16: 
Chris@16:     // handle special case of non-skew normal distribution.
Chris@16:     if(shape == 0)
Chris@16:       return result;
Chris@16: 
Chris@16:     // refine the result by numerically searching the root of (p-cdf)
Chris@16: 
Chris@16:     const RealType search_min = range(dist).first;
Chris@16:     const RealType search_max = range(dist).second;
Chris@16: 
Chris@16:     const int get_digits = policies::digits<RealType, Policy>();// get digits from policy, 
Chris@16:     boost::uintmax_t m = policies::get_max_root_iterations<Policy>(); // and max iterations.
Chris@16: 
Chris@16:     result = tools::newton_raphson_iterate(detail::skew_normal_quantile_functor<RealType, Policy>(dist, p), result,
Chris@16:       search_min, search_max, get_digits, m);
Chris@16: 
Chris@16:     return result;
Chris@16:   } // quantile
Chris@16: 
Chris@16:   template <class RealType, class Policy>
Chris@16:   inline RealType quantile(const complemented2_type<skew_normal_distribution<RealType, Policy>, RealType>& c)
Chris@16:   {
Chris@16:     const RealType scale = c.dist.scale();
Chris@16:     const RealType location = c.dist.location();
Chris@16:     const RealType shape = c.dist.shape();
Chris@16: 
Chris@16:     static const char* function = "boost::math::quantile(const complement(skew_normal_distribution<%1%>&), %1%)";
Chris@16:     RealType result = 0;
Chris@16:     if(false == detail::check_scale(function, scale, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_location(function, location, &result, Policy()))
Chris@16:       return result;
Chris@16:     if(false == detail::check_skew_normal_shape(function, shape, &result, Policy()))
Chris@16:       return result;
Chris@16:     RealType q = c.param;
Chris@16:     if(false == detail::check_probability(function, q, &result, Policy()))
Chris@16:       return result;
Chris@16: 
Chris@16:     skew_normal_distribution<RealType, Policy> D(-location, scale, -shape);
Chris@16: 
Chris@16:     result = -quantile(D, q);
Chris@16: 
Chris@16:     return result;
Chris@16:   } // quantile
Chris@16: 
Chris@16: 
Chris@16: } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: // This include must be at the end, *after* the accessors
Chris@16: // for this distribution have been defined, in order to
Chris@16: // keep compilers that support two-phase lookup happy.
Chris@16: #include <boost/math/distributions/detail/derived_accessors.hpp>
Chris@16: 
Chris@16: #endif // BOOST_STATS_SKEW_NORMAL_HPP
Chris@16: 
Chris@16: