/External/boost/boost/math/distributions/inverse_gamma.hpp

// inverse_gamma.hpp



//  Copyright Paul A. Bristow 2010.

//  Copyright John Maddock 2010.

//  Use, modification and distribution are subject to the

//  Boost Software License, Version 1.0. (See accompanying file

//  LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)



#ifndef BOOST_STATS_INVERSE_GAMMA_HPP

#define BOOST_STATS_INVERSE_GAMMA_HPP



// Inverse Gamma Distribution is a two-parameter family

// of continuous probability distributions

// on the positive real line, which is the distribution of

// the reciprocal of a variable distributed according to the gamma distribution.



// http://en.wikipedia.org/wiki/Inverse-gamma_distribution

// http://rss.acs.unt.edu/Rdoc/library/pscl/html/igamma.html



// See also gamma distribution at gamma.hpp:

// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm

// http://mathworld.wolfram.com/GammaDistribution.html

// http://en.wikipedia.org/wiki/Gamma_distribution



#include <boost/math/distributions/fwd.hpp>

#include <boost/math/special_functions/gamma.hpp>

#include <boost/math/distributions/detail/common_error_handling.hpp>

#include <boost/math/distributions/complement.hpp>



#include <utility>



namespace boost{ namespace math

{

namespace detail

{



template <class RealType, class Policy>

inline bool check_inverse_gamma_shape(

      const char* function, // inverse_gamma

      RealType shape, // shape aka alpha

      RealType* result, // to update, perhaps with NaN

      const Policy& pol)

{  // Sources say shape argument must be > 0

   // but seems logical to allow shape zero as special case,

   // returning pdf and cdf zero (but not < 0).

   // (Functions like mean, variance with other limits on shape are checked

   // in version including an operator & limit below).

   if((shape < 0) || !(boost::math::isfinite)(shape))

   {

      *result = policies::raise_domain_error<RealType>(

         function,

         "Shape parameter is %1%, but must be >= 0 !", shape, pol);

      return false;

   }

   return true;

} //bool check_inverse_gamma_shape



template <class RealType, class Policy>

inline bool check_inverse_gamma_x(

      const char* function,

      RealType const& x,

      RealType* result, const Policy& pol)

{

   if((x < 0) || !(boost::math::isfinite)(x))

   {

      *result = policies::raise_domain_error<RealType>(

         function,

         "Random variate is %1% but must be >= 0 !", x, pol);

      return false;

   }

   return true;

}



template <class RealType, class Policy>

inline bool check_inverse_gamma(

      const char* function, // TODO swap these over, so shape is first.

      RealType scale,  // scale aka beta

      RealType shape, // shape aka alpha

      RealType* result, const Policy& pol)

{

   return check_scale(function, scale, result, pol)

     && check_inverse_gamma_shape(function, shape, result, pol);

} // bool check_inverse_gamma



} // namespace detail



template <class RealType = double, class Policy = policies::policy<> >

class inverse_gamma_distribution

{

public:

   typedef RealType value_type;

   typedef Policy policy_type;



   inverse_gamma_distribution(RealType shape = 1, RealType scale = 1)

      : m_shape(shape), m_scale(scale)

   {

      RealType result;

      detail::check_inverse_gamma(

        "boost::math::inverse_gamma_distribution<%1%>::inverse_gamma_distribution",

        scale, shape, &result, Policy());

   }



   RealType shape()const

   {

      return m_shape;

   }



   RealType scale()const

   {

      return m_scale;

   }

private:

   //

   // Data members:

   //

   RealType m_shape;     // distribution shape

   RealType m_scale;     // distribution scale

};



typedef inverse_gamma_distribution<double> inverse_gamma;

// typedef - but potential clash with name of inverse gamma *function*.

// but there is a typedef for gamma

//   typedef boost::math::gamma_distribution<Type, Policy> gamma;



// Allow random variable x to be zero, treated as a special case (unlike some definitions).



template <class RealType, class Policy>

inline const std::pair<RealType, RealType> range(const inverse_gamma_distribution<RealType, Policy>& /* dist */)

{  // Range of permissible values for random variable x.

   using boost::math::tools::max_value;

   return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>());

}



template <class RealType, class Policy>

inline const std::pair<RealType, RealType> support(const inverse_gamma_distribution<RealType, Policy>& /* dist */)

{  // Range of supported values for random variable x.

   // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.

   using boost::math::tools::max_value;

   using boost::math::tools::min_value;

   return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>());

}



template <class RealType, class Policy>

inline RealType pdf(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& x)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::pdf(const inverse_gamma_distribution<%1%>&, %1%)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

   { // distribution parameters bad.

      return result;

   } 

   if(x == 0)

   { // Treat random variate zero as a special case.

      return 0;

   }

   else if(false == detail::check_inverse_gamma_x(function, x, &result, Policy()))

   { // x bad.

      return result;

   }

   result = scale / x;

   if(result < tools::min_value<RealType>())

      return 0;  // random variable is infinite or so close as to make no difference.

   result = gamma_p_derivative(shape, result, Policy()) * scale;

   if(0 != result)

   {

      if(x < 0)

      {

         // x * x may under or overflow, likewise our result,

         // so be extra careful about the arithmetic:

         RealType lim = tools::max_value<RealType>() * x;

         if(lim < result)

            return policies::raise_overflow_error<RealType, Policy>(function, "PDF is infinite.", Policy());

         result /= x;

         if(lim < result)

            return policies::raise_overflow_error<RealType, Policy>(function, "PDF is infinite.", Policy());

         result /= x;

      }

      result /= (x * x);

   }

   // better than naive

   // result = (pow(scale, shape) * pow(x, (-shape -1)) * exp(-scale/x) ) / tgamma(shape);

   return result;

} // pdf



template <class RealType, class Policy>

inline RealType cdf(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& x)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::cdf(const inverse_gamma_distribution<%1%>&, %1%)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

   { // distribution parameters bad.

      return result;

   }

   if (x == 0)

   { // Treat zero as a special case.

     return 0;

   }

   else if(false == detail::check_inverse_gamma_x(function, x, &result, Policy()))

   { // x bad

      return result;

   }

   result = boost::math::gamma_q(shape, scale / x, Policy());

   // result = tgamma(shape, scale / x) / tgamma(shape); // naive using tgamma

   return result;

} // cdf



template <class RealType, class Policy>

inline RealType quantile(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& p)

{

   BOOST_MATH_STD_USING  // for ADL of std functions

   using boost::math::gamma_q_inv;



   static const char* function = "boost::math::quantile(const inverse_gamma_distribution<%1%>&, %1%)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

      return result;

   if(false == detail::check_probability(function, p, &result, Policy()))

      return result;

   if(p == 1)

   {

      return policies::raise_overflow_error<RealType>(function, 0, Policy());

   }

   result = gamma_q_inv(shape, p, Policy());

   if((result < 1) && (result * tools::max_value<RealType>() < scale))

      return policies::raise_overflow_error<RealType, Policy>(function, "Value of random variable in inverse gamma distribution quantile is infinite.", Policy());

   result = scale / result;

   return result;

}



template <class RealType, class Policy>

inline RealType cdf(const complemented2_type<inverse_gamma_distribution<RealType, Policy>, RealType>& c)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::quantile(const gamma_distribution<%1%>&, %1%)";



   RealType shape = c.dist.shape();

   RealType scale = c.dist.scale();



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

      return result;

   if(false == detail::check_inverse_gamma_x(function, c.param, &result, Policy()))

      return result;



   if(c.param == 0)

      return 1; // Avoid division by zero



   //result = 1. - gamma_q(shape, c.param / scale, Policy());

   result = gamma_p(shape, scale/c.param, Policy());

   return result;

}



template <class RealType, class Policy>

inline RealType quantile(const complemented2_type<inverse_gamma_distribution<RealType, Policy>, RealType>& c)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::quantile(const inverse_gamma_distribution<%1%>&, %1%)";



   RealType shape = c.dist.shape();

   RealType scale = c.dist.scale();

   RealType q = c.param;



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

      return result;

   if(false == detail::check_probability(function, q, &result, Policy()))

      return result;



   if(q == 0)

   {

      return policies::raise_overflow_error<RealType>(function, 0, Policy());

   }

   result = gamma_p_inv(shape, q, Policy());

   if((result < 1) && (result * tools::max_value<RealType>() < scale))

      return policies::raise_overflow_error<RealType, Policy>(function, "Value of random variable in inverse gamma distribution quantile is infinite.", Policy());

   result = scale / result;

   return result;

}



template <class RealType, class Policy>

inline RealType mean(const inverse_gamma_distribution<RealType, Policy>& dist)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::mean(const inverse_gamma_distribution<%1%>&)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;



   if(false == detail::check_scale(function, scale, &result, Policy()))

   {

     return result;

   }

   if((shape <= 1) || !(boost::math::isfinite)(shape))

   {

     result = policies::raise_domain_error<RealType>(

       function,

       "Shape parameter is %1%, but for a defined mean it must be > 1", shape, Policy());

     return result;

   }

  result = scale / (shape - 1);

  return result;

} // mean



template <class RealType, class Policy>

inline RealType variance(const inverse_gamma_distribution<RealType, Policy>& dist)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::variance(const inverse_gamma_distribution<%1%>&)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

      if(false == detail::check_scale(function, scale, &result, Policy()))

   {

     return result;

   }

   if((shape <= 2) || !(boost::math::isfinite)(shape))

   {

     result = policies::raise_domain_error<RealType>(

       function,

       "Shape parameter is %1%, but for a defined variance it must be > 2", shape, Policy());

     return result;

   }

   result = (scale * scale) / ((shape - 1) * (shape -1) * (shape -2));

   return result;

}



template <class RealType, class Policy>

inline RealType mode(const inverse_gamma_distribution<RealType, Policy>& dist)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::mode(const inverse_gamma_distribution<%1%>&)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

   if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))

   {

      return result;

   }

   // Only defined for shape >= 0, but is checked by check_inverse_gamma.

   result = scale / (shape + 1);

   return result;

}



//template <class RealType, class Policy>

//inline RealType median(const gamma_distribution<RealType, Policy>& dist)

//{  // Wikipedia does not define median,

     // so rely on default definition quantile(0.5) in derived accessors.

//  return result.

//}



template <class RealType, class Policy>

inline RealType skewness(const inverse_gamma_distribution<RealType, Policy>& dist)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::skewness(const inverse_gamma_distribution<%1%>&)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();

   RealType result = 0;



   if(false == detail::check_scale(function, scale, &result, Policy()))

   {

     return result;

   }

   if((shape <= 3) || !(boost::math::isfinite)(shape))

   {

     result = policies::raise_domain_error<RealType>(

       function,

       "Shape parameter is %1%, but for a defined skewness it must be > 3", shape, Policy());

     return result;

   }

   result = (4 * sqrt(shape - 2) ) / (shape - 3);

   return result;

}



template <class RealType, class Policy>

inline RealType kurtosis_excess(const inverse_gamma_distribution<RealType, Policy>& dist)

{

   BOOST_MATH_STD_USING  // for ADL of std functions



   static const char* function = "boost::math::kurtosis_excess(const inverse_gamma_distribution<%1%>&)";



   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;

   if(false == detail::check_scale(function, scale, &result, Policy()))

   {

     return result;

   }

   if((shape <= 4) || !(boost::math::isfinite)(shape))

   {

     result = policies::raise_domain_error<RealType>(

       function,

       "Shape parameter is %1%, but for a defined kurtosis excess it must be > 4", shape, Policy());

     return result;

   }

   result = (30 * shape - 66) / ((shape - 3) * (shape - 4));

   return result;

}



template <class RealType, class Policy>

inline RealType kurtosis(const inverse_gamma_distribution<RealType, Policy>& dist)

{

  static const char* function = "boost::math::kurtosis(const inverse_gamma_distribution<%1%>&)";

   RealType shape = dist.shape();

   RealType scale = dist.scale();



   RealType result = 0;



  if(false == detail::check_scale(function, scale, &result, Policy()))

   {

     return result;

   }

   if((shape <= 4) || !(boost::math::isfinite)(shape))

   {

     result = policies::raise_domain_error<RealType>(

       function,

       "Shape parameter is %1%, but for a defined kurtosis it must be > 4", shape, Policy());

     return result;

   }

  return kurtosis_excess(dist) + 3;

}



} // namespace math

} // namespace boost



// This include must be at the end, *after* the accessors

// for this distribution have been defined, in order to

// keep compilers that support two-phase lookup happy.

#include <boost/math/distributions/detail/derived_accessors.hpp>



#endif // BOOST_STATS_INVERSE_GAMMA_HPP