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// boost\math\distributions\beta.hpp

// Copyright John Maddock 2006.

// Copyright Paul A. Bristow 2006.

// 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)

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

// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm

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

// The Beta Distribution is a continuous probability distribution.

// The beta distribution is used to model events which are constrained to take place

// within an interval defined by maxima and minima,

// so is used extensively in PERT and other project management systems

// to describe the time to completion.

// The cdf of the beta distribution is used as a convenient way

// of obtaining the sum over a set of binomial outcomes.

// The beta distribution is also used in Bayesian statistics.

#ifndef BOOST_MATH_DIST_BETA_HPP

#define BOOST_MATH_DIST_BETA_HPP

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

#include <boost/math/special_functions/beta.hpp> // for beta.

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

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

#include <boost/math/special_functions/fpclassify.hpp> // isnan.

#include <boost/math/tools/roots.hpp> // for root finding.

#if defined (BOOST_MSVC)

#  pragma warning(push)

#  pragma warning(disable: 4702) // unreachable code

// in domain_error_imp in error_handling

#endif

#include <utility>

namespace boost

{

  namespace math

  {

    namespace beta_detail

    {

      // Common error checking routines for beta distribution functions:

      template <class RealType, class Policy>

      inline bool check_alpha(const char* function, const RealType& alpha, RealType* result, const Policy& pol)

      {

        if(!(boost::math::isfinite)(alpha) || (alpha <= 0))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "Alpha argument is %1%, but must be > 0 !", alpha, pol);

          return false;

        }

        return true;

      } // bool check_alpha

      template <class RealType, class Policy>

      inline bool check_beta(const char* function, const RealType& beta, RealType* result, const Policy& pol)

      {

        if(!(boost::math::isfinite)(beta) || (beta <= 0))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "Beta argument is %1%, but must be > 0 !", beta, pol);

          return false;

        }

        return true;

      } // bool check_beta

      template <class RealType, class Policy>

      inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)

      {

        if((p < 0) || (p > 1) || !(boost::math::isfinite)(p))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);

          return false;

        }

        return true;

      } // bool check_prob

      template <class RealType, class Policy>

      inline bool check_x(const char* function, const RealType& x, RealType* result, const Policy& pol)

      {

        if(!(boost::math::isfinite)(x) || (x < 0) || (x > 1))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "x argument is %1%, but must be >= 0 and <= 1 !", x, pol);

          return false;

        }

        return true;

      } // bool check_x

      template <class RealType, class Policy>

      inline bool check_dist(const char* function, const RealType& alpha, const RealType& beta, RealType* result, const Policy& pol)

      { // Check both alpha and beta.

        return check_alpha(function, alpha, result, pol)

          && check_beta(function, beta, result, pol);

      } // bool check_dist

      template <class RealType, class Policy>

      inline bool check_dist_and_x(const char* function, const RealType& alpha, const RealType& beta, RealType x, RealType* result, const Policy& pol)

      {

        return check_dist(function, alpha, beta, result, pol)

          && beta_detail::check_x(function, x, result, pol);

      } // bool check_dist_and_x

      template <class RealType, class Policy>

      inline bool check_dist_and_prob(const char* function, const RealType& alpha, const RealType& beta, RealType p, RealType* result, const Policy& pol)

      {

        return check_dist(function, alpha, beta, result, pol)

          && check_prob(function, p, result, pol);

      } // bool check_dist_and_prob

      template <class RealType, class Policy>

      inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)

      {

        if(!(boost::math::isfinite)(mean) || (mean <= 0))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "mean argument is %1%, but must be > 0 !", mean, pol);

          return false;

        }

        return true;

      } // bool check_mean

      template <class RealType, class Policy>

      inline bool check_variance(const char* function, const RealType& variance, RealType* result, const Policy& pol)

      {

        if(!(boost::math::isfinite)(variance) || (variance <= 0))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "variance argument is %1%, but must be > 0 !", variance, pol);

          return false;

        }

        return true;

      } // bool check_variance

    } // namespace beta_detail

    // typedef beta_distribution<double> beta;

    // is deliberately NOT included to avoid a name clash with the beta function.

    // Use beta_distribution<> mybeta(...) to construct type double.

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

    class beta_distribution

    {

    public:

      typedef RealType value_type;

      typedef Policy policy_type;

      beta_distribution(RealType l_alpha = 1, RealType l_beta = 1) : m_alpha(l_alpha), m_beta(l_beta)

      {

        RealType result;

        beta_detail::check_dist(

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

          m_alpha,

          m_beta,

          &result, Policy());

      } // beta_distribution constructor.

      // Accessor functions:

      RealType alpha() const

      {

        return m_alpha;

      }

      RealType beta() const

      { // .

        return m_beta;

      }

      // Estimation of the alpha & beta parameters.

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

      // gives formulae in section on parameter estimation.

      // Also NIST EDA page 3 & 4 give the same.

      // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm

      // http://www.epi.ucdavis.edu/diagnostictests/betabuster.html

      static RealType find_alpha(

        RealType mean, // Expected value of mean.

        RealType variance) // Expected value of variance.

      {

        static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";

        RealType result = 0; // of error checks.

        if(false ==

            (

              beta_detail::check_mean(function, mean, &result, Policy())

              && beta_detail::check_variance(function, variance, &result, Policy())

            )

          )

        {

          return result;

        }

        return mean * (( (mean * (1 - mean)) / variance)- 1);

      } // RealType find_alpha

      static RealType find_beta(

        RealType mean, // Expected value of mean.

        RealType variance) // Expected value of variance.

      {

        static const char* function = "boost::math::beta_distribution<%1%>::find_beta";

        RealType result = 0; // of error checks.

        if(false ==

            (

              beta_detail::check_mean(function, mean, &result, Policy())

              &&

              beta_detail::check_variance(function, variance, &result, Policy())

            )

          )

        {

          return result;

        }

        return (1 - mean) * (((mean * (1 - mean)) /variance)-1);

      } //  RealType find_beta

      // Estimate alpha & beta from either alpha or beta, and x and probability.

      // Uses for these parameter estimators are unclear.

      static RealType find_alpha(

        RealType beta, // from beta.

        RealType x, //  x.

        RealType probability) // cdf

      {

        static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";

        RealType result = 0; // of error checks.

        if(false ==

            (

             beta_detail::check_prob(function, probability, &result, Policy())

             &&

             beta_detail::check_beta(function, beta, &result, Policy())

             &&

             beta_detail::check_x(function, x, &result, Policy())

            )

          )

        {

          return result;

        }

        return ibeta_inva(beta, x, probability, Policy());

      } // RealType find_alpha(beta, a, probability)

      static RealType find_beta(

        // ibeta_invb(T b, T x, T p); (alpha, x, cdf,)

        RealType alpha, // alpha.

        RealType x, // probability x.

        RealType probability) // probability cdf.

      {

        static const char* function = "boost::math::beta_distribution<%1%>::find_beta";

        RealType result = 0; // of error checks.

        if(false ==

            (

              beta_detail::check_prob(function, probability, &result, Policy())

              &&

              beta_detail::check_alpha(function, alpha, &result, Policy())

              &&

              beta_detail::check_x(function, x, &result, Policy())

            )

          )

        {

          return result;

        }

        return ibeta_invb(alpha, x, probability, Policy());

      } //  RealType find_beta(alpha, x, probability)

    private:

      RealType m_alpha; // Two parameters of the beta distribution.

      RealType m_beta;

    }; // template <class RealType, class Policy> class beta_distribution

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> range(const beta_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), static_cast<RealType>(1));

    }

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> support(const beta_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.

      return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));

    }

    template <class RealType, class Policy>

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

    { // Mean of beta distribution = np.

      return  dist.alpha() / (dist.alpha() + dist.beta());

    } // mean

    template <class RealType, class Policy>

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

    { // Variance of beta distribution = np(1-p).

      RealType a = dist.alpha();

      RealType b = dist.beta();

      return  (a * b) / ((a + b ) * (a + b) * (a + b + 1));

    } // variance

    template <class RealType, class Policy>

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

    {

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

      RealType result;

      if ((dist.alpha() <= 1))

      {

        result = policies::raise_domain_error<RealType>(

          function,

          "mode undefined for alpha = %1%, must be > 1!", dist.alpha(), Policy());

        return result;

      }

      if ((dist.beta() <= 1))

      {

        result = policies::raise_domain_error<RealType>(

          function,

          "mode undefined for beta = %1%, must be > 1!", dist.beta(), Policy());

        return result;

      }

      RealType a = dist.alpha();

      RealType b = dist.beta();

      return (a-1) / (a + b - 2);

    } // mode

    //template <class RealType, class Policy>

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

    //{ // Median of beta distribution is not defined.

    //  return tools::domain_error<RealType>(function, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());

    //} // median

    //But WILL be provided by the derived accessor as quantile(0.5).

    template <class RealType, class Policy>

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

    {

      BOOST_MATH_STD_USING // ADL of std functions.

      RealType a = dist.alpha();

      RealType b = dist.beta();

      return (2 * (b-a) * sqrt(a + b + 1)) / ((a + b + 2) * sqrt(a * b));

    } // skewness

    template <class RealType, class Policy>

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

    {

      RealType a = dist.alpha();

      RealType b = dist.beta();

      RealType a_2 = a * a;

      RealType n = 6 * (a_2 * a - a_2 * (2 * b - 1) + b * b * (b + 1) - 2 * a * b * (b + 2));

      RealType d = a * b * (a + b + 2) * (a + b + 3);

      return  n / d;

    } // kurtosis_excess

    template <class RealType, class Policy>

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

    {

      return 3 + kurtosis_excess(dist);

    } // kurtosis

    template <class RealType, class Policy>

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

    { // Probability Density/Mass Function.

      BOOST_FPU_EXCEPTION_GUARD

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

      BOOST_MATH_STD_USING // for ADL of std functions

      RealType a = dist.alpha();

      RealType b = dist.beta();

      // Argument checks:

      RealType result = 0;

      if(false == beta_detail::check_dist_and_x(

        function,

        a, b, x,

        &result, Policy()))

      {

        return result;

      }

      using boost::math::beta;

      return ibeta_derivative(a, b, x, Policy());

    } // pdf

    template <class RealType, class Policy>

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

    { // Cumulative Distribution Function beta.

      BOOST_MATH_STD_USING // for ADL of std functions

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

      RealType a = dist.alpha();

      RealType b = dist.beta();

      // Argument checks:

      RealType result = 0;

      if(false == beta_detail::check_dist_and_x(

        function,

        a, b, x,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if (x == 0)

      {

        return 0;

      }

      else if (x == 1)

      {

        return 1;

      }

      return ibeta(a, b, x, Policy());

    } // beta cdf

    template <class RealType, class Policy>

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

    { // Complemented Cumulative Distribution Function beta.

      BOOST_MATH_STD_USING // for ADL of std functions

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

      RealType const& x = c.param;

      beta_distribution<RealType, Policy> const& dist = c.dist;

      RealType a = dist.alpha();

      RealType b = dist.beta();

      // Argument checks:

      RealType result = 0;

      if(false == beta_detail::check_dist_and_x(

        function,

        a, b, x,

        &result, Policy()))

      {

        return result;

      }

      if (x == 0)

      {

        return 1;

      }

      else if (x == 1)

      {

        return 0;

      }

      // Calculate cdf beta using the incomplete beta function.

      // Use of ibeta here prevents cancellation errors in calculating

      // 1 - x if x is very small, perhaps smaller than machine epsilon.

      return ibetac(a, b, x, Policy());

    } // beta cdf

    template <class RealType, class Policy>

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

    { // Quantile or Percent Point beta function or

      // Inverse Cumulative probability distribution function CDF.

      // Return x (0 <= x <= 1),

      // for a given probability p (0 <= p <= 1).

      // These functions take a probability as an argument

      // and return a value such that the probability that a random variable x

      // will be less than or equal to that value

      // is whatever probability you supplied as an argument.

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

      RealType result = 0; // of argument checks:

      RealType a = dist.alpha();

      RealType b = dist.beta();

      if(false == beta_detail::check_dist_and_prob(

        function,

        a, b, p,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if (p == 0)

      {

        return 0;

      }

      if (p == 1)

      {

        return 1;

      }

      return ibeta_inv(a, b, p, static_cast<RealType*>(0), Policy());

    } // quantile

    template <class RealType, class Policy>

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

    { // Complement Quantile or Percent Point beta function .

      // Return the number of expected x for a given

      // complement of the probability q.

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

      //

      // Error checks:

      RealType q = c.param;

      const beta_distribution<RealType, Policy>& dist = c.dist;

      RealType result = 0;

      RealType a = dist.alpha();

      RealType b = dist.beta();

      if(false == beta_detail::check_dist_and_prob(

        function,

        a,

        b,

        q,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if(q == 1)

      {

        return 0;

      }

      if(q == 0)

      {

        return 1;

      }

      return ibetac_inv(a, b, q, static_cast<RealType*>(0), Policy());

    } // Quantile Complement

  } // namespace math

} // namespace boost

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

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

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

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

#if defined (BOOST_MSVC)

# pragma warning(pop)

#endif

#endif // BOOST_MATH_DIST_BETA_HPP