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// boost/math/distributions/arcsine.hpp

// Copyright John Maddock 2014.

// Copyright Paul A. Bristow 2014.

// 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/arcsine_distribution

// The arcsine Distribution is a continuous probability distribution.

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

// http://www.wolframalpha.com/input/?i=ArcSinDistribution

// Standard arcsine distribution is a special case of beta distribution with both a & b = one half,

// and 0 <= x <= 1.

// It is generalized to include any bounded support a <= x <= b from 0 <= x <= 1

// by Wolfram and Wikipedia,

// but using location and scale parameters by

// Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/index.html

// http://www.math.uah.edu/stat/special/Arcsine.html

// The end-point version is simpler and more obvious, so we implement that.

// TODO Perhaps provide location and scale functions?

#ifndef BOOST_MATH_DIST_ARCSINE_HPP

#define BOOST_MATH_DIST_ARCSINE_HPP

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

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

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

#include <boost/math/constants/constants.hpp>

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

#if defined (BOOST_MSVC)

#  pragma warning(push)

#  pragma warning(disable: 4702) // Unreachable code,

// in domain_error_imp in error_handling.

#endif

#include <utility>

#include <exception>  // For std::domain_error.

namespace boost

{

  namespace math

  {

    namespace arcsine_detail

    {

      // Common error checking routines for arcsine distribution functions:

      // Duplicating for x_min and x_max provides specific error messages.

      template <class RealType, class Policy>

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

      {

        if (!(boost::math::isfinite)(x))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "x_min argument is %1%, but must be finite !", x, pol);

          return false;

        }

        return true;

      } // bool check_x_min

      template <class RealType, class Policy>

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

      {

        if (!(boost::math::isfinite)(x))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "x_max argument is %1%, but must be finite !", x, pol);

          return false;

        }

        return true;

      } // bool check_x_max

      template <class RealType, class Policy>

      inline bool check_x_minmax(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)

      { // Check x_min < x_max

        if (x_min >= x_max)

        {

          std::string msg = "x_max argument is %1%, but must be > x_min = " + lexical_cast<std::string>(x_min) + "!";

          *result = policies::raise_domain_error<RealType>(

            function,

            msg.c_str(), x_max, pol);

           // "x_max argument is %1%, but must be > x_min !", x_max, pol);

            //  "x_max argument is %1%, but must be > x_min %2!", x_max, x_min, pol); would be better. 

          // But would require replication of all helpers functions in /policies/error_handling.hpp for two values,

          // as well as two value versions of raise_error, raise_domain_error and do_format ...

          // so use slightly hacky lexical_cast to string instead.

          return false;

        }

        return true;

      } // bool check_x_minmax

      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_min, const RealType& x_max, const RealType& x, RealType* result, const Policy& pol)

      { // Check x finite and x_min < x < x_max.

        if (!(boost::math::isfinite)(x))

        {

          *result = policies::raise_domain_error<RealType>(

            function,

            "x argument is %1%, but must be finite !", x, pol);

          return false;

        }

        if ((x < x_min) || (x > x_max))

        {

          // std::cout << x_min << ' ' << x << x_max << std::endl;

          *result = policies::raise_domain_error<RealType>(

            function,

            "x argument is %1%, but must be x_min < x < x_max !", x, pol);

          // For example:

          // Error in function boost::math::pdf(arcsine_distribution<double> const&, double) : x argument is -1.01, but must be x_min < x < x_max !

          // TODO Perhaps show values of x_min and x_max?

          return false;

        }

        return true;

      } // bool check_x

      template <class RealType, class Policy>

      inline bool check_dist(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)

      { // Check both x_min and x_max finite, and x_min  < x_max.

        return check_x_min(function, x_min, result, pol)

            && check_x_max(function, x_max, result, pol)

            && check_x_minmax(function, x_min, x_max, result, pol);

      } // bool check_dist

      template <class RealType, class Policy>

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

      {

        return check_dist(function, x_min, x_max, result, pol)

          && arcsine_detail::check_x(function, x_min, x_max, x, result, pol);

      } // bool check_dist_and_x

      template <class RealType, class Policy>

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

      {

        return check_dist(function, x_min, x_max, result, pol)

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

      } // bool check_dist_and_prob

    } // namespace arcsine_detail

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

    class arcsine_distribution

    {

    public:

      typedef RealType value_type;

      typedef Policy policy_type;

      arcsine_distribution(RealType x_min = 0, RealType x_max = 1) : m_x_min(x_min), m_x_max(x_max)

      { // Default beta (alpha = beta = 0.5) is standard arcsine with x_min = 0, x_max = 1.

        // Generalized to allow x_min and x_max to be specified.

        RealType result;

        arcsine_detail::check_dist(

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

          m_x_min,

          m_x_max,

          &result, Policy());

      } // arcsine_distribution constructor.

      // Accessor functions:

      RealType x_min() const

      {

        return m_x_min;

      }

      RealType x_max() const

      {

        return m_x_max;

      }

    private:

      RealType m_x_min; // Two x min and x max parameters of the arcsine distribution.

      RealType m_x_max;

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

    // Convenient typedef to construct double version.

    typedef arcsine_distribution<double> arcsine;

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> range(const arcsine_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>(dist.x_min()), static_cast<RealType>(dist.x_max()));

    }

    template <class RealType, class Policy>

    inline const std::pair<RealType, RealType> support(const arcsine_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>(dist.x_min()), static_cast<RealType>(dist.x_max()));

    }

    template <class RealType, class Policy>

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

    { // Mean of arcsine distribution .

      RealType result;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist(

        "boost::math::mean(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      return  (x_min + x_max) / 2;

    } // mean

    template <class RealType, class Policy>

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

    { // Variance of standard arcsine distribution = (1-0)/8 = 0.125.

      RealType result;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist(

        "boost::math::variance(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      return  (x_max - x_min) * (x_max - x_min) / 8;

    } // variance

    template <class RealType, class Policy>

    inline RealType mode(const arcsine_distribution<RealType, Policy>& /* dist */)

    { //There are always [*two] values for the mode, at ['x_min] and at ['x_max], default 0 and 1,

      // so instead we raise the exception domain_error.

      return policies::raise_domain_error<RealType>(

        "boost::math::mode(arcsine_distribution<%1%>&)",

        "The arcsine distribution has two modes at x_min and x_max: "

        "so the return value is %1%.",

        std::numeric_limits<RealType>::quiet_NaN(), Policy());

    } // mode

    template <class RealType, class Policy>

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

    { // Median of arcsine distribution (a + b) / 2 == mean.

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      RealType result;

      if (false == arcsine_detail::check_dist(

        "boost::math::median(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      return  (x_min + x_max) / 2;

    }

    template <class RealType, class Policy>

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

    {

      RealType result;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist(

        "boost::math::skewness(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      return 0;

    } // skewness

    template <class RealType, class Policy>

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

    {

      RealType result;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist(

        "boost::math::kurtosis_excess(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      result = -3;

      return  result / 2;

    } // kurtosis_excess

    template <class RealType, class Policy>

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

    {

      RealType result;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist(

        "boost::math::kurtosis(arcsine_distribution<%1%> const&, %1% )",

        x_min,

        x_max,

        &result, Policy())

        )

      {

        return result;

      }

      return 3 + kurtosis_excess(dist);

    } // kurtosis

    template <class RealType, class Policy>

    inline RealType pdf(const arcsine_distribution<RealType, Policy>& dist, const RealType& xx)

    { // Probability Density/Mass Function arcsine.

      BOOST_FPU_EXCEPTION_GUARD

      BOOST_MATH_STD_USING // For ADL of std functions.

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

      RealType lo = dist.x_min();

      RealType hi = dist.x_max();

      RealType x = xx;

      // Argument checks:

      RealType result = 0; 

      if (false == arcsine_detail::check_dist_and_x(

        function,

        lo, hi, x,

        &result, Policy()))

      {

        return result;

      }

      using boost::math::constants::pi;

      result = static_cast<RealType>(1) / (pi<RealType>() * sqrt((x - lo) * (hi - x)));

      return result;

    } // pdf

    template <class RealType, class Policy>

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

    { // Cumulative Distribution Function arcsine.

      BOOST_MATH_STD_USING // For ADL of std functions.

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

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      // Argument checks:

      RealType result = 0;

      if (false == arcsine_detail::check_dist_and_x(

        function,

        x_min, x_max, x,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if (x == x_min)

      {

        return 0;

      }

      else if (x == x_max)

      {

        return 1;

      }

      using boost::math::constants::pi;

      result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();

      return result;

    } // arcsine cdf

    template <class RealType, class Policy>

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

    { // Complemented Cumulative Distribution Function arcsine.

      BOOST_MATH_STD_USING // For ADL of std functions.

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

      RealType x = c.param;

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

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      // Argument checks:

      RealType result = 0;

      if (false == arcsine_detail::check_dist_and_x(

        function,

        x_min, x_max, x,

        &result, Policy()))

      {

        return result;

      }

      if (x == x_min)

      {

        return 0;

      }

      else if (x == x_max)

      {

        return 1;

      }

      using boost::math::constants::pi;

      // Naive version x = 1 - x;

      // result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();

      // is less accurate, so use acos instead of asin for complement.

      result = static_cast<RealType>(2) * acos(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();

      return result;

    } // arcine ccdf

    template <class RealType, class Policy>

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

    { 

      // Quantile or Percent Point arcsine 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.

      BOOST_MATH_STD_USING // For ADL of std functions.

      using boost::math::constants::half_pi;

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

      RealType result = 0; // of argument checks:

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist_and_prob(

        function,

        x_min, x_max, p,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if (p == 0)

      {

        return 0;

      }

      if (p == 1)

      {

        return 1;

      }

      RealType sin2hpip = sin(half_pi<RealType>() * p);

      RealType sin2hpip2 = sin2hpip * sin2hpip;

      result = -x_min * sin2hpip2 + x_min + x_max * sin2hpip2;

      return result;

    } // quantile

    template <class RealType, class Policy>

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

    { 

      // Complement Quantile or Percent Point arcsine function.

      // Return the number of expected x for a given

      // complement of the probability q.

      BOOST_MATH_STD_USING // For ADL of std functions.

      using boost::math::constants::half_pi;

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

      // Error checks:

      RealType q = c.param;

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

      RealType result = 0;

      RealType x_min = dist.x_min();

      RealType x_max = dist.x_max();

      if (false == arcsine_detail::check_dist_and_prob(

        function,

        x_min,

        x_max,

        q,

        &result, Policy()))

      {

        return result;

      }

      // Special cases:

      if (q == 1)

      {

        return 0;

      }

      if (q == 0)

      {

        return 1;

      }

      // Naive RealType p = 1 - q; result = sin(half_pi<RealType>() * p); loses accuracy, so use a cos alternative instead.

      //result = cos(half_pi<RealType>() * q); // for arcsine(0,1)

      //result = result * result;

      // For generalized arcsine:

      RealType cos2hpip = cos(half_pi<RealType>() * q);

      RealType cos2hpip2 = cos2hpip * cos2hpip;

      result = -x_min * cos2hpip2 + x_min + x_max * cos2hpip2;

      return result;

    } // 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_ARCSINE_HPP