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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;
- } // arcsine 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
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