// boost/math/distributions/arcsine.hpp
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// Copyright John Maddock 2014.
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// Copyright Paul A. Bristow 2014.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// http://en.wikipedia.org/wiki/arcsine_distribution
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// The arcsine Distribution is a continuous probability distribution.
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// http://en.wikipedia.org/wiki/Arcsine_distribution
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// http://www.wolframalpha.com/input/?i=ArcSinDistribution
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// Standard arcsine distribution is a special case of beta distribution with both a & b = one half,
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// and 0 <= x <= 1.
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// It is generalized to include any bounded support a <= x <= b from 0 <= x <= 1
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// by Wolfram and Wikipedia,
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// but using location and scale parameters by
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// Virtual Laboratories in Probability and Statistics http://www.math.uah.edu/stat/index.html
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// http://www.math.uah.edu/stat/special/Arcsine.html
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// The end-point version is simpler and more obvious, so we implement that.
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// TODO Perhaps provide location and scale functions?
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#ifndef BOOST_MATH_DIST_ARCSINE_HPP
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#define BOOST_MATH_DIST_ARCSINE_HPP
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#include <boost/math/distributions/fwd.hpp>
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#include <boost/math/distributions/complement.hpp> // complements.
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#include <boost/math/distributions/detail/common_error_handling.hpp> // error checks.
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#include <boost/math/constants/constants.hpp>
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#include <boost/math/special_functions/fpclassify.hpp> // isnan.
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#if defined (BOOST_MSVC)
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# pragma warning(push)
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# pragma warning(disable: 4702) // Unreachable code,
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// in domain_error_imp in error_handling.
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#endif
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#include <utility>
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#include <exception> // For std::domain_error.
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namespace boost
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{
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namespace math
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{
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namespace arcsine_detail
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{
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// Common error checking routines for arcsine distribution functions:
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// Duplicating for x_min and x_max provides specific error messages.
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template <class RealType, class Policy>
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inline bool check_x_min(const char* function, const RealType& x, RealType* result, const Policy& pol)
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{
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if (!(boost::math::isfinite)(x))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"x_min argument is %1%, but must be finite !", x, pol);
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return false;
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}
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return true;
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} // bool check_x_min
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template <class RealType, class Policy>
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inline bool check_x_max(const char* function, const RealType& x, RealType* result, const Policy& pol)
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{
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if (!(boost::math::isfinite)(x))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"x_max argument is %1%, but must be finite !", x, pol);
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return false;
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}
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return true;
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} // bool check_x_max
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template <class RealType, class Policy>
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inline bool check_x_minmax(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)
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{ // Check x_min < x_max
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if (x_min >= x_max)
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{
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std::string msg = "x_max argument is %1%, but must be > x_min = " + lexical_cast<std::string>(x_min) + "!";
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*result = policies::raise_domain_error<RealType>(
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function,
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msg.c_str(), x_max, pol);
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// "x_max argument is %1%, but must be > x_min !", x_max, pol);
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// "x_max argument is %1%, but must be > x_min %2!", x_max, x_min, pol); would be better.
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// But would require replication of all helpers functions in /policies/error_handling.hpp for two values,
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// as well as two value versions of raise_error, raise_domain_error and do_format ...
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// so use slightly hacky lexical_cast to string instead.
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return false;
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}
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return true;
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} // bool check_x_minmax
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template <class RealType, class Policy>
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inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
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{
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if ((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
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return false;
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}
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return true;
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} // bool check_prob
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template <class RealType, class Policy>
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inline bool check_x(const char* function, const RealType& x_min, const RealType& x_max, const RealType& x, RealType* result, const Policy& pol)
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{ // Check x finite and x_min < x < x_max.
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if (!(boost::math::isfinite)(x))
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{
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*result = policies::raise_domain_error<RealType>(
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function,
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"x argument is %1%, but must be finite !", x, pol);
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return false;
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}
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if ((x < x_min) || (x > x_max))
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{
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// std::cout << x_min << ' ' << x << x_max << std::endl;
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*result = policies::raise_domain_error<RealType>(
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function,
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"x argument is %1%, but must be x_min < x < x_max !", x, pol);
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// For example:
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// Error in function boost::math::pdf(arcsine_distribution<double> const&, double) : x argument is -1.01, but must be x_min < x < x_max !
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// TODO Perhaps show values of x_min and x_max?
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return false;
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}
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return true;
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} // bool check_x
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template <class RealType, class Policy>
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inline bool check_dist(const char* function, const RealType& x_min, const RealType& x_max, RealType* result, const Policy& pol)
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{ // Check both x_min and x_max finite, and x_min < x_max.
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return check_x_min(function, x_min, result, pol)
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&& check_x_max(function, x_max, result, pol)
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&& check_x_minmax(function, x_min, x_max, result, pol);
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} // bool check_dist
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template <class RealType, class Policy>
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inline bool check_dist_and_x(const char* function, const RealType& x_min, const RealType& x_max, RealType x, RealType* result, const Policy& pol)
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{
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return check_dist(function, x_min, x_max, result, pol)
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&& arcsine_detail::check_x(function, x_min, x_max, x, result, pol);
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} // bool check_dist_and_x
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template <class RealType, class Policy>
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inline bool check_dist_and_prob(const char* function, const RealType& x_min, const RealType& x_max, RealType p, RealType* result, const Policy& pol)
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{
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return check_dist(function, x_min, x_max, result, pol)
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&& check_prob(function, p, result, pol);
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} // bool check_dist_and_prob
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} // namespace arcsine_detail
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template <class RealType = double, class Policy = policies::policy<> >
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class arcsine_distribution
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{
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public:
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typedef RealType value_type;
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typedef Policy policy_type;
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arcsine_distribution(RealType x_min = 0, RealType x_max = 1) : m_x_min(x_min), m_x_max(x_max)
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{ // Default beta (alpha = beta = 0.5) is standard arcsine with x_min = 0, x_max = 1.
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// Generalized to allow x_min and x_max to be specified.
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RealType result;
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arcsine_detail::check_dist(
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"boost::math::arcsine_distribution<%1%>::arcsine_distribution",
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m_x_min,
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m_x_max,
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&result, Policy());
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} // arcsine_distribution constructor.
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// Accessor functions:
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RealType x_min() const
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{
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return m_x_min;
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}
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RealType x_max() const
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{
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return m_x_max;
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}
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private:
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RealType m_x_min; // Two x min and x max parameters of the arcsine distribution.
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RealType m_x_max;
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}; // template <class RealType, class Policy> class arcsine_distribution
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// Convenient typedef to construct double version.
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typedef arcsine_distribution<double> arcsine;
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> range(const arcsine_distribution<RealType, Policy>& dist)
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{ // Range of permissible values for random variable x.
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using boost::math::tools::max_value;
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return std::pair<RealType, RealType>(static_cast<RealType>(dist.x_min()), static_cast<RealType>(dist.x_max()));
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}
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template <class RealType, class Policy>
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inline const std::pair<RealType, RealType> support(const arcsine_distribution<RealType, Policy>& dist)
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{ // Range of supported values for random variable x.
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// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
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return std::pair<RealType, RealType>(static_cast<RealType>(dist.x_min()), static_cast<RealType>(dist.x_max()));
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}
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template <class RealType, class Policy>
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inline RealType mean(const arcsine_distribution<RealType, Policy>& dist)
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{ // Mean of arcsine distribution .
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RealType result;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist(
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"boost::math::mean(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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return (x_min + x_max) / 2;
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} // mean
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template <class RealType, class Policy>
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inline RealType variance(const arcsine_distribution<RealType, Policy>& dist)
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{ // Variance of standard arcsine distribution = (1-0)/8 = 0.125.
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RealType result;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist(
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"boost::math::variance(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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return (x_max - x_min) * (x_max - x_min) / 8;
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} // variance
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template <class RealType, class Policy>
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inline RealType mode(const arcsine_distribution<RealType, Policy>& /* dist */)
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{ //There are always [*two] values for the mode, at ['x_min] and at ['x_max], default 0 and 1,
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// so instead we raise the exception domain_error.
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return policies::raise_domain_error<RealType>(
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"boost::math::mode(arcsine_distribution<%1%>&)",
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"The arcsine distribution has two modes at x_min and x_max: "
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"so the return value is %1%.",
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std::numeric_limits<RealType>::quiet_NaN(), Policy());
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} // mode
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template <class RealType, class Policy>
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inline RealType median(const arcsine_distribution<RealType, Policy>& dist)
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{ // Median of arcsine distribution (a + b) / 2 == mean.
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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RealType result;
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if (false == arcsine_detail::check_dist(
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"boost::math::median(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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return (x_min + x_max) / 2;
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}
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template <class RealType, class Policy>
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inline RealType skewness(const arcsine_distribution<RealType, Policy>& dist)
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{
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RealType result;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist(
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"boost::math::skewness(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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return 0;
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} // skewness
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template <class RealType, class Policy>
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inline RealType kurtosis_excess(const arcsine_distribution<RealType, Policy>& dist)
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{
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RealType result;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist(
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"boost::math::kurtosis_excess(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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result = -3;
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return result / 2;
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} // kurtosis_excess
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template <class RealType, class Policy>
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inline RealType kurtosis(const arcsine_distribution<RealType, Policy>& dist)
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{
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RealType result;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist(
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"boost::math::kurtosis(arcsine_distribution<%1%> const&, %1% )",
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x_min,
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x_max,
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&result, Policy())
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)
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{
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return result;
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}
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return 3 + kurtosis_excess(dist);
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} // kurtosis
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template <class RealType, class Policy>
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inline RealType pdf(const arcsine_distribution<RealType, Policy>& dist, const RealType& xx)
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{ // Probability Density/Mass Function arcsine.
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BOOST_FPU_EXCEPTION_GUARD
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BOOST_MATH_STD_USING // For ADL of std functions.
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static const char* function = "boost::math::pdf(arcsine_distribution<%1%> const&, %1%)";
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RealType lo = dist.x_min();
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RealType hi = dist.x_max();
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RealType x = xx;
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// Argument checks:
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RealType result = 0;
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if (false == arcsine_detail::check_dist_and_x(
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function,
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lo, hi, x,
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&result, Policy()))
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{
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return result;
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}
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using boost::math::constants::pi;
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result = static_cast<RealType>(1) / (pi<RealType>() * sqrt((x - lo) * (hi - x)));
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return result;
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} // pdf
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template <class RealType, class Policy>
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inline RealType cdf(const arcsine_distribution<RealType, Policy>& dist, const RealType& x)
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{ // Cumulative Distribution Function arcsine.
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BOOST_MATH_STD_USING // For ADL of std functions.
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static const char* function = "boost::math::cdf(arcsine_distribution<%1%> const&, %1%)";
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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// Argument checks:
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RealType result = 0;
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if (false == arcsine_detail::check_dist_and_x(
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function,
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x_min, x_max, x,
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&result, Policy()))
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{
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return result;
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}
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// Special cases:
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if (x == x_min)
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{
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return 0;
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}
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else if (x == x_max)
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{
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return 1;
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}
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using boost::math::constants::pi;
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result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
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return result;
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} // arcsine cdf
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template <class RealType, class Policy>
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inline RealType cdf(const complemented2_type<arcsine_distribution<RealType, Policy>, RealType>& c)
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{ // Complemented Cumulative Distribution Function arcsine.
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BOOST_MATH_STD_USING // For ADL of std functions.
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static const char* function = "boost::math::cdf(arcsine_distribution<%1%> const&, %1%)";
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RealType x = c.param;
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arcsine_distribution<RealType, Policy> const& dist = c.dist;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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// Argument checks:
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RealType result = 0;
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if (false == arcsine_detail::check_dist_and_x(
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function,
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x_min, x_max, x,
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&result, Policy()))
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{
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return result;
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}
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if (x == x_min)
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{
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return 0;
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}
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else if (x == x_max)
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{
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return 1;
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}
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using boost::math::constants::pi;
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// Naive version x = 1 - x;
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// result = static_cast<RealType>(2) * asin(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
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// is less accurate, so use acos instead of asin for complement.
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result = static_cast<RealType>(2) * acos(sqrt((x - x_min) / (x_max - x_min))) / pi<RealType>();
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return result;
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} // arcsine ccdf
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template <class RealType, class Policy>
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inline RealType quantile(const arcsine_distribution<RealType, Policy>& dist, const RealType& p)
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{
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// Quantile or Percent Point arcsine function or
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// Inverse Cumulative probability distribution function CDF.
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// Return x (0 <= x <= 1),
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// for a given probability p (0 <= p <= 1).
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// These functions take a probability as an argument
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// and return a value such that the probability that a random variable x
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// will be less than or equal to that value
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// is whatever probability you supplied as an argument.
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BOOST_MATH_STD_USING // For ADL of std functions.
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using boost::math::constants::half_pi;
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static const char* function = "boost::math::quantile(arcsine_distribution<%1%> const&, %1%)";
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RealType result = 0; // of argument checks:
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist_and_prob(
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function,
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x_min, x_max, p,
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&result, Policy()))
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{
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return result;
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}
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// Special cases:
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if (p == 0)
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{
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return 0;
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}
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if (p == 1)
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{
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return 1;
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}
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RealType sin2hpip = sin(half_pi<RealType>() * p);
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RealType sin2hpip2 = sin2hpip * sin2hpip;
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result = -x_min * sin2hpip2 + x_min + x_max * sin2hpip2;
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return result;
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} // quantile
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template <class RealType, class Policy>
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inline RealType quantile(const complemented2_type<arcsine_distribution<RealType, Policy>, RealType>& c)
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{
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// Complement Quantile or Percent Point arcsine function.
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// Return the number of expected x for a given
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// complement of the probability q.
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BOOST_MATH_STD_USING // For ADL of std functions.
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using boost::math::constants::half_pi;
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static const char* function = "boost::math::quantile(arcsine_distribution<%1%> const&, %1%)";
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// Error checks:
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RealType q = c.param;
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const arcsine_distribution<RealType, Policy>& dist = c.dist;
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RealType result = 0;
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RealType x_min = dist.x_min();
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RealType x_max = dist.x_max();
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if (false == arcsine_detail::check_dist_and_prob(
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function,
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x_min,
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x_max,
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q,
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&result, Policy()))
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{
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return result;
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}
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// Special cases:
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if (q == 1)
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{
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return 0;
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}
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if (q == 0)
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{
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return 1;
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}
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// Naive RealType p = 1 - q; result = sin(half_pi<RealType>() * p); loses accuracy, so use a cos alternative instead.
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//result = cos(half_pi<RealType>() * q); // for arcsine(0,1)
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//result = result * result;
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// For generalized arcsine:
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RealType cos2hpip = cos(half_pi<RealType>() * q);
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RealType cos2hpip2 = cos2hpip * cos2hpip;
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result = -x_min * cos2hpip2 + x_min + x_max * cos2hpip2;
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return result;
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} // Quantile Complement
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} // namespace math
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} // namespace boost
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// This include must be at the end, *after* the accessors
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// for this distribution have been defined, in order to
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// keep compilers that support two-phase lookup happy.
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#include <boost/math/distributions/detail/derived_accessors.hpp>
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#if defined (BOOST_MSVC)
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# pragma warning(pop)
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#endif
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#endif // BOOST_MATH_DIST_ARCSINE_HPP
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