| // 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)
|
|
|
| #ifndef BOOST_STATS_STUDENTS_T_HPP
|
| #define BOOST_STATS_STUDENTS_T_HPP
|
|
|
| // http://en.wikipedia.org/wiki/Student%27s_t_distribution
|
| // http://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm
|
|
|
| #include <boost/math/distributions/fwd.hpp>
|
| #include <boost/math/special_functions/beta.hpp> // for ibeta(a, b, x).
|
| #include <boost/math/distributions/complement.hpp>
|
| #include <boost/math/distributions/detail/common_error_handling.hpp>
|
|
|
| #include <utility>
|
|
|
| #ifdef BOOST_MSVC
|
| # pragma warning(push)
|
| # pragma warning(disable: 4702) // unreachable code (return after domain_error throw).
|
| #endif
|
|
|
| namespace boost{ namespace math{
|
|
|
| template <class RealType = double, class Policy = policies::policy<> >
|
| class students_t_distribution
|
| {
|
| public:
|
| typedef RealType value_type;
|
| typedef Policy policy_type;
|
|
|
| students_t_distribution(RealType i) : m_df(i)
|
| { // Constructor.
|
| RealType result;
|
| detail::check_df(
|
| "boost::math::students_t_distribution<%1%>::students_t_distribution", m_df, &result, Policy());
|
| } // students_t_distribution
|
|
|
| RealType degrees_of_freedom()const
|
| {
|
| return m_df;
|
| }
|
|
|
| // Parameter estimation:
|
| static RealType find_degrees_of_freedom(
|
| RealType difference_from_mean,
|
| RealType alpha,
|
| RealType beta,
|
| RealType sd,
|
| RealType hint = 100);
|
|
|
| private:
|
| //
|
| // Data members:
|
| //
|
| RealType m_df; // degrees of freedom are a real number.
|
| };
|
|
|
| typedef students_t_distribution<double> students_t;
|
|
|
| template <class RealType, class Policy>
|
| inline const std::pair<RealType, RealType> range(const students_t_distribution<RealType, Policy>& /*dist*/)
|
| { // Range of permissible values for random variable x.
|
| using boost::math::tools::max_value;
|
| return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline const std::pair<RealType, RealType> support(const students_t_distribution<RealType, Policy>& /*dist*/)
|
| { // Range of supported values for random variable x.
|
| // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
|
| using boost::math::tools::max_value;
|
| return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType pdf(const students_t_distribution<RealType, Policy>& dist, const RealType& t)
|
| {
|
| BOOST_FPU_EXCEPTION_GUARD
|
| BOOST_MATH_STD_USING // for ADL of std functions
|
|
|
| RealType degrees_of_freedom = dist.degrees_of_freedom();
|
| // Error check:
|
| RealType error_result;
|
| if(false == detail::check_df(
|
| "boost::math::pdf(const students_t_distribution<%1%>&, %1%)", degrees_of_freedom, &error_result, Policy()))
|
| return error_result;
|
| // Might conceivably permit df = +infinity and use normal distribution.
|
| RealType result;
|
| RealType basem1 = t * t / degrees_of_freedom;
|
| if(basem1 < 0.125)
|
| {
|
| result = exp(-boost::math::log1p(basem1, Policy()) * (1+degrees_of_freedom) / 2);
|
| }
|
| else
|
| {
|
| result = pow(1 / (1 + basem1), (degrees_of_freedom + 1) / 2);
|
| }
|
| result /= sqrt(degrees_of_freedom) * boost::math::beta(degrees_of_freedom / 2, RealType(0.5f), Policy());
|
| return result;
|
| } // pdf
|
|
|
| template <class RealType, class Policy>
|
| inline RealType cdf(const students_t_distribution<RealType, Policy>& dist, const RealType& t)
|
| {
|
| RealType degrees_of_freedom = dist.degrees_of_freedom();
|
| // Error check:
|
| RealType error_result;
|
| if(false == detail::check_df(
|
| "boost::math::cdf(const students_t_distribution<%1%>&, %1%)", degrees_of_freedom, &error_result, Policy()))
|
| return error_result;
|
|
|
| if (t == 0)
|
| {
|
| return 0.5;
|
| }
|
| //
|
| // Calculate probability of Student's t using the incomplete beta function.
|
| // probability = ibeta(degrees_of_freedom / 2, 1/2, degrees_of_freedom / (degrees_of_freedom + t*t))
|
| //
|
| // However when t is small compared to the degrees of freedom, that formula
|
| // suffers from rounding error, use the identity formula to work around
|
| // the problem:
|
| //
|
| // I[x](a,b) = 1 - I[1-x](b,a)
|
| //
|
| // and:
|
| //
|
| // x = df / (df + t^2)
|
| //
|
| // so:
|
| //
|
| // 1 - x = t^2 / (df + t^2)
|
| //
|
| RealType t2 = t * t;
|
| RealType probability;
|
| if(degrees_of_freedom > 2 * t2)
|
| {
|
| RealType z = t2 / (degrees_of_freedom + t2);
|
| probability = ibetac(static_cast<RealType>(0.5), degrees_of_freedom / 2, z, Policy()) / 2;
|
| }
|
| else
|
| {
|
| RealType z = degrees_of_freedom / (degrees_of_freedom + t2);
|
| probability = ibeta(degrees_of_freedom / 2, static_cast<RealType>(0.5), z, Policy()) / 2;
|
| }
|
| return (t > 0 ? 1 - probability : probability);
|
| } // cdf
|
|
|
| template <class RealType, class Policy>
|
| inline RealType quantile(const students_t_distribution<RealType, Policy>& dist, const RealType& p)
|
| {
|
| BOOST_MATH_STD_USING // for ADL of std functions
|
| //
|
| // Obtain parameters:
|
| //
|
| RealType degrees_of_freedom = dist.degrees_of_freedom();
|
| RealType probability = p;
|
| //
|
| // Check for domain errors:
|
| //
|
| static const char* function = "boost::math::quantile(const students_t_distribution<%1%>&, %1%)";
|
| RealType error_result;
|
| if(false == detail::check_df(
|
| function, degrees_of_freedom, &error_result, Policy())
|
| && detail::check_probability(function, probability, &error_result, Policy()))
|
| return error_result;
|
|
|
| // Special cases, regardless of degrees_of_freedom.
|
| if (probability == 0)
|
| return -policies::raise_overflow_error<RealType>(function, 0, Policy());
|
| if (probability == 1)
|
| return policies::raise_overflow_error<RealType>(function, 0, Policy());
|
| if (probability == static_cast<RealType>(0.5))
|
| return 0;
|
| //
|
| // This next block is disabled in favour of a faster method than
|
| // incomplete beta inverse, code retained for future reference:
|
| //
|
| #if 0
|
| //
|
| // Calculate quantile of Student's t using the incomplete beta function inverse:
|
| //
|
| probability = (probability > 0.5) ? 1 - probability : probability;
|
| RealType t, x, y;
|
| x = ibeta_inv(degrees_of_freedom / 2, RealType(0.5), 2 * probability, &y);
|
| if(degrees_of_freedom * y > tools::max_value<RealType>() * x)
|
| t = tools::overflow_error<RealType>(function);
|
| else
|
| t = sqrt(degrees_of_freedom * y / x);
|
| //
|
| // Figure out sign based on the size of p:
|
| //
|
| if(p < 0.5)
|
| t = -t;
|
|
|
| return t;
|
| #endif
|
| //
|
| // Depending on how many digits RealType has, this may forward
|
| // to the incomplete beta inverse as above. Otherwise uses a
|
| // faster method that is accurate to ~15 digits everywhere
|
| // and a couple of epsilon at double precision and in the central
|
| // region where most use cases will occur...
|
| //
|
| return boost::math::detail::fast_students_t_quantile(degrees_of_freedom, probability, Policy());
|
| } // quantile
|
|
|
| template <class RealType, class Policy>
|
| inline RealType cdf(const complemented2_type<students_t_distribution<RealType, Policy>, RealType>& c)
|
| {
|
| return cdf(c.dist, -c.param);
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType quantile(const complemented2_type<students_t_distribution<RealType, Policy>, RealType>& c)
|
| {
|
| return -quantile(c.dist, c.param);
|
| }
|
|
|
| //
|
| // Parameter estimation follows:
|
| //
|
| namespace detail{
|
| //
|
| // Functors for finding degrees of freedom:
|
| //
|
| template <class RealType, class Policy>
|
| struct sample_size_func
|
| {
|
| sample_size_func(RealType a, RealType b, RealType s, RealType d)
|
| : alpha(a), beta(b), ratio(s*s/(d*d)) {}
|
|
|
| RealType operator()(const RealType& df)
|
| {
|
| if(df <= tools::min_value<RealType>())
|
| return 1;
|
| students_t_distribution<RealType, Policy> t(df);
|
| RealType qa = quantile(complement(t, alpha));
|
| RealType qb = quantile(complement(t, beta));
|
| qa += qb;
|
| qa *= qa;
|
| qa *= ratio;
|
| qa -= (df + 1);
|
| return qa;
|
| }
|
| RealType alpha, beta, ratio;
|
| };
|
|
|
| } // namespace detail
|
|
|
| template <class RealType, class Policy>
|
| RealType students_t_distribution<RealType, Policy>::find_degrees_of_freedom(
|
| RealType difference_from_mean,
|
| RealType alpha,
|
| RealType beta,
|
| RealType sd,
|
| RealType hint)
|
| {
|
| static const char* function = "boost::math::students_t_distribution<%1%>::find_degrees_of_freedom";
|
| //
|
| // Check for domain errors:
|
| //
|
| RealType error_result;
|
| if(false == detail::check_probability(
|
| function, alpha, &error_result, Policy())
|
| && detail::check_probability(function, beta, &error_result, Policy()))
|
| return error_result;
|
|
|
| if(hint <= 0)
|
| hint = 1;
|
|
|
| detail::sample_size_func<RealType, Policy> f(alpha, beta, sd, difference_from_mean);
|
| tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>());
|
| boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>();
|
| std::pair<RealType, RealType> r = tools::bracket_and_solve_root(f, hint, RealType(2), false, tol, max_iter, Policy());
|
| RealType result = r.first + (r.second - r.first) / 2;
|
| if(max_iter >= policies::get_max_root_iterations<Policy>())
|
| {
|
| policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:"
|
| " either there is no answer to how many degrees of freedom are required"
|
| " or the answer is infinite. Current best guess is %1%", result, Policy());
|
| }
|
| return result;
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType mean(const students_t_distribution<RealType, Policy>& )
|
| {
|
| return 0;
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType variance(const students_t_distribution<RealType, Policy>& dist)
|
| {
|
| // Error check:
|
| RealType error_result;
|
| if(false == detail::check_df(
|
| "boost::math::variance(students_t_distribution<%1%> const&, %1%)", dist.degrees_of_freedom(), &error_result, Policy()))
|
| return error_result;
|
|
|
| RealType v = dist.degrees_of_freedom();
|
| return v / (v - 2);
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType mode(const students_t_distribution<RealType, Policy>& /*dist*/)
|
| {
|
| return 0;
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType median(const students_t_distribution<RealType, Policy>& /*dist*/)
|
| {
|
| return 0;
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType skewness(const students_t_distribution<RealType, Policy>& dist)
|
| {
|
| if(dist.degrees_of_freedom() <= 3)
|
| {
|
| policies::raise_domain_error<RealType>(
|
| "boost::math::skewness(students_t_distribution<%1%> const&, %1%)",
|
| "Skewness is undefined for degrees of freedom <= 3, but got %1%.",
|
| dist.degrees_of_freedom(), Policy());
|
| }
|
| return 0;
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType kurtosis(const students_t_distribution<RealType, Policy>& dist)
|
| {
|
| RealType df = dist.degrees_of_freedom();
|
| if(df <= 3)
|
| {
|
| policies::raise_domain_error<RealType>(
|
| "boost::math::kurtosis(students_t_distribution<%1%> const&, %1%)",
|
| "Skewness is undefined for degrees of freedom <= 3, but got %1%.",
|
| df, Policy());
|
| }
|
| return 3 * (df - 2) / (df - 4);
|
| }
|
|
|
| template <class RealType, class Policy>
|
| inline RealType kurtosis_excess(const students_t_distribution<RealType, Policy>& dist)
|
| {
|
| // see http://mathworld.wolfram.com/Kurtosis.html
|
| RealType df = dist.degrees_of_freedom();
|
| if(df <= 3)
|
| {
|
| policies::raise_domain_error<RealType>(
|
| "boost::math::kurtosis_excess(students_t_distribution<%1%> const&, %1%)",
|
| "Skewness is undefined for degrees of freedom <= 3, but got %1%.",
|
| df, Policy());
|
| }
|
| return 6 / (df - 4);
|
| }
|
|
|
| } // namespace math
|
| } // namespace boost
|
|
|
| #ifdef BOOST_MSVC
|
| # pragma warning(pop)
|
| #endif
|
|
|
| // This include must be at the end, *after* the accessors
|
| // for this distribution have been defined, in order to
|
| // keep compilers that support two-phase lookup happy.
|
| #include <boost/math/distributions/detail/derived_accessors.hpp>
|
|
|
| #endif // BOOST_STATS_STUDENTS_T_HPP
|
