--- /dev/null
+// Copyright John Maddock 2006.
+// Copyright Paul A. Bristow 2006, 2012, 2017.
+// Copyright Thomas Mang 2012.
+
+// 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 <boost/math/distributions/normal.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 df) : df_(df)
+ { // Constructor.
+ RealType result;
+ detail::check_df_gt0_to_inf( // Checks that df > 0 or df == inf.
+ "boost::math::students_t_distribution<%1%>::students_t_distribution", df_, &result, Policy());
+ } // students_t_distribution
+
+ RealType degrees_of_freedom()const
+ {
+ return df_;
+ }
+
+ // Parameter estimation:
+ static RealType find_degrees_of_freedom(
+ RealType difference_from_mean,
+ RealType alpha,
+ RealType beta,
+ RealType sd,
+ RealType hint = 100);
+
+private:
+ // Data member:
+ RealType df_; // degrees of freedom is a real number > 0 or +infinity.
+};
+
+typedef students_t_distribution<double> students_t; // Convenience typedef for double version.
+
+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.
+ // Now including infinity.
+ using boost::math::tools::max_value;
+ //return std::pair<RealType, RealType>(-max_value<RealType>(), max_value<RealType>());
+ return std::pair<RealType, RealType>(((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>()), ((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? +std::numeric_limits<RealType>::infinity() : +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.
+ // Now including infinity.
+ // 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>());
+ return std::pair<RealType, RealType>(((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? -std::numeric_limits<RealType>::infinity() : -max_value<RealType>()), ((::std::numeric_limits<RealType>::is_specialized & ::std::numeric_limits<RealType>::has_infinity) ? +std::numeric_limits<RealType>::infinity() : +max_value<RealType>()));
+}
+
+template <class RealType, class Policy>
+inline RealType pdf(const students_t_distribution<RealType, Policy>& dist, const RealType& x)
+{
+ BOOST_FPU_EXCEPTION_GUARD
+ BOOST_MATH_STD_USING // for ADL of std functions.
+
+ RealType error_result;
+ if(false == detail::check_x_not_NaN(
+ "boost::math::pdf(const students_t_distribution<%1%>&, %1%)", x, &error_result, Policy()))
+ return error_result;
+ RealType df = dist.degrees_of_freedom();
+ if(false == detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
+ "boost::math::pdf(const students_t_distribution<%1%>&, %1%)", df, &error_result, Policy()))
+ return error_result;
+
+ RealType result;
+ if ((boost::math::isinf)(x))
+ { // - or +infinity.
+ result = static_cast<RealType>(0);
+ return result;
+ }
+ RealType limit = policies::get_epsilon<RealType, Policy>();
+ // Use policies so that if policy requests lower precision,
+ // then get the normal distribution approximation earlier.
+ limit = static_cast<RealType>(1) / limit; // 1/eps
+ // for 64-bit double 1/eps = 4503599627370496
+ if (df > limit)
+ { // Special case for really big degrees_of_freedom > 1 / eps
+ // - use normal distribution which is much faster and more accurate.
+ normal_distribution<RealType, Policy> n(0, 1);
+ result = pdf(n, x);
+ }
+ else
+ { //
+ RealType basem1 = x * x / df;
+ if(basem1 < 0.125)
+ {
+ result = exp(-boost::math::log1p(basem1, Policy()) * (1+df) / 2);
+ }
+ else
+ {
+ result = pow(1 / (1 + basem1), (df + 1) / 2);
+ }
+ result /= sqrt(df) * boost::math::beta(df / 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& x)
+{
+ RealType error_result;
+ // degrees_of_freedom > 0 or infinity check:
+ RealType df = dist.degrees_of_freedom();
+ if (false == detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
+ "boost::math::cdf(const students_t_distribution<%1%>&, %1%)", df, &error_result, Policy()))
+ {
+ return error_result;
+ }
+ // Check for bad x first.
+ if(false == detail::check_x_not_NaN(
+ "boost::math::cdf(const students_t_distribution<%1%>&, %1%)", x, &error_result, Policy()))
+ {
+ return error_result;
+ }
+ if (x == 0)
+ { // Special case with exact result.
+ return static_cast<RealType>(0.5);
+ }
+ if ((boost::math::isinf)(x))
+ { // x == - or + infinity, regardless of df.
+ return ((x < 0) ? static_cast<RealType>(0) : static_cast<RealType>(1));
+ }
+
+ RealType limit = policies::get_epsilon<RealType, Policy>();
+ // Use policies so that if policy requests lower precision,
+ // then get the normal distribution approximation earlier.
+ limit = static_cast<RealType>(1) / limit; // 1/eps
+ // for 64-bit double 1/eps = 4503599627370496
+ if (df > limit)
+ { // Special case for really big degrees_of_freedom > 1 / eps (perhaps infinite?)
+ // - use normal distribution which is much faster and more accurate.
+ normal_distribution<RealType, Policy> n(0, 1);
+ RealType result = cdf(n, x);
+ return result;
+ }
+ else
+ { // normal df case.
+ //
+ // 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 x2 = x * x;
+ RealType probability;
+ if(df > 2 * x2)
+ {
+ RealType z = x2 / (df + x2);
+ probability = ibetac(static_cast<RealType>(0.5), df / 2, z, Policy()) / 2;
+ }
+ else
+ {
+ RealType z = df / (df + x2);
+ probability = ibeta(df / 2, static_cast<RealType>(0.5), z, Policy()) / 2;
+ }
+ return (x > 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 probability = p;
+
+ // Check for domain errors:
+ RealType df = dist.degrees_of_freedom();
+ static const char* function = "boost::math::quantile(const students_t_distribution<%1%>&, %1%)";
+ RealType error_result;
+ if(false == (detail::check_df_gt0_to_inf( // Check that df > 0 or == +infinity.
+ function, df, &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; //
+ //
+#if 0
+ // This next block is disabled in favour of a faster method than
+ // incomplete beta inverse, but code retained for future reference:
+ //
+ // 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(df, 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>())
+ {
+ return 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 mode(const students_t_distribution<RealType, Policy>& /*dist*/)
+{
+ // Assume no checks on degrees of freedom are useful (unlike mean).
+ return 0; // Always zero by definition.
+}
+
+template <class RealType, class Policy>
+inline RealType median(const students_t_distribution<RealType, Policy>& /*dist*/)
+{
+ // Assume no checks on degrees of freedom are useful (unlike mean).
+ return 0; // Always zero by definition.
+}
+
+// See section 5.1 on moments at http://en.wikipedia.org/wiki/Student%27s_t-distribution
+
+template <class RealType, class Policy>
+inline RealType mean(const students_t_distribution<RealType, Policy>& dist)
+{ // Revised for https://svn.boost.org/trac/boost/ticket/7177
+ RealType df = dist.degrees_of_freedom();
+ if(((boost::math::isnan)(df)) || (df <= 1) )
+ { // mean is undefined for moment <= 1!
+ return policies::raise_domain_error<RealType>(
+ "boost::math::mean(students_t_distribution<%1%> const&, %1%)",
+ "Mean is undefined for degrees of freedom < 1 but got %1%.", df, Policy());
+ return std::numeric_limits<RealType>::quiet_NaN();
+ }
+ return 0;
+} // mean
+
+template <class RealType, class Policy>
+inline RealType variance(const students_t_distribution<RealType, Policy>& dist)
+{ // http://en.wikipedia.org/wiki/Student%27s_t-distribution
+ // Revised for https://svn.boost.org/trac/boost/ticket/7177
+ RealType df = dist.degrees_of_freedom();
+ if ((boost::math::isnan)(df) || (df <= 2))
+ { // NaN or undefined for <= 2.
+ return policies::raise_domain_error<RealType>(
+ "boost::math::variance(students_t_distribution<%1%> const&, %1%)",
+ "variance is undefined for degrees of freedom <= 2, but got %1%.",
+ df, Policy());
+ return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
+ }
+ if ((boost::math::isinf)(df))
+ { // +infinity.
+ return 1;
+ }
+ RealType limit = policies::get_epsilon<RealType, Policy>();
+ // Use policies so that if policy requests lower precision,
+ // then get the normal distribution approximation earlier.
+ limit = static_cast<RealType>(1) / limit; // 1/eps
+ // for 64-bit double 1/eps = 4503599627370496
+ if (df > limit)
+ { // Special case for really big degrees_of_freedom > 1 / eps.
+ return 1;
+ }
+ else
+ {
+ return df / (df - 2);
+ }
+} // variance
+
+template <class RealType, class Policy>
+inline RealType skewness(const students_t_distribution<RealType, Policy>& dist)
+{
+ RealType df = dist.degrees_of_freedom();
+ if( ((boost::math::isnan)(df)) || (dist.degrees_of_freedom() <= 3))
+ { // Undefined for moment k = 3.
+ return 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 std::numeric_limits<RealType>::quiet_NaN();
+ }
+ return 0; // For all valid df, including infinity.
+} // skewness
+
+template <class RealType, class Policy>
+inline RealType kurtosis(const students_t_distribution<RealType, Policy>& dist)
+{
+ RealType df = dist.degrees_of_freedom();
+ if(((boost::math::isnan)(df)) || (df <= 4))
+ { // Undefined or infinity for moment k = 4.
+ return policies::raise_domain_error<RealType>(
+ "boost::math::kurtosis(students_t_distribution<%1%> const&, %1%)",
+ "Kurtosis is undefined for degrees of freedom <= 4, but got %1%.",
+ df, Policy());
+ return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
+ }
+ if ((boost::math::isinf)(df))
+ { // +infinity.
+ return 3;
+ }
+ RealType limit = policies::get_epsilon<RealType, Policy>();
+ // Use policies so that if policy requests lower precision,
+ // then get the normal distribution approximation earlier.
+ limit = static_cast<RealType>(1) / limit; // 1/eps
+ // for 64-bit double 1/eps = 4503599627370496
+ if (df > limit)
+ { // Special case for really big degrees_of_freedom > 1 / eps.
+ return 3;
+ }
+ else
+ {
+ //return 3 * (df - 2) / (df - 4); re-arranged to
+ return 6 / (df - 4) + 3;
+ }
+} // kurtosis
+
+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(((boost::math::isnan)(df)) || (df <= 4))
+ { // Undefined or infinity for moment k = 4.
+ return policies::raise_domain_error<RealType>(
+ "boost::math::kurtosis_excess(students_t_distribution<%1%> const&, %1%)",
+ "Kurtosis_excess is undefined for degrees of freedom <= 4, but got %1%.",
+ df, Policy());
+ return std::numeric_limits<RealType>::quiet_NaN(); // Undefined.
+ }
+ if ((boost::math::isinf)(df))
+ { // +infinity.
+ return 0;
+ }
+ RealType limit = policies::get_epsilon<RealType, Policy>();
+ // Use policies so that if policy requests lower precision,
+ // then get the normal distribution approximation earlier.
+ limit = static_cast<RealType>(1) / limit; // 1/eps
+ // for 64-bit double 1/eps = 4503599627370496
+ if (df > limit)
+ { // Special case for really big degrees_of_freedom > 1 / eps.
+ return 0;
+ }
+ else
+ {
+ 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
projects / ceph.git / blobdiff