/usr/include/boost/math/distributions
Edit: /usr/include/boost/math/distributions/students_t.hpp (18476B)
// 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
#include // for ibeta(a, b, x).
#include
#include
#include
#include
#include
#ifdef BOOST_MSVC
# pragma warning(push)
# pragma warning(disable: 4702) // unreachable code (return after domain_error throw).
#endif
namespace boost { namespace math {
template >
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 students_t; // Convenience typedef for double version.
template
inline const std::pair range(const students_t_distribution& /*dist*/)
{ // Range of permissible values for random variable x.
// Now including infinity.
using boost::math::tools::max_value;
//return std::pair(-max_value(), max_value());
return std::pair(((::std::numeric_limits::is_specialized & ::std::numeric_limits::has_infinity) ? -std::numeric_limits::infinity() : -max_value()), ((::std::numeric_limits::is_specialized & ::std::numeric_limits::has_infinity) ? +std::numeric_limits::infinity() : +max_value()));
}
template
inline const std::pair support(const students_t_distribution& /*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(-max_value(), max_value());
return std::pair(((::std::numeric_limits::is_specialized & ::std::numeric_limits::has_infinity) ? -std::numeric_limits::infinity() : -max_value()), ((::std::numeric_limits::is_specialized & ::std::numeric_limits::has_infinity) ? +std::numeric_limits::infinity() : +max_value()));
}
template
inline RealType pdf(const students_t_distribution& 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(0);
return result;
}
RealType limit = policies::get_epsilon();
// Use policies so that if policy requests lower precision,
// then get the normal distribution approximation earlier.
limit = static_cast(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 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
inline RealType cdf(const students_t_distribution& 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(0.5);
}
if ((boost::math::isinf)(x))
{ // x == - or + infinity, regardless of df.
return ((x < 0) ? static_cast(0) : static_cast(1));
}
RealType limit = policies::get_epsilon();
// Use policies so that if policy requests lower precision,
// then get the normal distribution approximation earlier.
limit = static_cast(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 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(0.5), df / 2, z, Policy()) / 2;
}
else
{
RealType z = df / (df + x2);
probability = ibeta(df / 2, static_cast(0.5), z, Policy()) / 2;
}
return (x > 0 ? 1 - probability : probability);
}
} // cdf
template
inline RealType quantile(const students_t_distribution& 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(function, 0, Policy());
if (probability == 1)
return policies::raise_overflow_error(function, 0, Policy());
if (probability == static_cast(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() * x)
t = tools::overflow_error(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
inline RealType cdf(const complemented2_type, RealType>& c)
{
return cdf(c.dist, -c.param);
}
template
inline RealType quantile(const complemented2_type, RealType>& c)
{
return -quantile(c.dist, c.param);
}
//
// Parameter estimation follows:
//
namespace detail{
//
// Functors for finding degrees of freedom:
//
template
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())
{ //
return 1;
}
students_t_distribution 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
RealType students_t_distribution::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 f(alpha, beta, sd, difference_from_mean);
tools::eps_tolerance tol(policies::digits());
boost::uintmax_t max_iter = policies::get_max_root_iterations();
std::pair 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())
{
return policies::raise_evaluation_error(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
inline RealType mode(const students_t_distribution& /*dist*/)
{
// Assume no checks on degrees of freedom are useful (unlike mean).
return 0; // Always zero by definition.
}
template
inline RealType median(const students_t_distribution& /*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
inline RealType mean(const students_t_distribution& 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(
"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::quiet_NaN();
}
return 0;
} // mean
template
inline RealType variance(const students_t_distribution& 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(
"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::quiet_NaN(); // Undefined.
}
if ((boost::math::isinf)(df))
{ // +infinity.
return 1;
}
RealType limit = policies::get_epsilon();
// Use policies so that if policy requests lower precision,
// then get the normal distribution approximation earlier.
limit = static_cast(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
inline RealType skewness(const students_t_distribution& 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(
"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::quiet_NaN();
}
return 0; // For all valid df, including infinity.
} // skewness
template
inline RealType kurtosis(const students_t_distribution& 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(
"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::quiet_NaN(); // Undefined.
}
if ((boost::math::isinf)(df))
{ // +infinity.
return 3;
}
RealType limit = policies::get_epsilon();
// Use policies so that if policy requests lower precision,
// then get the normal distribution approximation earlier.
limit = static_cast(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
inline RealType kurtosis_excess(const students_t_distribution& 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(
"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::quiet_NaN(); // Undefined.
}
if ((boost::math::isinf)(df))
{ // +infinity.
return 0;
}
RealType limit = policies::get_epsilon();
// Use policies so that if policy requests lower precision,
// then get the normal distribution approximation earlier.
limit = static_cast(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);
}
}
template
inline RealType entropy(const students_t_distribution& dist)
{
using std::log;
using std::sqrt;
RealType v = dist.degrees_of_freedom();
RealType vp1 = (v+1)/2;
RealType vd2 = v/2;
return vp1*(digamma(vp1) - digamma(vd2)) + log(sqrt(v)*beta(vd2, RealType(1)/RealType(2)));
}
} // 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
#endif // BOOST_STATS_STUDENTS_T_HPP