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// Copyright Paul A. Bristow 2006.
// Copyright John Maddock 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)
// test_students_t.cpp
// http://en.wikipedia.org/wiki/Student%27s_t_distribution
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3664.htm
// Basic sanity test for Student's t probability (quantile) (0. < p < 1).
// and Student's t probability Quantile (0. < p < 1).
#include <boost/test/included/test_exec_monitor.hpp> // Boost.Test
#include <boost/test/floating_point_comparison.hpp>
#include <boost/math/distributions/students_t.hpp>
using boost::math::students_t_distribution;
#include <boost/math/concepts/real_concept.hpp> // for real_concept
#include <boost/math/tools/test.hpp> // for real_concept
#include <iostream>
using std::cout;
using std::endl;
using std::setprecision;
#include <limits>
using std::numeric_limits;
template <class RealType>
RealType naive_pdf(RealType v, RealType t)
{
// Calculate the pdf of the students t in a deliberately
// naive way, using equation (5) from
// http://mathworld.wolfram.com/Studentst-Distribution.html
// This is equivalent to, but a different method
// to the one in the actual implementation, so can be used as
// a very basic sanity check. However some published values
// would be nice....
using namespace std; // for ADL
using boost::math::beta;
//return pow(v / (v + t*t), (1+v) / 2) / (sqrt(v) * beta(v/2, RealType(0.5f)));
RealType result = boost::math::tgamma_ratio((v+1)/2, v/2);
result /= sqrt(v * boost::math::constants::pi<RealType>());
result /= pow(1 + t*t/v, (v+1)/2);
return result;
}
template <class RealType>
void test_spots(RealType)
{
// Basic sanity checks
RealType tolerance = 1e-4; // 1e-6 (as %)
// Some tests only pass at 1e-5 because probability value is less accurate,
// a digit in 6th decimal place, although calculated using
// a t-distribution generator (claimed 6 decimal digits) at
// http://faculty.vassar.edu/lowry/VassarStats.html
// http://faculty.vassar.edu/lowry/tsamp.html
// df = 5, +/-t = 2.0, 1-tailed = 0.050970, 2-tailed = 0.101939
cout << "Tolerance for type " << typeid(RealType).name() << " is " << tolerance << " %" << endl;
// http://en.wikipedia.org/wiki/Student%27s_t_distribution#Table_of_selected_values
// Using tabulated value of t = 3.182 for 0.975, 3 df, one-sided.
// http://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf refers to:
// A lookup table of quantiles of the RealType distribution
// for 1 to 25 in steps of 0.1 is provided in CSV form at:
// www.mth.kcl.ac.uk/~shaww/web_page/papers/Tsupp/tquantiles.csv
// gives accurate t of -3.1824463052837 and 3 degrees of freedom.
// Values below are from this source, saved as tquantiles.xls.
// DF are across the columns, probabilities down the rows
// and the t- values (quantiles) are shown.
// These values are probably accurate to nearly 64-bit double
// (perhaps 14 decimal digits).
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(2), // degrees_of_freedom
static_cast<RealType>(-6.96455673428326)), // t
static_cast<RealType>(0.01), // probability.
tolerance); // %
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(5), // degrees_of_freedom
static_cast<RealType>(-3.36492999890721)), // t
static_cast<RealType>(0.01), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(1), // degrees_of_freedom
static_cast<RealType>(-31830.988607907)), // t
static_cast<RealType>(0.00001), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(25.), // degrees_of_freedom
static_cast<RealType>(-5.2410429995425)), // t
static_cast<RealType>(0.00001), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(1), // degrees_of_freedom
static_cast<RealType>(-63661.97723)), // t
static_cast<RealType>(0.000005), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(-17.89686614)), // t
static_cast<RealType>(0.000005), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(25.), // degrees_of_freedom
static_cast<RealType>(-5.510848412)), // t
static_cast<RealType>(0.000005), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(10.), // degrees_of_freedom
static_cast<RealType>(-1.812461123)), // t
static_cast<RealType>(0.05), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(10), // degrees_of_freedom
static_cast<RealType>(1.812461123)), // t
static_cast<RealType>(0.95), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
complement(
students_t_distribution<RealType>(10), // degrees_of_freedom
static_cast<RealType>(1.812461123))), // t
static_cast<RealType>(0.05), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(10), // degrees_of_freedom
static_cast<RealType>(9.751995491)), // t
static_cast<RealType>(0.999999), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(10.), // degrees_of_freedom - for ALL degrees_of_freedom!
static_cast<RealType>(0.)), // t
static_cast<RealType>(0.5), // probability.
tolerance);
// Student's t Inverse function tests.
// Special cases
BOOST_CHECK_THROW(boost::math::quantile(
students_t_distribution<RealType>(1.), // degrees_of_freedom (ignored).
static_cast<RealType>(0)), std::overflow_error); // t == -infinity.
BOOST_CHECK_THROW(boost::math::quantile(
students_t_distribution<RealType>(1.), // degrees_of_freedom (ignored).
static_cast<RealType>(1)), std::overflow_error); // t == +infinity.
BOOST_CHECK_EQUAL(boost::math::quantile(
students_t_distribution<RealType>(1.), // degrees_of_freedom (ignored).
static_cast<RealType>(0.5)), // probability == half - special case.
static_cast<RealType>(0)); // t == zero.
BOOST_CHECK_EQUAL(boost::math::quantile(
complement(
students_t_distribution<RealType>(1.), // degrees_of_freedom (ignored).
static_cast<RealType>(0.5))), // probability == half - special case.
static_cast<RealType>(0)); // t == zero.
BOOST_CHECK_CLOSE(boost::math::quantile(
students_t_distribution<RealType>(1.), // degrees_of_freedom (ignored).
static_cast<RealType>(0.5)), // probability == half - special case.
static_cast<RealType>(0), // t == zero.
tolerance);
BOOST_CHECK_CLOSE( // Tests of p middling.
::boost::math::cdf(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(-0.559429644)), // t
static_cast<RealType>(0.3), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::quantile(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(0.3)), // probability.
static_cast<RealType>(-0.559429644), // t
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::quantile(
complement(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(0.7))), // probability.
static_cast<RealType>(-0.559429644), // t
tolerance);
BOOST_CHECK_CLOSE( // Tests of p high.
::boost::math::cdf(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(1.475884049)), // t
static_cast<RealType>(0.9), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::quantile(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(0.9)), // probability.
static_cast<RealType>(1.475884049), // t
tolerance);
BOOST_CHECK_CLOSE( // Tests of p low.
::boost::math::cdf(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(-1.475884049)), // t
static_cast<RealType>(0.1), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::quantile(
students_t_distribution<RealType>(5.), // degrees_of_freedom
static_cast<RealType>(0.1)), // probability.
static_cast<RealType>(-1.475884049), // t
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::cdf(
students_t_distribution<RealType>(2.), // degrees_of_freedom
static_cast<RealType>(-6.96455673428326)), // t
static_cast<RealType>(0.01), // probability.
tolerance);
BOOST_CHECK_CLOSE(
::boost::math::quantile(
students_t_distribution<RealType>(2.), // degrees_of_freedom
static_cast<RealType>(0.01)), // probability.
static_cast<RealType>(-6.96455673428326), // t
tolerance);
// Student's t pdf tests.
// for PDF checks, use 100 eps tolerance expressed as a percent:
tolerance = boost::math::tools::epsilon<RealType>() * 10000;
for(unsigned i = 1; i < 20; i += 3)
{
for(RealType r = -10; r < 10; r += 0.125)
{
//std::cout << "df=" << i << " t=" << r << std::endl;
BOOST_CHECK_CLOSE(
boost::math::pdf(
students_t_distribution<RealType>(static_cast<RealType>(i)),
r),
naive_pdf<RealType>(static_cast<RealType>(i), r),
tolerance);
}
}
RealType tol2 = boost::math::tools::epsilon<RealType>() * 5;
students_t_distribution<RealType> dist(8);
RealType x = static_cast<RealType>(0.125);
using namespace std; // ADL of std names.
// mean:
BOOST_CHECK_CLOSE(
mean(dist)
, static_cast<RealType>(0), tol2);
// variance:
BOOST_CHECK_CLOSE(
variance(dist)
, static_cast<RealType>(8.0L / 6.0L), tol2);
// std deviation:
BOOST_CHECK_CLOSE(
standard_deviation(dist)
, static_cast<RealType>(sqrt(8.0L / 6.0L)), tol2);
// hazard:
BOOST_CHECK_CLOSE(
hazard(dist, x)
, pdf(dist, x) / cdf(complement(dist, x)), tol2);
// cumulative hazard:
BOOST_CHECK_CLOSE(
chf(dist, x)
, -log(cdf(complement(dist, x))), tol2);
// coefficient_of_variation:
BOOST_CHECK_THROW(
coefficient_of_variation(dist),
std::overflow_error);
// mode:
BOOST_CHECK_CLOSE(
mean(dist)
, static_cast<RealType>(0), tol2);
// median:
BOOST_CHECK_CLOSE(
median(dist)
, static_cast<RealType>(0), tol2);
// skewness:
BOOST_CHECK_CLOSE(
skewness(dist)
, static_cast<RealType>(0), tol2);
// kurtosis:
BOOST_CHECK_CLOSE(
kurtosis(dist)
, static_cast<RealType>(4.5), tol2);
// kurtosis excess:
BOOST_CHECK_CLOSE(
kurtosis_excess(dist)
, static_cast<RealType>(1.5), tol2);
// Parameter estimation. These results are close to but
// not identical to those reported on the NIST website at
// http://www.itl.nist.gov/div898/handbook/prc/section2/prc222.htm
// the NIST results appear to be calculated using a normal
// approximation, which slightly under-estimates the degrees of
// freedom required, particularly when the result is small.
//
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(0.5),
static_cast<RealType>(0.005),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
99);
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(1.5),
static_cast<RealType>(0.005),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
14);
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(0.5),
static_cast<RealType>(0.025),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
76);
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(1.5),
static_cast<RealType>(0.025),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
11);
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(0.5),
static_cast<RealType>(0.05),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
65);
BOOST_CHECK_EQUAL(
ceil(students_t_distribution<RealType>::find_degrees_of_freedom(
static_cast<RealType>(1.5),
static_cast<RealType>(0.05),
static_cast<RealType>(0.01),
static_cast<RealType>(1.0))),
9);
} // template <class RealType>void test_spots(RealType)
int test_main(int, char* [])
{
// Check that can construct students_t distribution using the two convenience methods:
using namespace boost::math;
students_t myst1(2); // Using typedef
students_t_distribution<> myst2(2); // Using default RealType double.
//students_t_distribution<double> myst3(2); // Using explicit RealType double.
// Basic sanity-check spot values.
// (Parameter value, arbitrarily zero, only communicates the floating point type).
test_spots(0.0F); // Test float. OK at decdigits = 0 tolerance = 0.0001 %
test_spots(0.0); // Test double. OK at decdigits 7, tolerance = 1e07 %
#ifndef BOOST_MATH_NO_LONG_DOUBLE_MATH_FUNCTIONS
test_spots(0.0L); // Test long double.
#if !BOOST_WORKAROUND(__BORLANDC__, BOOST_TESTED_AT(0x582))
test_spots(boost::math::concepts::real_concept(0.)); // Test real concept.
#endif
#else
std::cout << "<note>The long double tests have been disabled on this platform "
"either because the long double overloads of the usual math functions are "
"not available at all, or because they are too inaccurate for these tests "
"to pass.</note>" << std::cout;
#endif
return 0;
} // int test_main(int, char* [])
/*
Autorun "i:\boost-06-05-03-1300\libs\math\test\Math_test\debug\test_students_t.exe"
Running 1 test case...
Tolerance for type float is 0.0001 %
Tolerance for type double is 0.0001 %
Tolerance for type long double is 0.0001 %
Tolerance for type class boost::math::concepts::real_concept is 0.0001 %
*** No errors detected
*/
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