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Diffstat (limited to 'src/third_party/boost-1.56.0/boost/math/special_functions/detail/t_distribution_inv.hpp')
-rw-r--r-- | src/third_party/boost-1.56.0/boost/math/special_functions/detail/t_distribution_inv.hpp | 549 |
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diff --git a/src/third_party/boost-1.56.0/boost/math/special_functions/detail/t_distribution_inv.hpp b/src/third_party/boost-1.56.0/boost/math/special_functions/detail/t_distribution_inv.hpp deleted file mode 100644 index 72f6f0c6468..00000000000 --- a/src/third_party/boost-1.56.0/boost/math/special_functions/detail/t_distribution_inv.hpp +++ /dev/null @@ -1,549 +0,0 @@ -// Copyright John Maddock 2007. -// Copyright Paul A. Bristow 2007 -// 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_MATH_SF_DETAIL_INV_T_HPP -#define BOOST_MATH_SF_DETAIL_INV_T_HPP - -#ifdef _MSC_VER -#pragma once -#endif - -#include <boost/math/special_functions/cbrt.hpp> -#include <boost/math/special_functions/round.hpp> -#include <boost/math/special_functions/trunc.hpp> - -namespace boost{ namespace math{ namespace detail{ - -// -// The main method used is due to Hill: -// -// G. W. Hill, Algorithm 396, Student's t-Quantiles, -// Communications of the ACM, 13(10): 619-620, Oct., 1970. -// -template <class T, class Policy> -T inverse_students_t_hill(T ndf, T u, const Policy& pol) -{ - BOOST_MATH_STD_USING - BOOST_ASSERT(u <= 0.5); - - T a, b, c, d, q, x, y; - - if (ndf > 1e20f) - return -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); - - a = 1 / (ndf - 0.5f); - b = 48 / (a * a); - c = ((20700 * a / b - 98) * a - 16) * a + 96.36f; - d = ((94.5f / (b + c) - 3) / b + 1) * sqrt(a * constants::pi<T>() / 2) * ndf; - y = pow(d * 2 * u, 2 / ndf); - - if (y > (0.05f + a)) - { - // - // Asymptotic inverse expansion about normal: - // - x = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); - y = x * x; - - if (ndf < 5) - c += 0.3f * (ndf - 4.5f) * (x + 0.6f); - c += (((0.05f * d * x - 5) * x - 7) * x - 2) * x + b; - y = (((((0.4f * y + 6.3f) * y + 36) * y + 94.5f) / c - y - 3) / b + 1) * x; - y = boost::math::expm1(a * y * y, pol); - } - else - { - y = ((1 / (((ndf + 6) / (ndf * y) - 0.089f * d - 0.822f) - * (ndf + 2) * 3) + 0.5 / (ndf + 4)) * y - 1) - * (ndf + 1) / (ndf + 2) + 1 / y; - } - q = sqrt(ndf * y); - - return -q; -} -// -// Tail and body series are due to Shaw: -// -// www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf -// -// Shaw, W.T., 2006, "Sampling Student's T distribution - use of -// the inverse cumulative distribution function." -// Journal of Computational Finance, Vol 9 Issue 4, pp 37-73, Summer 2006 -// -template <class T, class Policy> -T inverse_students_t_tail_series(T df, T v, const Policy& pol) -{ - BOOST_MATH_STD_USING - // Tail series expansion, see section 6 of Shaw's paper. - // w is calculated using Eq 60: - T w = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol) - * sqrt(df * constants::pi<T>()) * v; - // define some variables: - T np2 = df + 2; - T np4 = df + 4; - T np6 = df + 6; - // - // Calculate the coefficients d(k), these depend only on the - // number of degrees of freedom df, so at least in theory - // we could tabulate these for fixed df, see p15 of Shaw: - // - T d[7] = { 1, }; - d[1] = -(df + 1) / (2 * np2); - np2 *= (df + 2); - d[2] = -df * (df + 1) * (df + 3) / (8 * np2 * np4); - np2 *= df + 2; - d[3] = -df * (df + 1) * (df + 5) * (((3 * df) + 7) * df -2) / (48 * np2 * np4 * np6); - np2 *= (df + 2); - np4 *= (df + 4); - d[4] = -df * (df + 1) * (df + 7) * - ( (((((15 * df) + 154) * df + 465) * df + 286) * df - 336) * df + 64 ) - / (384 * np2 * np4 * np6 * (df + 8)); - np2 *= (df + 2); - d[5] = -df * (df + 1) * (df + 3) * (df + 9) - * (((((((35 * df + 452) * df + 1573) * df + 600) * df - 2020) * df) + 928) * df -128) - / (1280 * np2 * np4 * np6 * (df + 8) * (df + 10)); - np2 *= (df + 2); - np4 *= (df + 4); - np6 *= (df + 6); - d[6] = -df * (df + 1) * (df + 11) - * ((((((((((((945 * df) + 31506) * df + 425858) * df + 2980236) * df + 11266745) * df + 20675018) * df + 7747124) * df - 22574632) * df - 8565600) * df + 18108416) * df - 7099392) * df + 884736) - / (46080 * np2 * np4 * np6 * (df + 8) * (df + 10) * (df +12)); - // - // Now bring everthing together to provide the result, - // this is Eq 62 of Shaw: - // - T rn = sqrt(df); - T div = pow(rn * w, 1 / df); - T power = div * div; - T result = tools::evaluate_polynomial<7, T, T>(d, power); - result *= rn; - result /= div; - return -result; -} - -template <class T, class Policy> -T inverse_students_t_body_series(T df, T u, const Policy& pol) -{ - BOOST_MATH_STD_USING - // - // Body series for small N: - // - // Start with Eq 56 of Shaw: - // - T v = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol) - * sqrt(df * constants::pi<T>()) * (u - constants::half<T>()); - // - // Workspace for the polynomial coefficients: - // - T c[11] = { 0, 1, }; - // - // Figure out what the coefficients are, note these depend - // only on the degrees of freedom (Eq 57 of Shaw): - // - T in = 1 / df; - c[2] = 0.16666666666666666667 + 0.16666666666666666667 * in; - c[3] = (0.0083333333333333333333 * in - + 0.066666666666666666667) * in - + 0.058333333333333333333; - c[4] = ((0.00019841269841269841270 * in - + 0.0017857142857142857143) * in - + 0.026785714285714285714) * in - + 0.025198412698412698413; - c[5] = (((2.7557319223985890653e-6 * in - + 0.00037477954144620811287) * in - - 0.0011078042328042328042) * in - + 0.010559964726631393298) * in - + 0.012039792768959435626; - c[6] = ((((2.5052108385441718775e-8 * in - - 0.000062705427288760622094) * in - + 0.00059458674042007375341) * in - - 0.0016095979637646304313) * in - + 0.0061039211560044893378) * in - + 0.0038370059724226390893; - c[7] = (((((1.6059043836821614599e-10 * in - + 0.000015401265401265401265) * in - - 0.00016376804137220803887) * in - + 0.00069084207973096861986) * in - - 0.0012579159844784844785) * in - + 0.0010898206731540064873) * in - + 0.0032177478835464946576; - c[8] = ((((((7.6471637318198164759e-13 * in - - 3.9851014346715404916e-6) * in - + 0.000049255746366361445727) * in - - 0.00024947258047043099953) * in - + 0.00064513046951456342991) * in - - 0.00076245135440323932387) * in - + 0.000033530976880017885309) * in - + 0.0017438262298340009980; - c[9] = (((((((2.8114572543455207632e-15 * in - + 1.0914179173496789432e-6) * in - - 0.000015303004486655377567) * in - + 0.000090867107935219902229) * in - - 0.00029133414466938067350) * in - + 0.00051406605788341121363) * in - - 0.00036307660358786885787) * in - - 0.00031101086326318780412) * in - + 0.00096472747321388644237; - c[10] = ((((((((8.2206352466243297170e-18 * in - - 3.1239569599829868045e-7) * in - + 4.8903045291975346210e-6) * in - - 0.000033202652391372058698) * in - + 0.00012645437628698076975) * in - - 0.00028690924218514613987) * in - + 0.00035764655430568632777) * in - - 0.00010230378073700412687) * in - - 0.00036942667800009661203) * in - + 0.00054229262813129686486; - // - // The result is then a polynomial in v (see Eq 56 of Shaw): - // - return tools::evaluate_odd_polynomial<11, T, T>(c, v); -} - -template <class T, class Policy> -T inverse_students_t(T df, T u, T v, const Policy& pol, bool* pexact = 0) -{ - // - // df = number of degrees of freedom. - // u = probablity. - // v = 1 - u. - // l = lanczos type to use. - // - BOOST_MATH_STD_USING - bool invert = false; - T result = 0; - if(pexact) - *pexact = false; - if(u > v) - { - // function is symmetric, invert it: - std::swap(u, v); - invert = true; - } - if((floor(df) == df) && (df < 20)) - { - // - // we have integer degrees of freedom, try for the special - // cases first: - // - T tolerance = ldexp(1.0f, (2 * policies::digits<T, Policy>()) / 3); - - switch(itrunc(df, Policy())) - { - case 1: - { - // - // df = 1 is the same as the Cauchy distribution, see - // Shaw Eq 35: - // - if(u == 0.5) - result = 0; - else - result = -cos(constants::pi<T>() * u) / sin(constants::pi<T>() * u); - if(pexact) - *pexact = true; - break; - } - case 2: - { - // - // df = 2 has an exact result, see Shaw Eq 36: - // - result =(2 * u - 1) / sqrt(2 * u * v); - if(pexact) - *pexact = true; - break; - } - case 4: - { - // - // df = 4 has an exact result, see Shaw Eq 38 & 39: - // - T alpha = 4 * u * v; - T root_alpha = sqrt(alpha); - T r = 4 * cos(acos(root_alpha) / 3) / root_alpha; - T x = sqrt(r - 4); - result = u - 0.5f < 0 ? (T)-x : x; - if(pexact) - *pexact = true; - break; - } - case 6: - { - // - // We get numeric overflow in this area: - // - if(u < 1e-150) - return (invert ? -1 : 1) * inverse_students_t_hill(df, u, pol); - // - // Newton-Raphson iteration of a polynomial case, - // choice of seed value is taken from Shaw's online - // supplement: - // - T a = 4 * (u - u * u);//1 - 4 * (u - 0.5f) * (u - 0.5f); - T b = boost::math::cbrt(a); - static const T c = 0.85498797333834849467655443627193; - T p = 6 * (1 + c * (1 / b - 1)); - T p0; - do{ - T p2 = p * p; - T p4 = p2 * p2; - T p5 = p * p4; - p0 = p; - // next term is given by Eq 41: - p = 2 * (8 * a * p5 - 270 * p2 + 2187) / (5 * (4 * a * p4 - 216 * p - 243)); - }while(fabs((p - p0) / p) > tolerance); - // - // Use Eq 45 to extract the result: - // - p = sqrt(p - df); - result = (u - 0.5f) < 0 ? (T)-p : p; - break; - } -#if 0 - // - // These are Shaw's "exact" but iterative solutions - // for even df, the numerical accuracy of these is - // rather less than Hill's method, so these are disabled - // for now, which is a shame because they are reasonably - // quick to evaluate... - // - case 8: - { - // - // Newton-Raphson iteration of a polynomial case, - // choice of seed value is taken from Shaw's online - // supplement: - // - static const T c8 = 0.85994765706259820318168359251872L; - T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f); - T b = pow(a, T(1) / 4); - T p = 8 * (1 + c8 * (1 / b - 1)); - T p0 = p; - do{ - T p5 = p * p; - p5 *= p5 * p; - p0 = p; - // Next term is given by Eq 42: - p = 2 * (3 * p + (640 * (160 + p * (24 + p * (p + 4)))) / (-5120 + p * (-2048 - 960 * p + a * p5))) / 7; - }while(fabs((p - p0) / p) > tolerance); - // - // Use Eq 45 to extract the result: - // - p = sqrt(p - df); - result = (u - 0.5f) < 0 ? -p : p; - break; - } - case 10: - { - // - // Newton-Raphson iteration of a polynomial case, - // choice of seed value is taken from Shaw's online - // supplement: - // - static const T c10 = 0.86781292867813396759105692122285L; - T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f); - T b = pow(a, T(1) / 5); - T p = 10 * (1 + c10 * (1 / b - 1)); - T p0; - do{ - T p6 = p * p; - p6 *= p6 * p6; - p0 = p; - // Next term given by Eq 43: - p = (8 * p) / 9 + (218750 * (21875 + 4 * p * (625 + p * (75 + 2 * p * (5 + p))))) / - (9 * (-68359375 + 8 * p * (-2343750 + p * (-546875 - 175000 * p + 8 * a * p6)))); - }while(fabs((p - p0) / p) > tolerance); - // - // Use Eq 45 to extract the result: - // - p = sqrt(p - df); - result = (u - 0.5f) < 0 ? -p : p; - break; - } -#endif - default: - goto calculate_real; - } - } - else - { -calculate_real: - if(df > 0x10000000) - { - result = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>(); - if((pexact) && (df >= 1e20)) - *pexact = true; - } - else if(df < 3) - { - // - // Use a roughly linear scheme to choose between Shaw's - // tail series and body series: - // - T crossover = 0.2742f - df * 0.0242143f; - if(u > crossover) - { - result = boost::math::detail::inverse_students_t_body_series(df, u, pol); - } - else - { - result = boost::math::detail::inverse_students_t_tail_series(df, u, pol); - } - } - else - { - // - // Use Hill's method except in the exteme tails - // where we use Shaw's tail series. - // The crossover point is roughly exponential in -df: - // - T crossover = ldexp(1.0f, iround(T(df / -0.654f), typename policies::normalise<Policy, policies::rounding_error<policies::ignore_error> >::type())); - if(u > crossover) - { - result = boost::math::detail::inverse_students_t_hill(df, u, pol); - } - else - { - result = boost::math::detail::inverse_students_t_tail_series(df, u, pol); - } - } - } - return invert ? (T)-result : result; -} - -template <class T, class Policy> -inline T find_ibeta_inv_from_t_dist(T a, T p, T /*q*/, T* py, const Policy& pol) -{ - T u = p / 2; - T v = 1 - u; - T df = a * 2; - T t = boost::math::detail::inverse_students_t(df, u, v, pol); - *py = t * t / (df + t * t); - return df / (df + t * t); -} - -template <class T, class Policy> -inline T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::false_*) -{ - BOOST_MATH_STD_USING - // - // Need to use inverse incomplete beta to get - // required precision so not so fast: - // - T probability = (p > 0.5) ? 1 - p : p; - T t, x, y(0); - x = ibeta_inv(df / 2, T(0.5), 2 * probability, &y, pol); - if(df * y > tools::max_value<T>() * x) - t = policies::raise_overflow_error<T>("boost::math::students_t_quantile<%1%>(%1%,%1%)", 0, pol); - else - t = sqrt(df * y / x); - // - // Figure out sign based on the size of p: - // - if(p < 0.5) - t = -t; - return t; -} - -template <class T, class Policy> -T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::true_*) -{ - BOOST_MATH_STD_USING - bool invert = false; - if((df < 2) && (floor(df) != df)) - return boost::math::detail::fast_students_t_quantile_imp(df, p, pol, static_cast<mpl::false_*>(0)); - if(p > 0.5) - { - p = 1 - p; - invert = true; - } - // - // Get an estimate of the result: - // - bool exact; - T t = inverse_students_t(df, p, T(1-p), pol, &exact); - if((t == 0) || exact) - return invert ? -t : t; // can't do better! - // - // Change variables to inverse incomplete beta: - // - T t2 = t * t; - T xb = df / (df + t2); - T y = t2 / (df + t2); - T a = df / 2; - // - // t can be so large that x underflows, - // just return our estimate in that case: - // - if(xb == 0) - return t; - // - // Get incomplete beta and it's derivative: - // - T f1; - T f0 = xb < y ? ibeta_imp(a, constants::half<T>(), xb, pol, false, true, &f1) - : ibeta_imp(constants::half<T>(), a, y, pol, true, true, &f1); - - // Get cdf from incomplete beta result: - T p0 = f0 / 2 - p; - // Get pdf from derivative: - T p1 = f1 * sqrt(y * xb * xb * xb / df); - // - // Second derivative divided by p1: - // - // yacas gives: - // - // In> PrettyForm(Simplify(D(t) (1 + t^2/v) ^ (-(v+1)/2))) - // - // | | v + 1 | | - // | -| ----- + 1 | | - // | | 2 | | - // -| | 2 | | - // | | t | | - // | | -- + 1 | | - // | ( v + 1 ) * | v | * t | - // --------------------------------------------- - // v - // - // Which after some manipulation is: - // - // -p1 * t * (df + 1) / (t^2 + df) - // - T p2 = t * (df + 1) / (t * t + df); - // Halley step: - t = fabs(t); - t += p0 / (p1 + p0 * p2 / 2); - return !invert ? -t : t; -} - -template <class T, class Policy> -inline T fast_students_t_quantile(T df, T p, const Policy& pol) -{ - typedef typename policies::evaluation<T, Policy>::type value_type; - typedef typename policies::normalise< - Policy, - policies::promote_float<false>, - policies::promote_double<false>, - policies::discrete_quantile<>, - policies::assert_undefined<> >::type forwarding_policy; - - typedef mpl::bool_< - (std::numeric_limits<T>::digits <= 53) - && - (std::numeric_limits<T>::is_specialized) - && - (std::numeric_limits<T>::radix == 2) - > tag_type; - return policies::checked_narrowing_cast<T, forwarding_policy>(fast_students_t_quantile_imp(static_cast<value_type>(df), static_cast<value_type>(p), pol, static_cast<tag_type*>(0)), "boost::math::students_t_quantile<%1%>(%1%,%1%,%1%)"); -} - -}}} // namespaces - -#endif // BOOST_MATH_SF_DETAIL_INV_T_HPP - - - |