550 lines
18 KiB
Plaintext
550 lines
18 KiB
Plaintext
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// Copyright John Maddock 2007.
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// Copyright Paul A. Bristow 2007
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0. (See accompanying file
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// LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
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#ifndef BOOST_MATH_SF_DETAIL_INV_T_HPP
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#define BOOST_MATH_SF_DETAIL_INV_T_HPP
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#ifdef _MSC_VER
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#pragma once
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#endif
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#include <boost/math/special_functions/cbrt.hpp>
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#include <boost/math/special_functions/round.hpp>
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#include <boost/math/special_functions/trunc.hpp>
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namespace boost{ namespace math{ namespace detail{
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//
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// The main method used is due to Hill:
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//
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// G. W. Hill, Algorithm 396, Student's t-Quantiles,
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// Communications of the ACM, 13(10): 619-620, Oct., 1970.
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//
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template <class T, class Policy>
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T inverse_students_t_hill(T ndf, T u, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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BOOST_ASSERT(u <= 0.5);
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T a, b, c, d, q, x, y;
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if (ndf > 1e20f)
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return -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>();
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a = 1 / (ndf - 0.5f);
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b = 48 / (a * a);
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c = ((20700 * a / b - 98) * a - 16) * a + 96.36f;
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d = ((94.5f / (b + c) - 3) / b + 1) * sqrt(a * constants::pi<T>() / 2) * ndf;
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y = pow(d * 2 * u, 2 / ndf);
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if (y > (0.05f + a))
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{
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//
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// Asymptotic inverse expansion about normal:
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//
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x = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>();
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y = x * x;
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if (ndf < 5)
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c += 0.3f * (ndf - 4.5f) * (x + 0.6f);
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c += (((0.05f * d * x - 5) * x - 7) * x - 2) * x + b;
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y = (((((0.4f * y + 6.3f) * y + 36) * y + 94.5f) / c - y - 3) / b + 1) * x;
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y = boost::math::expm1(a * y * y, pol);
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}
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else
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{
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y = static_cast<T>(((1 / (((ndf + 6) / (ndf * y) - 0.089f * d - 0.822f)
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* (ndf + 2) * 3) + 0.5 / (ndf + 4)) * y - 1)
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* (ndf + 1) / (ndf + 2) + 1 / y);
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}
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q = sqrt(ndf * y);
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return -q;
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}
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//
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// Tail and body series are due to Shaw:
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//
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// www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf
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//
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// Shaw, W.T., 2006, "Sampling Student's T distribution - use of
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// the inverse cumulative distribution function."
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// Journal of Computational Finance, Vol 9 Issue 4, pp 37-73, Summer 2006
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//
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template <class T, class Policy>
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T inverse_students_t_tail_series(T df, T v, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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// Tail series expansion, see section 6 of Shaw's paper.
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// w is calculated using Eq 60:
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T w = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol)
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* sqrt(df * constants::pi<T>()) * v;
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// define some variables:
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T np2 = df + 2;
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T np4 = df + 4;
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T np6 = df + 6;
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//
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// Calculate the coefficients d(k), these depend only on the
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// number of degrees of freedom df, so at least in theory
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// we could tabulate these for fixed df, see p15 of Shaw:
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//
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T d[7] = { 1, };
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d[1] = -(df + 1) / (2 * np2);
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np2 *= (df + 2);
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d[2] = -df * (df + 1) * (df + 3) / (8 * np2 * np4);
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np2 *= df + 2;
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d[3] = -df * (df + 1) * (df + 5) * (((3 * df) + 7) * df -2) / (48 * np2 * np4 * np6);
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np2 *= (df + 2);
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np4 *= (df + 4);
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d[4] = -df * (df + 1) * (df + 7) *
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( (((((15 * df) + 154) * df + 465) * df + 286) * df - 336) * df + 64 )
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/ (384 * np2 * np4 * np6 * (df + 8));
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np2 *= (df + 2);
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d[5] = -df * (df + 1) * (df + 3) * (df + 9)
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* (((((((35 * df + 452) * df + 1573) * df + 600) * df - 2020) * df) + 928) * df -128)
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/ (1280 * np2 * np4 * np6 * (df + 8) * (df + 10));
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np2 *= (df + 2);
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np4 *= (df + 4);
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np6 *= (df + 6);
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d[6] = -df * (df + 1) * (df + 11)
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* ((((((((((((945 * df) + 31506) * df + 425858) * df + 2980236) * df + 11266745) * df + 20675018) * df + 7747124) * df - 22574632) * df - 8565600) * df + 18108416) * df - 7099392) * df + 884736)
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/ (46080 * np2 * np4 * np6 * (df + 8) * (df + 10) * (df +12));
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//
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// Now bring everthing together to provide the result,
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// this is Eq 62 of Shaw:
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//
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T rn = sqrt(df);
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T div = pow(rn * w, 1 / df);
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T power = div * div;
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T result = tools::evaluate_polynomial<7, T, T>(d, power);
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result *= rn;
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result /= div;
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return -result;
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}
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template <class T, class Policy>
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T inverse_students_t_body_series(T df, T u, const Policy& pol)
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{
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BOOST_MATH_STD_USING
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//
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// Body series for small N:
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//
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// Start with Eq 56 of Shaw:
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//
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T v = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol)
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* sqrt(df * constants::pi<T>()) * (u - constants::half<T>());
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//
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// Workspace for the polynomial coefficients:
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//
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T c[11] = { 0, 1, };
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//
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// Figure out what the coefficients are, note these depend
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// only on the degrees of freedom (Eq 57 of Shaw):
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//
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T in = 1 / df;
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c[2] = static_cast<T>(0.16666666666666666667 + 0.16666666666666666667 * in);
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c[3] = static_cast<T>((0.0083333333333333333333 * in
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+ 0.066666666666666666667) * in
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+ 0.058333333333333333333);
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c[4] = static_cast<T>(((0.00019841269841269841270 * in
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+ 0.0017857142857142857143) * in
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+ 0.026785714285714285714) * in
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+ 0.025198412698412698413);
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c[5] = static_cast<T>((((2.7557319223985890653e-6 * in
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+ 0.00037477954144620811287) * in
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- 0.0011078042328042328042) * in
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+ 0.010559964726631393298) * in
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+ 0.012039792768959435626);
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c[6] = static_cast<T>(((((2.5052108385441718775e-8 * in
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- 0.000062705427288760622094) * in
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+ 0.00059458674042007375341) * in
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- 0.0016095979637646304313) * in
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+ 0.0061039211560044893378) * in
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+ 0.0038370059724226390893);
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c[7] = static_cast<T>((((((1.6059043836821614599e-10 * in
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+ 0.000015401265401265401265) * in
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- 0.00016376804137220803887) * in
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+ 0.00069084207973096861986) * in
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- 0.0012579159844784844785) * in
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+ 0.0010898206731540064873) * in
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+ 0.0032177478835464946576);
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c[8] = static_cast<T>(((((((7.6471637318198164759e-13 * in
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- 3.9851014346715404916e-6) * in
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+ 0.000049255746366361445727) * in
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- 0.00024947258047043099953) * in
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+ 0.00064513046951456342991) * in
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- 0.00076245135440323932387) * in
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+ 0.000033530976880017885309) * in
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+ 0.0017438262298340009980);
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c[9] = static_cast<T>((((((((2.8114572543455207632e-15 * in
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+ 1.0914179173496789432e-6) * in
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- 0.000015303004486655377567) * in
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+ 0.000090867107935219902229) * in
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- 0.00029133414466938067350) * in
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+ 0.00051406605788341121363) * in
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- 0.00036307660358786885787) * in
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- 0.00031101086326318780412) * in
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+ 0.00096472747321388644237);
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c[10] = static_cast<T>(((((((((8.2206352466243297170e-18 * in
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- 3.1239569599829868045e-7) * in
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+ 4.8903045291975346210e-6) * in
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- 0.000033202652391372058698) * in
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+ 0.00012645437628698076975) * in
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- 0.00028690924218514613987) * in
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+ 0.00035764655430568632777) * in
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- 0.00010230378073700412687) * in
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- 0.00036942667800009661203) * in
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+ 0.00054229262813129686486);
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//
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// The result is then a polynomial in v (see Eq 56 of Shaw):
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//
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return tools::evaluate_odd_polynomial<11, T, T>(c, v);
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}
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template <class T, class Policy>
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T inverse_students_t(T df, T u, T v, const Policy& pol, bool* pexact = 0)
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{
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//
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// df = number of degrees of freedom.
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// u = probablity.
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// v = 1 - u.
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// l = lanczos type to use.
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//
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BOOST_MATH_STD_USING
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bool invert = false;
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T result = 0;
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if(pexact)
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*pexact = false;
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if(u > v)
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{
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// function is symmetric, invert it:
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std::swap(u, v);
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invert = true;
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}
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if((floor(df) == df) && (df < 20))
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{
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//
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// we have integer degrees of freedom, try for the special
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// cases first:
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//
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T tolerance = ldexp(1.0f, (2 * policies::digits<T, Policy>()) / 3);
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switch(itrunc(df, Policy()))
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{
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case 1:
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{
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//
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// df = 1 is the same as the Cauchy distribution, see
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// Shaw Eq 35:
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//
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if(u == 0.5)
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result = 0;
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else
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result = -cos(constants::pi<T>() * u) / sin(constants::pi<T>() * u);
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if(pexact)
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*pexact = true;
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break;
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}
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case 2:
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{
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//
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// df = 2 has an exact result, see Shaw Eq 36:
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//
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result =(2 * u - 1) / sqrt(2 * u * v);
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if(pexact)
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*pexact = true;
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break;
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}
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case 4:
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{
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//
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// df = 4 has an exact result, see Shaw Eq 38 & 39:
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//
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T alpha = 4 * u * v;
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T root_alpha = sqrt(alpha);
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T r = 4 * cos(acos(root_alpha) / 3) / root_alpha;
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T x = sqrt(r - 4);
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result = u - 0.5f < 0 ? (T)-x : x;
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if(pexact)
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*pexact = true;
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break;
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}
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case 6:
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{
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//
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// We get numeric overflow in this area:
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//
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if(u < 1e-150)
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return (invert ? -1 : 1) * inverse_students_t_hill(df, u, pol);
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//
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// Newton-Raphson iteration of a polynomial case,
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// choice of seed value is taken from Shaw's online
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// supplement:
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//
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T a = 4 * (u - u * u);//1 - 4 * (u - 0.5f) * (u - 0.5f);
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T b = boost::math::cbrt(a);
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static const T c = static_cast<T>(0.85498797333834849467655443627193);
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T p = 6 * (1 + c * (1 / b - 1));
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T p0;
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do{
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T p2 = p * p;
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T p4 = p2 * p2;
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T p5 = p * p4;
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p0 = p;
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// next term is given by Eq 41:
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p = 2 * (8 * a * p5 - 270 * p2 + 2187) / (5 * (4 * a * p4 - 216 * p - 243));
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}while(fabs((p - p0) / p) > tolerance);
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//
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// Use Eq 45 to extract the result:
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//
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p = sqrt(p - df);
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result = (u - 0.5f) < 0 ? (T)-p : p;
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break;
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}
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#if 0
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//
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// These are Shaw's "exact" but iterative solutions
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// for even df, the numerical accuracy of these is
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// rather less than Hill's method, so these are disabled
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// for now, which is a shame because they are reasonably
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// quick to evaluate...
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//
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case 8:
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{
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//
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// Newton-Raphson iteration of a polynomial case,
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// choice of seed value is taken from Shaw's online
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// supplement:
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//
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static const T c8 = 0.85994765706259820318168359251872L;
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T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f);
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T b = pow(a, T(1) / 4);
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T p = 8 * (1 + c8 * (1 / b - 1));
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T p0 = p;
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do{
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T p5 = p * p;
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p5 *= p5 * p;
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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
|
||
|
|
|
||
|
|
|
||
|
|
|