// 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 < 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), pol));
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 > q) ? T(0.5f - q) / T(2) : T(p / 2);
T v = 1 - u; // u < 0.5 so no cancellation error
T df = a * 2;
T t = boost::math::detail::inverse_students_t(df, u, v, pol);
T x = df / (df + t * t);
*py = t * t / (df + t * t);
return x;
}
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