1 // Copyright John Maddock 2007.
2 // Copyright Paul A. Bristow 2007
3 // Use, modification and distribution are subject to the
4 // Boost Software License, Version 1.0. (See accompanying file
5 // LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)
7 #ifndef BOOST_MATH_SF_DETAIL_INV_T_HPP
8 #define BOOST_MATH_SF_DETAIL_INV_T_HPP
14 #include <boost/math/special_functions/cbrt.hpp>
15 #include <boost/math/special_functions/round.hpp>
16 #include <boost/math/special_functions/trunc.hpp>
18 namespace boost{ namespace math{ namespace detail{
21 // The main method used is due to Hill:
23 // G. W. Hill, Algorithm 396, Student's t-Quantiles,
24 // Communications of the ACM, 13(10): 619-620, Oct., 1970.
26 template <class T, class Policy>
27 T inverse_students_t_hill(T ndf, T u, const Policy& pol)
30 BOOST_ASSERT(u <= 0.5);
32 T a, b, c, d, q, x, y;
35 return -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>();
39 c = ((20700 * a / b - 98) * a - 16) * a + 96.36f;
40 d = ((94.5f / (b + c) - 3) / b + 1) * sqrt(a * constants::pi<T>() / 2) * ndf;
41 y = pow(d * 2 * u, 2 / ndf);
46 // Asymptotic inverse expansion about normal:
48 x = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>();
52 c += 0.3f * (ndf - 4.5f) * (x + 0.6f);
53 c += (((0.05f * d * x - 5) * x - 7) * x - 2) * x + b;
54 y = (((((0.4f * y + 6.3f) * y + 36) * y + 94.5f) / c - y - 3) / b + 1) * x;
55 y = boost::math::expm1(a * y * y, pol);
59 y = static_cast<T>(((1 / (((ndf + 6) / (ndf * y) - 0.089f * d - 0.822f)
60 * (ndf + 2) * 3) + 0.5 / (ndf + 4)) * y - 1)
61 * (ndf + 1) / (ndf + 2) + 1 / y);
68 // Tail and body series are due to Shaw:
70 // www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf
72 // Shaw, W.T., 2006, "Sampling Student's T distribution - use of
73 // the inverse cumulative distribution function."
74 // Journal of Computational Finance, Vol 9 Issue 4, pp 37-73, Summer 2006
76 template <class T, class Policy>
77 T inverse_students_t_tail_series(T df, T v, const Policy& pol)
80 // Tail series expansion, see section 6 of Shaw's paper.
81 // w is calculated using Eq 60:
82 T w = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol)
83 * sqrt(df * constants::pi<T>()) * v;
84 // define some variables:
89 // Calculate the coefficients d(k), these depend only on the
90 // number of degrees of freedom df, so at least in theory
91 // we could tabulate these for fixed df, see p15 of Shaw:
94 d[1] = -(df + 1) / (2 * np2);
96 d[2] = -df * (df + 1) * (df + 3) / (8 * np2 * np4);
98 d[3] = -df * (df + 1) * (df + 5) * (((3 * df) + 7) * df -2) / (48 * np2 * np4 * np6);
101 d[4] = -df * (df + 1) * (df + 7) *
102 ( (((((15 * df) + 154) * df + 465) * df + 286) * df - 336) * df + 64 )
103 / (384 * np2 * np4 * np6 * (df + 8));
105 d[5] = -df * (df + 1) * (df + 3) * (df + 9)
106 * (((((((35 * df + 452) * df + 1573) * df + 600) * df - 2020) * df) + 928) * df -128)
107 / (1280 * np2 * np4 * np6 * (df + 8) * (df + 10));
111 d[6] = -df * (df + 1) * (df + 11)
112 * ((((((((((((945 * df) + 31506) * df + 425858) * df + 2980236) * df + 11266745) * df + 20675018) * df + 7747124) * df - 22574632) * df - 8565600) * df + 18108416) * df - 7099392) * df + 884736)
113 / (46080 * np2 * np4 * np6 * (df + 8) * (df + 10) * (df +12));
115 // Now bring everthing together to provide the result,
116 // this is Eq 62 of Shaw:
119 T div = pow(rn * w, 1 / df);
121 T result = tools::evaluate_polynomial<7, T, T>(d, power);
127 template <class T, class Policy>
128 T inverse_students_t_body_series(T df, T u, const Policy& pol)
132 // Body series for small N:
134 // Start with Eq 56 of Shaw:
136 T v = boost::math::tgamma_delta_ratio(df / 2, constants::half<T>(), pol)
137 * sqrt(df * constants::pi<T>()) * (u - constants::half<T>());
139 // Workspace for the polynomial coefficients:
143 // Figure out what the coefficients are, note these depend
144 // only on the degrees of freedom (Eq 57 of Shaw):
147 c[2] = static_cast<T>(0.16666666666666666667 + 0.16666666666666666667 * in);
148 c[3] = static_cast<T>((0.0083333333333333333333 * in
149 + 0.066666666666666666667) * in
150 + 0.058333333333333333333);
151 c[4] = static_cast<T>(((0.00019841269841269841270 * in
152 + 0.0017857142857142857143) * in
153 + 0.026785714285714285714) * in
154 + 0.025198412698412698413);
155 c[5] = static_cast<T>((((2.7557319223985890653e-6 * in
156 + 0.00037477954144620811287) * in
157 - 0.0011078042328042328042) * in
158 + 0.010559964726631393298) * in
159 + 0.012039792768959435626);
160 c[6] = static_cast<T>(((((2.5052108385441718775e-8 * in
161 - 0.000062705427288760622094) * in
162 + 0.00059458674042007375341) * in
163 - 0.0016095979637646304313) * in
164 + 0.0061039211560044893378) * in
165 + 0.0038370059724226390893);
166 c[7] = static_cast<T>((((((1.6059043836821614599e-10 * in
167 + 0.000015401265401265401265) * in
168 - 0.00016376804137220803887) * in
169 + 0.00069084207973096861986) * in
170 - 0.0012579159844784844785) * in
171 + 0.0010898206731540064873) * in
172 + 0.0032177478835464946576);
173 c[8] = static_cast<T>(((((((7.6471637318198164759e-13 * in
174 - 3.9851014346715404916e-6) * in
175 + 0.000049255746366361445727) * in
176 - 0.00024947258047043099953) * in
177 + 0.00064513046951456342991) * in
178 - 0.00076245135440323932387) * in
179 + 0.000033530976880017885309) * in
180 + 0.0017438262298340009980);
181 c[9] = static_cast<T>((((((((2.8114572543455207632e-15 * in
182 + 1.0914179173496789432e-6) * in
183 - 0.000015303004486655377567) * in
184 + 0.000090867107935219902229) * in
185 - 0.00029133414466938067350) * in
186 + 0.00051406605788341121363) * in
187 - 0.00036307660358786885787) * in
188 - 0.00031101086326318780412) * in
189 + 0.00096472747321388644237);
190 c[10] = static_cast<T>(((((((((8.2206352466243297170e-18 * in
191 - 3.1239569599829868045e-7) * in
192 + 4.8903045291975346210e-6) * in
193 - 0.000033202652391372058698) * in
194 + 0.00012645437628698076975) * in
195 - 0.00028690924218514613987) * in
196 + 0.00035764655430568632777) * in
197 - 0.00010230378073700412687) * in
198 - 0.00036942667800009661203) * in
199 + 0.00054229262813129686486);
201 // The result is then a polynomial in v (see Eq 56 of Shaw):
203 return tools::evaluate_odd_polynomial<11, T, T>(c, v);
206 template <class T, class Policy>
207 T inverse_students_t(T df, T u, T v, const Policy& pol, bool* pexact = 0)
210 // df = number of degrees of freedom.
213 // l = lanczos type to use.
222 // function is symmetric, invert it:
226 if((floor(df) == df) && (df < 20))
229 // we have integer degrees of freedom, try for the special
232 T tolerance = ldexp(1.0f, (2 * policies::digits<T, Policy>()) / 3);
234 switch(itrunc(df, Policy()))
239 // df = 1 is the same as the Cauchy distribution, see
245 result = -cos(constants::pi<T>() * u) / sin(constants::pi<T>() * u);
253 // df = 2 has an exact result, see Shaw Eq 36:
255 result =(2 * u - 1) / sqrt(2 * u * v);
263 // df = 4 has an exact result, see Shaw Eq 38 & 39:
266 T root_alpha = sqrt(alpha);
267 T r = 4 * cos(acos(root_alpha) / 3) / root_alpha;
269 result = u - 0.5f < 0 ? (T)-x : x;
277 // We get numeric overflow in this area:
280 return (invert ? -1 : 1) * inverse_students_t_hill(df, u, pol);
282 // Newton-Raphson iteration of a polynomial case,
283 // choice of seed value is taken from Shaw's online
286 T a = 4 * (u - u * u);//1 - 4 * (u - 0.5f) * (u - 0.5f);
287 T b = boost::math::cbrt(a);
288 static const T c = static_cast<T>(0.85498797333834849467655443627193);
289 T p = 6 * (1 + c * (1 / b - 1));
296 // next term is given by Eq 41:
297 p = 2 * (8 * a * p5 - 270 * p2 + 2187) / (5 * (4 * a * p4 - 216 * p - 243));
298 }while(fabs((p - p0) / p) > tolerance);
300 // Use Eq 45 to extract the result:
303 result = (u - 0.5f) < 0 ? (T)-p : p;
308 // These are Shaw's "exact" but iterative solutions
309 // for even df, the numerical accuracy of these is
310 // rather less than Hill's method, so these are disabled
311 // for now, which is a shame because they are reasonably
312 // quick to evaluate...
317 // Newton-Raphson iteration of a polynomial case,
318 // choice of seed value is taken from Shaw's online
321 static const T c8 = 0.85994765706259820318168359251872L;
322 T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f);
323 T b = pow(a, T(1) / 4);
324 T p = 8 * (1 + c8 * (1 / b - 1));
330 // Next term is given by Eq 42:
331 p = 2 * (3 * p + (640 * (160 + p * (24 + p * (p + 4)))) / (-5120 + p * (-2048 - 960 * p + a * p5))) / 7;
332 }while(fabs((p - p0) / p) > tolerance);
334 // Use Eq 45 to extract the result:
337 result = (u - 0.5f) < 0 ? -p : p;
343 // Newton-Raphson iteration of a polynomial case,
344 // choice of seed value is taken from Shaw's online
347 static const T c10 = 0.86781292867813396759105692122285L;
348 T a = 4 * (u - u * u); //1 - 4 * (u - 0.5f) * (u - 0.5f);
349 T b = pow(a, T(1) / 5);
350 T p = 10 * (1 + c10 * (1 / b - 1));
356 // Next term given by Eq 43:
357 p = (8 * p) / 9 + (218750 * (21875 + 4 * p * (625 + p * (75 + 2 * p * (5 + p))))) /
358 (9 * (-68359375 + 8 * p * (-2343750 + p * (-546875 - 175000 * p + 8 * a * p6))));
359 }while(fabs((p - p0) / p) > tolerance);
361 // Use Eq 45 to extract the result:
364 result = (u - 0.5f) < 0 ? -p : p;
377 result = -boost::math::erfc_inv(2 * u, pol) * constants::root_two<T>();
378 if((pexact) && (df >= 1e20))
384 // Use a roughly linear scheme to choose between Shaw's
385 // tail series and body series:
387 T crossover = 0.2742f - df * 0.0242143f;
390 result = boost::math::detail::inverse_students_t_body_series(df, u, pol);
394 result = boost::math::detail::inverse_students_t_tail_series(df, u, pol);
400 // Use Hill's method except in the exteme tails
401 // where we use Shaw's tail series.
402 // The crossover point is roughly exponential in -df:
404 T crossover = ldexp(1.0f, iround(T(df / -0.654f), typename policies::normalise<Policy, policies::rounding_error<policies::ignore_error> >::type()));
407 result = boost::math::detail::inverse_students_t_hill(df, u, pol);
411 result = boost::math::detail::inverse_students_t_tail_series(df, u, pol);
415 return invert ? (T)-result : result;
418 template <class T, class Policy>
419 inline T find_ibeta_inv_from_t_dist(T a, T p, T /*q*/, T* py, const Policy& pol)
424 T t = boost::math::detail::inverse_students_t(df, u, v, pol);
425 *py = t * t / (df + t * t);
426 return df / (df + t * t);
429 template <class T, class Policy>
430 inline T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::false_*)
434 // Need to use inverse incomplete beta to get
435 // required precision so not so fast:
437 T probability = (p > 0.5) ? 1 - p : p;
439 x = ibeta_inv(df / 2, T(0.5), 2 * probability, &y, pol);
440 if(df * y > tools::max_value<T>() * x)
441 t = policies::raise_overflow_error<T>("boost::math::students_t_quantile<%1%>(%1%,%1%)", 0, pol);
443 t = sqrt(df * y / x);
445 // Figure out sign based on the size of p:
452 template <class T, class Policy>
453 T fast_students_t_quantile_imp(T df, T p, const Policy& pol, const mpl::true_*)
457 if((df < 2) && (floor(df) != df))
458 return boost::math::detail::fast_students_t_quantile_imp(df, p, pol, static_cast<mpl::false_*>(0));
465 // Get an estimate of the result:
468 T t = inverse_students_t(df, p, T(1-p), pol, &exact);
469 if((t == 0) || exact)
470 return invert ? -t : t; // can't do better!
472 // Change variables to inverse incomplete beta:
475 T xb = df / (df + t2);
476 T y = t2 / (df + t2);
479 // t can be so large that x underflows,
480 // just return our estimate in that case:
485 // Get incomplete beta and it's derivative:
488 T f0 = xb < y ? ibeta_imp(a, constants::half<T>(), xb, pol, false, true, &f1)
489 : ibeta_imp(constants::half<T>(), a, y, pol, true, true, &f1);
491 // Get cdf from incomplete beta result:
493 // Get pdf from derivative:
494 T p1 = f1 * sqrt(y * xb * xb * xb / df);
496 // Second derivative divided by p1:
500 // In> PrettyForm(Simplify(D(t) (1 + t^2/v) ^ (-(v+1)/2)))
503 // | -| ----- + 1 | |
508 // | ( v + 1 ) * | v | * t |
509 // ---------------------------------------------
512 // Which after some manipulation is:
514 // -p1 * t * (df + 1) / (t^2 + df)
516 T p2 = t * (df + 1) / (t * t + df);
519 t += p0 / (p1 + p0 * p2 / 2);
520 return !invert ? -t : t;
523 template <class T, class Policy>
524 inline T fast_students_t_quantile(T df, T p, const Policy& pol)
526 typedef typename policies::evaluation<T, Policy>::type value_type;
527 typedef typename policies::normalise<
529 policies::promote_float<false>,
530 policies::promote_double<false>,
531 policies::discrete_quantile<>,
532 policies::assert_undefined<> >::type forwarding_policy;
535 (std::numeric_limits<T>::digits <= 53)
537 (std::numeric_limits<T>::is_specialized)
539 (std::numeric_limits<T>::radix == 2)
541 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%)");
546 #endif // BOOST_MATH_SF_DETAIL_INV_T_HPP