// Copyright 2017 The Abseil Authors. // // Licensed under the Apache License, Version 2.0 (the "License"); // you may not use this file except in compliance with the License. // You may obtain a copy of the License at // // https://www.apache.org/licenses/LICENSE-2.0 // // Unless required by applicable law or agreed to in writing, software // distributed under the License is distributed on an "AS IS" BASIS, // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. // See the License for the specific language governing permissions and // limitations under the License. #include "absl/random/internal/distribution_test_util.h" #include #include #include #include #include "absl/base/internal/raw_logging.h" #include "absl/base/macros.h" #include "absl/strings/str_cat.h" #include "absl/strings/str_format.h" namespace absl { inline namespace lts_2019_08_08 { namespace random_internal { namespace { #if defined(__EMSCRIPTEN__) // Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found. inline double fma(double x, double y, double z) { return (x * y) + z; } #endif } // namespace DistributionMoments ComputeDistributionMoments( absl::Span data_points) { DistributionMoments result; // Compute m1 for (double x : data_points) { result.n++; result.mean += x; } result.mean /= static_cast(result.n); // Compute m2, m3, m4 for (double x : data_points) { double v = x - result.mean; result.variance += v * v; result.skewness += v * v * v; result.kurtosis += v * v * v * v; } result.variance /= static_cast(result.n - 1); result.skewness /= static_cast(result.n); result.skewness /= std::pow(result.variance, 1.5); result.kurtosis /= static_cast(result.n); result.kurtosis /= std::pow(result.variance, 2.0); return result; // When validating the min/max count, the following confidence intervals may // be of use: // 3.291 * stddev = 99.9% CI // 2.576 * stddev = 99% CI // 1.96 * stddev = 95% CI // 1.65 * stddev = 90% CI } std::ostream& operator<<(std::ostream& os, const DistributionMoments& moments) { return os << absl::StrFormat("mean=%f, stddev=%f, skewness=%f, kurtosis=%f", moments.mean, std::sqrt(moments.variance), moments.skewness, moments.kurtosis); } double InverseNormalSurvival(double x) { // inv_sf(u) = -sqrt(2) * erfinv(2u-1) static constexpr double kSqrt2 = 1.4142135623730950488; return -kSqrt2 * absl::random_internal::erfinv(2 * x - 1.0); } bool Near(absl::string_view msg, double actual, double expected, double bound) { assert(bound > 0.0); double delta = fabs(expected - actual); if (delta < bound) { return true; } std::string formatted = absl::StrCat( msg, " actual=", actual, " expected=", expected, " err=", delta / bound); ABSL_RAW_LOG(INFO, "%s", formatted.c_str()); return false; } // TODO(absl-team): Replace with an "ABSL_HAVE_SPECIAL_MATH" and try // to use std::beta(). As of this writing P0226R1 is not implemented // in libc++: http://libcxx.llvm.org/cxx1z_status.html double beta(double p, double q) { // Beta(x, y) = Gamma(x) * Gamma(y) / Gamma(x+y) double lbeta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q); return std::exp(lbeta); } // Approximation to inverse of the Error Function in double precision. // (http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf) double erfinv(double x) { #if !defined(__EMSCRIPTEN__) using std::fma; #endif double w = 0.0; double p = 0.0; w = -std::log((1.0 - x) * (1.0 + x)); if (w < 6.250000) { w = w - 3.125000; p = -3.6444120640178196996e-21; p = fma(p, w, -1.685059138182016589e-19); p = fma(p, w, 1.2858480715256400167e-18); p = fma(p, w, 1.115787767802518096e-17); p = fma(p, w, -1.333171662854620906e-16); p = fma(p, w, 2.0972767875968561637e-17); p = fma(p, w, 6.6376381343583238325e-15); p = fma(p, w, -4.0545662729752068639e-14); p = fma(p, w, -8.1519341976054721522e-14); p = fma(p, w, 2.6335093153082322977e-12); p = fma(p, w, -1.2975133253453532498e-11); p = fma(p, w, -5.4154120542946279317e-11); p = fma(p, w, 1.051212273321532285e-09); p = fma(p, w, -4.1126339803469836976e-09); p = fma(p, w, -2.9070369957882005086e-08); p = fma(p, w, 4.2347877827932403518e-07); p = fma(p, w, -1.3654692000834678645e-06); p = fma(p, w, -1.3882523362786468719e-05); p = fma(p, w, 0.0001867342080340571352); p = fma(p, w, -0.00074070253416626697512); p = fma(p, w, -0.0060336708714301490533); p = fma(p, w, 0.24015818242558961693); p = fma(p, w, 1.6536545626831027356); } else if (w < 16.000000) { w = std::sqrt(w) - 3.250000; p = 2.2137376921775787049e-09; p = fma(p, w, 9.0756561938885390979e-08); p = fma(p, w, -2.7517406297064545428e-07); p = fma(p, w, 1.8239629214389227755e-08); p = fma(p, w, 1.5027403968909827627e-06); p = fma(p, w, -4.013867526981545969e-06); p = fma(p, w, 2.9234449089955446044e-06); p = fma(p, w, 1.2475304481671778723e-05); p = fma(p, w, -4.7318229009055733981e-05); p = fma(p, w, 6.8284851459573175448e-05); p = fma(p, w, 2.4031110387097893999e-05); p = fma(p, w, -0.0003550375203628474796); p = fma(p, w, 0.00095328937973738049703); p = fma(p, w, -0.0016882755560235047313); p = fma(p, w, 0.0024914420961078508066); p = fma(p, w, -0.0037512085075692412107); p = fma(p, w, 0.005370914553590063617); p = fma(p, w, 1.0052589676941592334); p = fma(p, w, 3.0838856104922207635); } else { w = std::sqrt(w) - 5.000000; p = -2.7109920616438573243e-11; p = fma(p, w, -2.5556418169965252055e-10); p = fma(p, w, 1.5076572693500548083e-09); p = fma(p, w, -3.7894654401267369937e-09); p = fma(p, w, 7.6157012080783393804e-09); p = fma(p, w, -1.4960026627149240478e-08); p = fma(p, w, 2.9147953450901080826e-08); p = fma(p, w, -6.7711997758452339498e-08); p = fma(p, w, 2.2900482228026654717e-07); p = fma(p, w, -9.9298272942317002539e-07); p = fma(p, w, 4.5260625972231537039e-06); p = fma(p, w, -1.9681778105531670567e-05); p = fma(p, w, 7.5995277030017761139e-05); p = fma(p, w, -0.00021503011930044477347); p = fma(p, w, -0.00013871931833623122026); p = fma(p, w, 1.0103004648645343977); p = fma(p, w, 4.8499064014085844221); } return p * x; } namespace { // Direct implementation of AS63, BETAIN() // https://www.jstor.org/stable/2346797?seq=3#page_scan_tab_contents. // // BETAIN(x, p, q, beta) // x: the value of the upper limit x. // p: the value of the parameter p. // q: the value of the parameter q. // beta: the value of ln B(p, q) // double BetaIncompleteImpl(const double x, const double p, const double q, const double beta) { if (p < (p + q) * x) { // Incomplete beta function is symmetrical, so return the complement. return 1. - BetaIncompleteImpl(1.0 - x, q, p, beta); } double psq = p + q; const double kErr = 1e-14; const double xc = 1. - x; const double pre = std::exp(p * std::log(x) + (q - 1.) * std::log(xc) - beta) / p; double term = 1.; double ai = 1.; double result = 1.; int ns = static_cast(q + xc * psq); // Use the soper reduction forumla. double rx = (ns == 0) ? x : x / xc; double temp = q - ai; for (;;) { term = term * temp * rx / (p + ai); result = result + term; temp = std::fabs(term); if (temp < kErr && temp < kErr * result) { return result * pre; } ai = ai + 1.; --ns; if (ns >= 0) { temp = q - ai; if (ns == 0) { rx = x; } } else { temp = psq; psq = psq + 1.; } } // NOTE: See also TOMS Alogrithm 708. // http://www.netlib.org/toms/index.html // // NOTE: The NWSC library also includes BRATIO / ISUBX (p87) // https://archive.org/details/DTIC_ADA261511/page/n75 } // Direct implementation of AS109, XINBTA(p, q, beta, alpha) // https://www.jstor.org/stable/2346798?read-now=1&seq=4#page_scan_tab_contents // https://www.jstor.org/stable/2346887?seq=1#page_scan_tab_contents // // XINBTA(p, q, beta, alhpa) // p: the value of the parameter p. // q: the value of the parameter q. // beta: the value of ln B(p, q) // alpha: the value of the lower tail area. // double BetaIncompleteInvImpl(const double p, const double q, const double beta, const double alpha) { if (alpha < 0.5) { // Inverse Incomplete beta function is symmetrical, return the complement. return 1. - BetaIncompleteInvImpl(q, p, beta, 1. - alpha); } const double kErr = 1e-14; double value = kErr; // Compute the initial estimate. { double r = std::sqrt(-std::log(alpha * alpha)); double y = r - fma(r, 0.27061, 2.30753) / fma(r, fma(r, 0.04481, 0.99229), 1.0); if (p > 1. && q > 1.) { r = (y * y - 3.) / 6.; double s = 1. / (p + p - 1.); double t = 1. / (q + q - 1.); double h = 2. / s + t; double w = y * std::sqrt(h + r) / h - (t - s) * (r + 5. / 6. - t / (3. * h)); value = p / (p + q * std::exp(w + w)); } else { r = q + q; double t = 1.0 / (9. * q); double u = 1.0 - t + y * std::sqrt(t); t = r * (u * u * u); if (t <= 0) { value = 1.0 - std::exp((std::log((1.0 - alpha) * q) + beta) / q); } else { t = (4.0 * p + r - 2.0) / t; if (t <= 1) { value = std::exp((std::log(alpha * p) + beta) / p); } else { value = 1.0 - 2.0 / (t + 1.0); } } } } // Solve for x using a modified newton-raphson method using the function // BetaIncomplete. { value = std::max(value, kErr); value = std::min(value, 1.0 - kErr); const double r = 1.0 - p; const double t = 1.0 - q; double y; double yprev = 0; double sq = 1; double prev = 1; for (;;) { if (value < 0 || value > 1.0) { // Error case; value went infinite. return std::numeric_limits::infinity(); } else if (value == 0 || value == 1) { y = value; } else { y = BetaIncompleteImpl(value, p, q, beta); if (!std::isfinite(y)) { return y; } } y = (y - alpha) * std::exp(beta + r * std::log(value) + t * std::log(1.0 - value)); if (y * yprev <= 0) { prev = std::max(sq, std::numeric_limits::min()); } double g = 1.0; for (;;) { const double adj = g * y; const double adj_sq = adj * adj; if (adj_sq >= prev) { g = g / 3.0; continue; } const double tx = value - adj; if (tx < 0 || tx > 1) { g = g / 3.0; continue; } if (prev < kErr) { return value; } if (y * y < kErr) { return value; } if (tx == value) { return value; } if (tx == 0 || tx == 1) { g = g / 3.0; continue; } value = tx; yprev = y; break; } } } // NOTES: See also: Asymptotic inversion of the incomplete beta function. // https://core.ac.uk/download/pdf/82140723.pdf // // NOTE: See the Boost library documentation as well: // https://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html } } // namespace double BetaIncomplete(const double x, const double p, const double q) { // Error cases. if (p < 0 || q < 0 || x < 0 || x > 1.0) { return std::numeric_limits::infinity(); } if (x == 0 || x == 1) { return x; } // ln(Beta(p, q)) double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q); return BetaIncompleteImpl(x, p, q, beta); } double BetaIncompleteInv(const double p, const double q, const double alpha) { // Error cases. if (p < 0 || q < 0 || alpha < 0 || alpha > 1.0) { return std::numeric_limits::infinity(); } if (alpha == 0 || alpha == 1) { return alpha; } // ln(Beta(p, q)) double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q); return BetaIncompleteInvImpl(p, q, beta, alpha); } // Given `num_trials` trials each with probability `p` of success, the // probability of no failures is `p^k`. To ensure the probability of a failure // is no more than `p_fail`, it must be that `p^k == 1 - p_fail`. This function // computes `p` from that equation. double RequiredSuccessProbability(const double p_fail, const int num_trials) { double p = std::exp(std::log(1.0 - p_fail) / static_cast(num_trials)); ABSL_ASSERT(p > 0); return p; } double ZScore(double expected_mean, const DistributionMoments& moments) { return (moments.mean - expected_mean) / (std::sqrt(moments.variance) / std::sqrt(static_cast(moments.n))); } double MaxErrorTolerance(double acceptance_probability) { double one_sided_pvalue = 0.5 * (1.0 - acceptance_probability); const double max_err = InverseNormalSurvival(one_sided_pvalue); ABSL_ASSERT(max_err > 0); return max_err; } } // namespace random_internal } // inline namespace lts_2019_08_08 } // namespace absl