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Diffstat (limited to 'absl/random/internal/distribution_test_util.cc')
| -rw-r--r-- | absl/random/internal/distribution_test_util.cc | 416 |
1 files changed, 416 insertions, 0 deletions
diff --git a/absl/random/internal/distribution_test_util.cc b/absl/random/internal/distribution_test_util.cc new file mode 100644 index 00000000..85c8d596 --- /dev/null +++ b/absl/random/internal/distribution_test_util.cc @@ -0,0 +1,416 @@ +// 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 <cassert> +#include <cmath> +#include <string> +#include <vector> + +#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 { +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<const double> data_points) { + DistributionMoments result; + + // Compute m1 + for (double x : data_points) { + result.n++; + result.mean += x; + } + result.mean /= static_cast<double>(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<double>(result.n - 1); + + result.skewness /= static_cast<double>(result.n); + result.skewness /= std::pow(result.variance, 1.5); + + result.kurtosis /= static_cast<double>(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<int>(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<double>::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<double>::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<double>::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<double>::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<double>(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<double>(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 +} // namespace absl |