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