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authorGravatar Abseil Team <absl-team@google.com>2019-08-08 10:56:58 -0700
committerGravatar CJ Johnson <johnsoncj@google.com>2019-08-08 14:19:45 -0400
commitaa844899c937bde5d2b24f276b59997e5b668bde (patch)
treecd18e64150abc74b85bbbf6abf990f66fa47cacd /absl/random/internal/distribution_test_util.cc
parentfcb104594b0bb4b8ac306cb2f55ecdad40974683 (diff)
Creation of LTS branch "lts_2019_08_08"20190808
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Diffstat (limited to 'absl/random/internal/distribution_test_util.cc')
-rw-r--r--absl/random/internal/distribution_test_util.cc418
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diff --git a/absl/random/internal/distribution_test_util.cc b/absl/random/internal/distribution_test_util.cc
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+++ b/absl/random/internal/distribution_test_util.cc
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+// 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 {
+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<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
+} // inline namespace lts_2019_08_08
+} // namespace absl