// 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/gaussian_distribution.h"
#include <algorithm>
#include <cmath>
#include <cstddef>
#include <ios>
#include <iterator>
#include <random>
#include <string>
#include <vector>
#include "gmock/gmock.h"
#include "gtest/gtest.h"
#include "absl/base/internal/raw_logging.h"
#include "absl/base/macros.h"
#include "absl/random/internal/chi_square.h"
#include "absl/random/internal/distribution_test_util.h"
#include "absl/random/internal/sequence_urbg.h"
#include "absl/random/random.h"
#include "absl/strings/str_cat.h"
#include "absl/strings/str_format.h"
#include "absl/strings/str_replace.h"
#include "absl/strings/strip.h"
namespace {
using absl::random_internal::kChiSquared;
template <typename RealType>
class GaussianDistributionInterfaceTest : public ::testing::Test {};
using RealTypes = ::testing::Types<float, double, long double>;
TYPED_TEST_CASE(GaussianDistributionInterfaceTest, RealTypes);
TYPED_TEST(GaussianDistributionInterfaceTest, SerializeTest) {
using param_type =
typename absl::gaussian_distribution<TypeParam>::param_type;
const TypeParam kParams[] = {
// Cases around 1.
1, //
std::nextafter(TypeParam(1), TypeParam(0)), // 1 - epsilon
std::nextafter(TypeParam(1), TypeParam(2)), // 1 + epsilon
// Arbitrary values.
TypeParam(1e-8), TypeParam(1e-4), TypeParam(2), TypeParam(1e4),
TypeParam(1e8), TypeParam(1e20), TypeParam(2.5),
// Boundary cases.
std::numeric_limits<TypeParam>::infinity(),
std::numeric_limits<TypeParam>::max(),
std::numeric_limits<TypeParam>::epsilon(),
std::nextafter(std::numeric_limits<TypeParam>::min(),
TypeParam(1)), // min + epsilon
std::numeric_limits<TypeParam>::min(), // smallest normal
// There are some errors dealing with denorms on apple platforms.
std::numeric_limits<TypeParam>::denorm_min(), // smallest denorm
std::numeric_limits<TypeParam>::min() / 2,
std::nextafter(std::numeric_limits<TypeParam>::min(),
TypeParam(0)), // denorm_max
};
constexpr int kCount = 1000;
absl::InsecureBitGen gen;
// Use a loop to generate the combinations of {+/-x, +/-y}, and assign x, y to
// all values in kParams,
for (const auto mod : {0, 1, 2, 3}) {
for (const auto x : kParams) {
if (!std::isfinite(x)) continue;
for (const auto y : kParams) {
const TypeParam mean = (mod & 0x1) ? -x : x;
const TypeParam stddev = (mod & 0x2) ? -y : y;
const param_type param(mean, stddev);
absl::gaussian_distribution<TypeParam> before(mean, stddev);
EXPECT_EQ(before.mean(), param.mean());
EXPECT_EQ(before.stddev(), param.stddev());
{
absl::gaussian_distribution<TypeParam> via_param(param);
EXPECT_EQ(via_param, before);
EXPECT_EQ(via_param.param(), before.param());
}
// Smoke test.
auto sample_min = before.max();
auto sample_max = before.min();
for (int i = 0; i < kCount; i++) {
auto sample = before(gen);
if (sample > sample_max) sample_max = sample;
if (sample < sample_min) sample_min = sample;
EXPECT_GE(sample, before.min()) << before;
EXPECT_LE(sample, before.max()) << before;
}
if (!std::is_same<TypeParam, long double>::value) {
ABSL_INTERNAL_LOG(
INFO, absl::StrFormat("Range{%f, %f}: %f, %f", mean, stddev,
sample_min, sample_max));
}
std::stringstream ss;
ss << before;
if (!std::isfinite(mean) || !std::isfinite(stddev)) {
// Streams do not parse inf/nan.
continue;
}
// Validate stream serialization.
absl::gaussian_distribution<TypeParam> after(-0.53f, 2.3456f);
EXPECT_NE(before.mean(), after.mean());
EXPECT_NE(before.stddev(), after.stddev());
EXPECT_NE(before.param(), after.param());
EXPECT_NE(before, after);
ss >> after;
#if defined(__powerpc64__) || defined(__PPC64__) || defined(__powerpc__) || \
defined(__ppc__) || defined(__PPC__) || defined(__EMSCRIPTEN__)
if (std::is_same<TypeParam, long double>::value) {
// Roundtripping floating point values requires sufficient precision
// to reconstruct the exact value. It turns out that long double
// has some errors doing this on ppc, particularly for values
// near {1.0 +/- epsilon}.
//
// Emscripten is even worse, implementing long double as a 128-bit
// type, but shipping with a strtold() that doesn't support that.
if (mean <= std::numeric_limits<double>::max() &&
mean >= std::numeric_limits<double>::lowest()) {
EXPECT_EQ(static_cast<double>(before.mean()),
static_cast<double>(after.mean()))
<< ss.str();
}
if (stddev <= std::numeric_limits<double>::max() &&
stddev >= std::numeric_limits<double>::lowest()) {
EXPECT_EQ(static_cast<double>(before.stddev()),
static_cast<double>(after.stddev()))
<< ss.str();
}
continue;
}
#endif
EXPECT_EQ(before.mean(), after.mean());
EXPECT_EQ(before.stddev(), after.stddev()) //
<< ss.str() << " " //
<< (ss.good() ? "good " : "") //
<< (ss.bad() ? "bad " : "") //
<< (ss.eof() ? "eof " : "") //
<< (ss.fail() ? "fail " : "");
}
}
}
}
// http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm
class GaussianModel {
public:
GaussianModel(double mean, double stddev) : mean_(mean), stddev_(stddev) {}
double mean() const { return mean_; }
double variance() const { return stddev() * stddev(); }
double stddev() const { return stddev_; }
double skew() const { return 0; }
double kurtosis() const { return 3.0; }
// The inverse CDF, or PercentPoint function.
double InverseCDF(double p) {
ABSL_ASSERT(p >= 0.0);
ABSL_ASSERT(p < 1.0);
return mean() + stddev() * -absl::random_internal::InverseNormalSurvival(p);
}
private:
const double mean_;
const double stddev_;
};
struct Param {
double mean;
double stddev;
double p_fail; // Z-Test probability of failure.
int trials; // Z-Test trials.
};
// GaussianDistributionTests implements a z-test for the gaussian
// distribution.
class GaussianDistributionTests : public testing::TestWithParam<Param>,
public GaussianModel {
public:
GaussianDistributionTests()
: GaussianModel(GetParam().mean, GetParam().stddev) {}
// SingleZTest provides a basic z-squared test of the mean vs. expected
// mean for data generated by the poisson distribution.
template <typename D>
bool SingleZTest(const double p, const size_t samples);
// SingleChiSquaredTest provides a basic chi-squared test of the normal
// distribution.
template <typename D>
double SingleChiSquaredTest();
// We use a fixed bit generator for distribution accuracy tests. This allows
// these tests to be deterministic, while still testing the qualify of the
// implementation.
absl::random_internal::pcg64_2018_engine rng_{0x2B7E151628AED2A6};
};
template <typename D>
bool GaussianDistributionTests::SingleZTest(const double p,
const size_t samples) {
D dis(mean(), stddev());
std::vector<double> data;
data.reserve(samples);
for (size_t i = 0; i < samples; i++) {
const double x = dis(rng_);
data.push_back(x);
}
const double max_err = absl::random_internal::MaxErrorTolerance(p);
const auto m = absl::random_internal::ComputeDistributionMoments(data);
const double z = absl::random_internal::ZScore(mean(), m);
const bool pass = absl::random_internal::Near("z", z, 0.0, max_err);
// NOTE: Informational statistical test:
//
// Compute the Jarque-Bera test statistic given the excess skewness
// and kurtosis. The statistic is drawn from a chi-square(2) distribution.
// https://en.wikipedia.org/wiki/Jarque%E2%80%93Bera_test
//
// The null-hypothesis (normal distribution) is rejected when
// (p = 0.05 => jb > 5.99)
// (p = 0.01 => jb > 9.21)
// NOTE: JB has a large type-I error rate, so it will reject the
// null-hypothesis even when it is true more often than the z-test.
//
const double jb =
static_cast<double>(m.n) / 6.0 *
(std::pow(m.skewness, 2.0) + std::pow(m.kurtosis - 3.0, 2.0) / 4.0);
if (!pass || jb > 9.21) {
ABSL_INTERNAL_LOG(
INFO, absl::StrFormat("p=%f max_err=%f\n"
" mean=%f vs. %f\n"
" stddev=%f vs. %f\n"
" skewness=%f vs. %f\n"
" kurtosis=%f vs. %f\n"
" z=%f vs. 0\n"
" jb=%f vs. 9.21",
p, max_err, m.mean, mean(), std::sqrt(m.variance),
stddev(), m.skewness, skew(), m.kurtosis,
kurtosis(), z, jb));
}
return pass;
}
template <typename D>
double GaussianDistributionTests::SingleChiSquaredTest() {
const size_t kSamples = 10000;
const int kBuckets = 50;
// The InverseCDF is the percent point function of the
// distribution, and can be used to assign buckets
// roughly uniformly.
std::vector<double> cutoffs;
const double kInc = 1.0 / static_cast<double>(kBuckets);
for (double p = kInc; p < 1.0; p += kInc) {
cutoffs.push_back(InverseCDF(p));
}
if (cutoffs.back() != std::numeric_limits<double>::infinity()) {
cutoffs.push_back(std::numeric_limits<double>::infinity());
}
D dis(mean(), stddev());
std::vector<int32_t> counts(cutoffs.size(), 0);
for (int j = 0; j < kSamples; j++) {
const double x = dis(rng_);
auto it = std::upper_bound(cutoffs.begin(), cutoffs.end(), x);
counts[std::distance(cutoffs.begin(), it)]++;
}
// Null-hypothesis is that the distribution is a gaussian distribution
// with the provided mean and stddev (not estimated from the data).
const int dof = static_cast<int>(counts.size()) - 1;
// Our threshold for logging is 1-in-50.
const double threshold = absl::random_internal::ChiSquareValue(dof, 0.98);
const double expected =
static_cast<double>(kSamples) / static_cast<double>(counts.size());
double chi_square = absl::random_internal::ChiSquareWithExpected(
std::begin(counts), std::end(counts), expected);
double p = absl::random_internal::ChiSquarePValue(chi_square, dof);
// Log if the chi_square value is above the threshold.
if (chi_square > threshold) {
for (int i = 0; i < cutoffs.size(); i++) {
ABSL_INTERNAL_LOG(
INFO, absl::StrFormat("%d : (%f) = %d", i, cutoffs[i], counts[i]));
}
ABSL_INTERNAL_LOG(
INFO, absl::StrCat("mean=", mean(), " stddev=", stddev(), "\n", //
" expected ", expected, "\n", //
kChiSquared, " ", chi_square, " (", p, ")\n", //
kChiSquared, " @ 0.98 = ", threshold));
}
return p;
}
TEST_P(GaussianDistributionTests, ZTest) {
// TODO(absl-team): Run these tests against std::normal_distribution<double>
// to validate outcomes are similar.
const size_t kSamples = 10000;
const auto& param = GetParam();
const int expected_failures =
std::max(1, static_cast<int>(std::ceil(param.trials * param.p_fail)));
const double p = absl::random_internal::RequiredSuccessProbability(
param.p_fail, param.trials);
int failures = 0;
for (int i = 0; i < param.trials; i++) {
failures +=
SingleZTest<absl::gaussian_distribution<double>>(p, kSamples) ? 0 : 1;
}
EXPECT_LE(failures, expected_failures);
}
TEST_P(GaussianDistributionTests, ChiSquaredTest) {
const int kTrials = 20;
int failures = 0;
for (int i = 0; i < kTrials; i++) {
double p_value =
SingleChiSquaredTest<absl::gaussian_distribution<double>>();
if (p_value < 0.0025) { // 1/400
failures++;
}
}
// There is a 0.05% chance of producing at least one failure, so raise the
// failure threshold high enough to allow for a flake rate of less than one in
// 10,000.
EXPECT_LE(failures, 4);
}
std::vector<Param> GenParams() {
return {
// Mean around 0.
Param{0.0, 1.0, 0.01, 100},
Param{0.0, 1e2, 0.01, 100},
Param{0.0, 1e4, 0.01, 100},
Param{0.0, 1e8, 0.01, 100},
Param{0.0, 1e16, 0.01, 100},
Param{0.0, 1e-3, 0.01, 100},
Param{0.0, 1e-5, 0.01, 100},
Param{0.0, 1e-9, 0.01, 100},
Param{0.0, 1e-17, 0.01, 100},
// Mean around 1.
Param{1.0, 1.0, 0.01, 100},
Param{1.0, 1e2, 0.01, 100},
Param{1.0, 1e-2, 0.01, 100},
// Mean around 100 / -100
Param{1e2, 1.0, 0.01, 100},
Param{-1e2, 1.0, 0.01, 100},
Param{1e2, 1e6, 0.01, 100},
Param{-1e2, 1e6, 0.01, 100},
// More extreme
Param{1e4, 1e4, 0.01, 100},
Param{1e8, 1e4, 0.01, 100},
Param{1e12, 1e4, 0.01, 100},
};
}
std::string ParamName(const ::testing::TestParamInfo<Param>& info) {
const auto& p = info.param;
std::string name = absl::StrCat("mean_", absl::SixDigits(p.mean), "__stddev_",
absl::SixDigits(p.stddev));
return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
}
INSTANTIATE_TEST_SUITE_P(All, GaussianDistributionTests,
::testing::ValuesIn(GenParams()), ParamName);
// NOTE: absl::gaussian_distribution is not guaranteed to be stable.
TEST(GaussianDistributionTest, StabilityTest) {
// absl::gaussian_distribution stability relies on the underlying zignor
// data, absl::random_interna::RandU64ToDouble, std::exp, std::log, and
// std::abs.
absl::random_internal::sequence_urbg urbg(
{0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull, 0xC332DDEFBE6C5AA5ull,
0x6558218568AB9702ull, 0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull,
0xECDD4775619F1510ull, 0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull,
0xB5735C904C70A239ull, 0xD59E9E0BCBAADE14ull, 0xEECC86BC60622CA7ull});
std::vector<int> output(11);
{
absl::gaussian_distribution<double> dist;
std::generate(std::begin(output), std::end(output),
[&] { return static_cast<int>(10000000.0 * dist(urbg)); });
EXPECT_EQ(13, urbg.invocations());
EXPECT_THAT(output, //
testing::ElementsAre(1494, 25518841, 9991550, 1351856,
-20373238, 3456682, 333530, -6804981,
-15279580, -16459654, 1494));
}
urbg.reset();
{
absl::gaussian_distribution<float> dist;
std::generate(std::begin(output), std::end(output),
[&] { return static_cast<int>(1000000.0f * dist(urbg)); });
EXPECT_EQ(13, urbg.invocations());
EXPECT_THAT(
output, //
testing::ElementsAre(149, 2551884, 999155, 135185, -2037323, 345668,
33353, -680498, -1527958, -1645965, 149));
}
}
// This is an implementation-specific test. If any part of the implementation
// changes, then it is likely that this test will change as well.
// Also, if dependencies of the distribution change, such as RandU64ToDouble,
// then this is also likely to change.
TEST(GaussianDistributionTest, AlgorithmBounds) {
absl::gaussian_distribution<double> dist;
// In ~95% of cases, a single value is used to generate the output.
// for all inputs where |x| < 0.750461021389 this should be the case.
//
// The exact constraints are based on the ziggurat tables, and any
// changes to the ziggurat tables may require adjusting these bounds.
//
// for i in range(0, len(X)-1):
// print i, X[i+1]/X[i], (X[i+1]/X[i] > 0.984375)
//
// 0.125 <= |values| <= 0.75
const uint64_t kValues[] = {
0x1000000000000100ull, 0x2000000000000100ull, 0x3000000000000100ull,
0x4000000000000100ull, 0x5000000000000100ull, 0x6000000000000100ull,
// negative values
0x9000000000000100ull, 0xa000000000000100ull, 0xb000000000000100ull,
0xc000000000000100ull, 0xd000000000000100ull, 0xe000000000000100ull};
// 0.875 <= |values| <= 0.984375
const uint64_t kExtraValues[] = {
0x7000000000000100ull, 0x7800000000000100ull, //
0x7c00000000000100ull, 0x7e00000000000100ull, //
// negative values
0xf000000000000100ull, 0xf800000000000100ull, //
0xfc00000000000100ull, 0xfe00000000000100ull};
auto make_box = [](uint64_t v, uint64_t box) {
return (v & 0xffffffffffffff80ull) | box;
};
// The box is the lower 7 bits of the value. When the box == 0, then
// the algorithm uses an escape hatch to select the result for large
// outputs.
for (uint64_t box = 0; box < 0x7f; box++) {
for (const uint64_t v : kValues) {
// Extra values are added to the sequence to attempt to avoid
// infinite loops from rejection sampling on bugs/errors.
absl::random_internal::sequence_urbg urbg(
{make_box(v, box), 0x0003eb76f6f7f755ull, 0x5FCEA50FDB2F953Bull});
auto a = dist(urbg);
EXPECT_EQ(1, urbg.invocations()) << box << " " << std::hex << v;
if (v & 0x8000000000000000ull) {
EXPECT_LT(a, 0.0) << box << " " << std::hex << v;
} else {
EXPECT_GT(a, 0.0) << box << " " << std::hex << v;
}
}
if (box > 10 && box < 100) {
// The center boxes use the fast algorithm for more
// than 98.4375% of values.
for (const uint64_t v : kExtraValues) {
absl::random_internal::sequence_urbg urbg(
{make_box(v, box), 0x0003eb76f6f7f755ull, 0x5FCEA50FDB2F953Bull});
auto a = dist(urbg);
EXPECT_EQ(1, urbg.invocations()) << box << " " << std::hex << v;
if (v & 0x8000000000000000ull) {
EXPECT_LT(a, 0.0) << box << " " << std::hex << v;
} else {
EXPECT_GT(a, 0.0) << box << " " << std::hex << v;
}
}
}
}
// When the box == 0, the fallback algorithm uses a ratio of uniforms,
// which consumes 2 additional values from the urbg.
// Fallback also requires that the initial value be > 0.9271586026096681.
auto make_fallback = [](uint64_t v) { return (v & 0xffffffffffffff80ull); };
double tail[2];
{
// 0.9375
absl::random_internal::sequence_urbg urbg(
{make_fallback(0x7800000000000000ull), 0x13CCA830EB61BD96ull,
0x00000076f6f7f755ull});
tail[0] = dist(urbg);
EXPECT_EQ(3, urbg.invocations());
EXPECT_GT(tail[0], 0);
}
{
// -0.9375
absl::random_internal::sequence_urbg urbg(
{make_fallback(0xf800000000000000ull), 0x13CCA830EB61BD96ull,
0x00000076f6f7f755ull});
tail[1] = dist(urbg);
EXPECT_EQ(3, urbg.invocations());
EXPECT_LT(tail[1], 0);
}
EXPECT_EQ(tail[0], -tail[1]);
EXPECT_EQ(418610, static_cast<int64_t>(tail[0] * 100000.0));
// When the box != 0, the fallback algorithm computes a wedge function.
// Depending on the box, the threshold for varies as high as
// 0.991522480228.
{
// 0.9921875, 0.875
absl::random_internal::sequence_urbg urbg(
{make_box(0x7f00000000000000ull, 120), 0xe000000000000001ull,
0x13CCA830EB61BD96ull});
tail[0] = dist(urbg);
EXPECT_EQ(2, urbg.invocations());
EXPECT_GT(tail[0], 0);
}
{
// -0.9921875, 0.875
absl::random_internal::sequence_urbg urbg(
{make_box(0xff00000000000000ull, 120), 0xe000000000000001ull,
0x13CCA830EB61BD96ull});
tail[1] = dist(urbg);
EXPECT_EQ(2, urbg.invocations());
EXPECT_LT(tail[1], 0);
}
EXPECT_EQ(tail[0], -tail[1]);
EXPECT_EQ(61948, static_cast<int64_t>(tail[0] * 100000.0));
// Fallback rejected, try again.
{
// -0.9921875, 0.0625
absl::random_internal::sequence_urbg urbg(
{make_box(0xff00000000000000ull, 120), 0x1000000000000001,
make_box(0x1000000000000100ull, 50), 0x13CCA830EB61BD96ull});
dist(urbg);
EXPECT_EQ(3, urbg.invocations());
}
}
} // namespace
