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poisson_distribution_test.cc (20469B)

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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/poisson_distribution.h"

#include <algorithm>
#include <cstddef>
#include <cstdint>
#include <iterator>
#include <random>
#include <sstream>
#include <string>
#include <vector>

#include "gmock/gmock.h"
#include "gtest/gtest.h"
#include "absl/base/macros.h"
#include "absl/container/flat_hash_map.h"
#include "absl/log/log.h"
#include "absl/random/internal/chi_square.h"
#include "absl/random/internal/distribution_test_util.h"
#include "absl/random/internal/pcg_engine.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"

// Notes about generating poisson variates:
//
// It is unlikely that any implementation of std::poisson_distribution
// will be stable over time and across library implementations. For instance
// the three different poisson variate generators listed below all differ:
//
// https://github.com/ampl/gsl/tree/master/randist/poisson.c
// * GSL uses a gamma + binomial + knuth method to compute poisson variates.
//
// https://github.com/gcc-mirror/gcc/blob/master/libstdc%2B%2B-v3/include/bits/random.tcc
// * GCC uses the Devroye rejection algorithm, based on
// Devroye, L. Non-Uniform Random Variates Generation. Springer-Verlag,
// New York, 1986, Ch. X, Sects. 3.3 & 3.4 (+ Errata!), ~p.511
//   http://www.nrbook.com/devroye/
//
// https://github.com/llvm-mirror/libcxx/blob/master/include/random
// * CLANG uses a different rejection method, which appears to include a
// normal-distribution approximation and an exponential distribution to
// compute the threshold, including a similar factorial approximation to this
// one, but it is unclear where the algorithm comes from, exactly.
//

namespace {

using absl::random_internal::kChiSquared;

// The PoissonDistributionInterfaceTest provides a basic test that
// absl::poisson_distribution conforms to the interface and serialization
// requirements imposed by [rand.req.dist] for the common integer types.

template <typename IntType>
class PoissonDistributionInterfaceTest : public ::testing::Test {};

using IntTypes = ::testing::Types<int, int8_t, int16_t, int32_t, int64_t,
                                 uint8_t, uint16_t, uint32_t, uint64_t>;
TYPED_TEST_SUITE(PoissonDistributionInterfaceTest, IntTypes);

TYPED_TEST(PoissonDistributionInterfaceTest, SerializeTest) {
 using param_type = typename absl::poisson_distribution<TypeParam>::param_type;
 const double kMax =
     std::min(1e10 /* assertion limit */,
              static_cast<double>(std::numeric_limits<TypeParam>::max()));

 const double kParams[] = {
     // Cases around 1.
     1,                         //
     std::nextafter(1.0, 0.0),  // 1 - epsilon
     std::nextafter(1.0, 2.0),  // 1 + epsilon
     // Arbitrary values.
     1e-8, 1e-4,
     0.0000005,  // ~7.2e-7
     0.2,        // ~0.2x
     0.5,        // 0.72
     2,          // ~2.8
     20,         // 3x ~9.6
     100, 1e4, 1e8, 1.5e9, 1e20,
     // Boundary cases.
     std::numeric_limits<double>::max(),
     std::numeric_limits<double>::epsilon(),
     std::nextafter(std::numeric_limits<double>::min(),
                    1.0),                        // min + epsilon
     std::numeric_limits<double>::min(),         // smallest normal
     std::numeric_limits<double>::denorm_min(),  // smallest denorm
     std::numeric_limits<double>::min() / 2,     // denorm
     std::nextafter(std::numeric_limits<double>::min(),
                    0.0),  // denorm_max
 };

 constexpr int kCount = 1000;
 absl::InsecureBitGen gen;
 for (const double m : kParams) {
   const double mean = std::min(kMax, m);
   const param_type param(mean);

   // Validate parameters.
   absl::poisson_distribution<TypeParam> before(mean);
   EXPECT_EQ(before.mean(), param.mean());

   {
     absl::poisson_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);
     EXPECT_GE(sample, before.min());
     EXPECT_LE(sample, before.max());
     if (sample > sample_max) sample_max = sample;
     if (sample < sample_min) sample_min = sample;
   }

   LOG(INFO) << "Range {" << param.mean() << "}: " << sample_min << ", "
             << sample_max;

   // Validate stream serialization.
   std::stringstream ss;
   ss << before;

   absl::poisson_distribution<TypeParam> after(3.8);

   EXPECT_NE(before.mean(), after.mean());
   EXPECT_NE(before.param(), after.param());
   EXPECT_NE(before, after);

   ss >> after;

   EXPECT_EQ(before.mean(), after.mean())  //
       << ss.str() << " "                  //
       << (ss.good() ? "good " : "")       //
       << (ss.bad() ? "bad " : "")         //
       << (ss.eof() ? "eof " : "")         //
       << (ss.fail() ? "fail " : "");
 }
}

// See http://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm

class PoissonModel {
public:
 explicit PoissonModel(double mean) : mean_(mean) {}

 double mean() const { return mean_; }
 double variance() const { return mean_; }
 double stddev() const { return std::sqrt(variance()); }
 double skew() const { return 1.0 / mean_; }
 double kurtosis() const { return 3.0 + 1.0 / mean_; }

 // InitCDF() initializes the CDF for the distribution parameters.
 void InitCDF();

 // The InverseCDF, or the Percent-point function returns x, P(x) < v.
 struct CDF {
   size_t index;
   double pmf;
   double cdf;
 };
 CDF InverseCDF(double p) {
   CDF target{0, 0, p};
   auto it = std::upper_bound(
       std::begin(cdf_), std::end(cdf_), target,
       [](const CDF& a, const CDF& b) { return a.cdf < b.cdf; });
   return *it;
 }

 void LogCDF() {
   LOG(INFO) << "CDF (mean = " << mean_ << ")";
   for (const auto c : cdf_) {
     LOG(INFO) << c.index << ": pmf=" << c.pmf << " cdf=" << c.cdf;
   }
 }

private:
 const double mean_;

 std::vector<CDF> cdf_;
};

// The goal is to compute an InverseCDF function, or percent point function for
// the poisson distribution, and use that to partition our output into equal
// range buckets.  However there is no closed form solution for the inverse cdf
// for poisson distributions (the closest is the incomplete gamma function).
// Instead, `InitCDF` iteratively computes the PMF and the CDF. This enables
// searching for the bucket points.
void PoissonModel::InitCDF() {
 if (!cdf_.empty()) {
   // State already initialized.
   return;
 }
 ABSL_ASSERT(mean_ < 201.0);

 const size_t max_i = 50 * stddev() + mean();
 const double e_neg_mean = std::exp(-mean());
 ABSL_ASSERT(e_neg_mean > 0);

 double d = 1;
 double last_result = e_neg_mean;
 double cumulative = e_neg_mean;
 if (e_neg_mean > 1e-10) {
   cdf_.push_back({0, e_neg_mean, cumulative});
 }
 for (size_t i = 1; i < max_i; i++) {
   d *= (mean() / i);
   double result = e_neg_mean * d;
   cumulative += result;
   if (result < 1e-10 && result < last_result && cumulative > 0.999999) {
     break;
   }
   if (result > 1e-7) {
     cdf_.push_back({i, result, cumulative});
   }
   last_result = result;
 }
 ABSL_ASSERT(!cdf_.empty());
}

// PoissonDistributionZTest implements a z-test for the poisson distribution.

struct ZParam {
 double mean;
 double p_fail;   // Z-Test probability of failure.
 int trials;      // Z-Test trials.
 size_t samples;  // Z-Test samples.
};

class PoissonDistributionZTest : public testing::TestWithParam<ZParam>,
                                public PoissonModel {
public:
 PoissonDistributionZTest() : PoissonModel(GetParam().mean) {}

 // ZTestImpl 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);

 // 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 PoissonDistributionZTest::SingleZTest(const double p,
                                          const size_t samples) {
 D dis(mean());

 absl::flat_hash_map<int32_t, int> buckets;
 std::vector<double> data;
 data.reserve(samples);
 for (int j = 0; j < samples; j++) {
   const auto x = dis(rng_);
   buckets[x]++;
   data.push_back(x);
 }

 // The null-hypothesis is that the distribution is a poisson distribution with
 // the provided mean (not estimated from the data).
 const auto m = absl::random_internal::ComputeDistributionMoments(data);
 const double max_err = absl::random_internal::MaxErrorTolerance(p);
 const double z = absl::random_internal::ZScore(mean(), m);
 const bool pass = absl::random_internal::Near("z", z, 0.0, max_err);

 if (!pass) {
   // clang-format off
   LOG(INFO)
       << "p=" << p << " max_err=" << max_err << "\n"
          " mean=" << m.mean << " vs. " << mean() << "\n"
          " stddev=" << std::sqrt(m.variance) << " vs. " << stddev() << "\n"
          " skewness=" << m.skewness << " vs. " << skew() << "\n"
          " kurtosis=" << m.kurtosis << " vs. " << kurtosis() << "\n"
          " z=" << z;
   // clang-format on
 }
 return pass;
}

TEST_P(PoissonDistributionZTest, AbslPoissonDistribution) {
 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::poisson_distribution<int32_t>>(p, param.samples) ? 0
                                                                          : 1;
 }
 EXPECT_LE(failures, expected_failures);
}

std::vector<ZParam> GetZParams() {
 // These values have been adjusted from the "exact" computed values to reduce
 // failure rates.
 //
 // It turns out that the actual values are not as close to the expected values
 // as would be ideal.
 return std::vector<ZParam>({
     // Knuth method.
     ZParam{0.5, 0.01, 100, 1000},
     ZParam{1.0, 0.01, 100, 1000},
     ZParam{10.0, 0.01, 100, 5000},
     // Split-knuth method.
     ZParam{20.0, 0.01, 100, 10000},
     ZParam{50.0, 0.01, 100, 10000},
     // Ratio of gaussians method.
     ZParam{51.0, 0.01, 100, 10000},
     ZParam{200.0, 0.05, 10, 100000},
     ZParam{100000.0, 0.05, 10, 1000000},
 });
}

std::string ZParamName(const ::testing::TestParamInfo<ZParam>& info) {
 const auto& p = info.param;
 std::string name = absl::StrCat("mean_", absl::SixDigits(p.mean));
 return absl::StrReplaceAll(name, {{"+", "_"}, {"-", "_"}, {".", "_"}});
}

INSTANTIATE_TEST_SUITE_P(All, PoissonDistributionZTest,
                        ::testing::ValuesIn(GetZParams()), ZParamName);

// The PoissonDistributionChiSquaredTest class provides a basic test framework
// for variates generated by a conforming poisson_distribution.
class PoissonDistributionChiSquaredTest : public testing::TestWithParam<double>,
                                         public PoissonModel {
public:
 PoissonDistributionChiSquaredTest() : PoissonModel(GetParam()) {}

 // The ChiSquaredTestImpl provides a chi-squared goodness of fit test for data
 // generated by the poisson distribution.
 template <typename D>
 double ChiSquaredTestImpl();

private:
 void InitChiSquaredTest(const double buckets);

 std::vector<size_t> cutoffs_;
 std::vector<double> expected_;

 // 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};
};

void PoissonDistributionChiSquaredTest::InitChiSquaredTest(
   const double buckets) {
 if (!cutoffs_.empty() && !expected_.empty()) {
   return;
 }
 InitCDF();

 // The code below finds cuttoffs that yield approximately equally-sized
 // buckets to the extent that it is possible. However for poisson
 // distributions this is particularly challenging for small mean parameters.
 // Track the expected proportion of items in each bucket.
 double last_cdf = 0;
 const double inc = 1.0 / buckets;
 for (double p = inc; p <= 1.0; p += inc) {
   auto result = InverseCDF(p);
   if (!cutoffs_.empty() && cutoffs_.back() == result.index) {
     continue;
   }
   double d = result.cdf - last_cdf;
   cutoffs_.push_back(result.index);
   expected_.push_back(d);
   last_cdf = result.cdf;
 }
 cutoffs_.push_back(std::numeric_limits<size_t>::max());
 expected_.push_back(std::max(0.0, 1.0 - last_cdf));
}

template <typename D>
double PoissonDistributionChiSquaredTest::ChiSquaredTestImpl() {
 const int kSamples = 2000;
 const int kBuckets = 50;

 // The poisson CDF fails for large mean values, since e^-mean exceeds the
 // machine precision. For these cases, using a normal approximation would be
 // appropriate.
 ABSL_ASSERT(mean() <= 200);
 InitChiSquaredTest(kBuckets);

 D dis(mean());

 std::vector<int32_t> counts(cutoffs_.size(), 0);
 for (int j = 0; j < kSamples; j++) {
   const size_t x = dis(rng_);
   auto it = std::lower_bound(std::begin(cutoffs_), std::end(cutoffs_), x);
   counts[std::distance(cutoffs_.begin(), it)]++;
 }

 // Normalize the counts.
 std::vector<int32_t> e(expected_.size(), 0);
 for (int i = 0; i < e.size(); i++) {
   e[i] = kSamples * expected_[i];
 }

 // The null-hypothesis is that the distribution is a poisson distribution with
 // the provided mean (not estimated from the data).
 const int dof = static_cast<int>(counts.size()) - 1;

 // The threshold for logging is 1-in-50.
 const double threshold = absl::random_internal::ChiSquareValue(dof, 0.98);

 const double chi_square = absl::random_internal::ChiSquare(
     std::begin(counts), std::end(counts), std::begin(e), std::end(e));

 const double p = absl::random_internal::ChiSquarePValue(chi_square, dof);

 // Log if the chi_squared value is above the threshold.
 if (chi_square > threshold) {
   LogCDF();

   LOG(INFO) << "VALUES  buckets=" << counts.size()
             << "  samples=" << kSamples;
   for (size_t i = 0; i < counts.size(); i++) {
     LOG(INFO) << cutoffs_[i] << ": " << counts[i] << " vs. E=" << e[i];
   }

   LOG(INFO) << kChiSquared << "(data, dof=" << dof << ") = " << chi_square
             << " (" << p << ")\n"
             << " vs.\n"
             << kChiSquared << " @ 0.98 = " << threshold;
 }
 return p;
}

TEST_P(PoissonDistributionChiSquaredTest, AbslPoissonDistribution) {
 const int kTrials = 20;

 // Large values are not yet supported -- this requires estimating the cdf
 // using the normal distribution instead of the poisson in this case.
 ASSERT_LE(mean(), 200.0);
 if (mean() > 200.0) {
   return;
 }

 int failures = 0;
 for (int i = 0; i < kTrials; i++) {
   double p_value = ChiSquaredTestImpl<absl::poisson_distribution<int32_t>>();
   if (p_value < 0.005) {
     failures++;
   }
 }
 // There is a 0.10% chance of producing at least one failure, so raise the
 // failure threshold high enough to allow for a flake rate < 10,000.
 EXPECT_LE(failures, 4);
}

INSTANTIATE_TEST_SUITE_P(All, PoissonDistributionChiSquaredTest,
                        ::testing::Values(0.5, 1.0, 2.0, 10.0, 50.0, 51.0,
                                          200.0));

// NOTE: absl::poisson_distribution is not guaranteed to be stable.
TEST(PoissonDistributionTest, StabilityTest) {
 using testing::ElementsAre;
 // absl::poisson_distribution stability relies on stability of
 // std::exp, std::log, std::sqrt, std::ceil, std::floor, and
 // absl::FastUniformBits, absl::StirlingLogFactorial, absl::RandU64ToDouble.
 absl::random_internal::sequence_urbg urbg({
     0x035b0dc7e0a18acfull, 0x06cebe0d2653682eull, 0x0061e9b23861596bull,
     0x0003eb76f6f7f755ull, 0xFFCEA50FDB2F953Bull, 0xC332DDEFBE6C5AA5ull,
     0x6558218568AB9702ull, 0x2AEF7DAD5B6E2F84ull, 0x1521B62829076170ull,
     0xECDD4775619F1510ull, 0x13CCA830EB61BD96ull, 0x0334FE1EAA0363CFull,
     0xB5735C904C70A239ull, 0xD59E9E0BCBAADE14ull, 0xEECC86BC60622CA7ull,
     0x4864f22c059bf29eull, 0x247856d8b862665cull, 0xe46e86e9a1337e10ull,
     0xd8c8541f3519b133ull, 0xe75b5162c567b9e4ull, 0xf732e5ded7009c5bull,
     0xb170b98353121eacull, 0x1ec2e8986d2362caull, 0x814c8e35fe9a961aull,
     0x0c3cd59c9b638a02ull, 0xcb3bb6478a07715cull, 0x1224e62c978bbc7full,
     0x671ef2cb04e81f6eull, 0x3c1cbd811eaf1808ull, 0x1bbc23cfa8fac721ull,
     0xa4c2cda65e596a51ull, 0xb77216fad37adf91ull, 0x836d794457c08849ull,
     0xe083df03475f49d7ull, 0xbc9feb512e6b0d6cull, 0xb12d74fdd718c8c5ull,
     0x12ff09653bfbe4caull, 0x8dd03a105bc4ee7eull, 0x5738341045ba0d85ull,
     0xf3fd722dc65ad09eull, 0xfa14fd21ea2a5705ull, 0xffe6ea4d6edb0c73ull,
     0xD07E9EFE2BF11FB4ull, 0x95DBDA4DAE909198ull, 0xEAAD8E716B93D5A0ull,
     0xD08ED1D0AFC725E0ull, 0x8E3C5B2F8E7594B7ull, 0x8FF6E2FBF2122B64ull,
     0x8888B812900DF01Cull, 0x4FAD5EA0688FC31Cull, 0xD1CFF191B3A8C1ADull,
     0x2F2F2218BE0E1777ull, 0xEA752DFE8B021FA1ull, 0xE5A0CC0FB56F74E8ull,
     0x18ACF3D6CE89E299ull, 0xB4A84FE0FD13E0B7ull, 0x7CC43B81D2ADA8D9ull,
     0x165FA26680957705ull, 0x93CC7314211A1477ull, 0xE6AD206577B5FA86ull,
     0xC75442F5FB9D35CFull, 0xEBCDAF0C7B3E89A0ull, 0xD6411BD3AE1E7E49ull,
     0x00250E2D2071B35Eull, 0x226800BB57B8E0AFull, 0x2464369BF009B91Eull,
     0x5563911D59DFA6AAull, 0x78C14389D95A537Full, 0x207D5BA202E5B9C5ull,
     0x832603766295CFA9ull, 0x11C819684E734A41ull, 0xB3472DCA7B14A94Aull,
 });

 std::vector<int> output(10);

 // Method 1.
 {
   absl::poisson_distribution<int> dist(5);
   std::generate(std::begin(output), std::end(output),
                 [&] { return dist(urbg); });
 }
 EXPECT_THAT(output,  // mean = 4.2
             ElementsAre(1, 0, 0, 4, 2, 10, 3, 3, 7, 12));

 // Method 2.
 {
   urbg.reset();
   absl::poisson_distribution<int> dist(25);
   std::generate(std::begin(output), std::end(output),
                 [&] { return dist(urbg); });
 }
 EXPECT_THAT(output,  // mean = 19.8
             ElementsAre(9, 35, 18, 10, 35, 18, 10, 35, 18, 10));

 // Method 3.
 {
   urbg.reset();
   absl::poisson_distribution<int> dist(121);
   std::generate(std::begin(output), std::end(output),
                 [&] { return dist(urbg); });
 }
 EXPECT_THAT(output,  // mean = 124.1
             ElementsAre(161, 122, 129, 124, 112, 112, 117, 120, 130, 114));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_1) {
 // This tests small values of the Knuth method.
 // The underlying uniform distribution will generate exactly 0.5.
 absl::random_internal::sequence_urbg urbg({0x8000000000000001ull});
 absl::poisson_distribution<int> dist(5);
 EXPECT_EQ(7, dist(urbg));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_2) {
 // This tests larger values of the Knuth method.
 // The underlying uniform distribution will generate exactly 0.5.
 absl::random_internal::sequence_urbg urbg({0x8000000000000001ull});
 absl::poisson_distribution<int> dist(25);
 EXPECT_EQ(36, dist(urbg));
}

TEST(PoissonDistributionTest, AlgorithmExpectedValue_3) {
 // This variant uses the ratio of uniforms method.
 absl::random_internal::sequence_urbg urbg(
     {0x7fffffffffffffffull, 0x8000000000000000ull});

 absl::poisson_distribution<int> dist(121);
 EXPECT_EQ(121, dist(urbg));
}

}  // namespace