// 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. #ifndef ABSL_RANDOM_POISSON_DISTRIBUTION_H_ #define ABSL_RANDOM_POISSON_DISTRIBUTION_H_ #include <cassert> #include <cmath> #include <istream> #include <limits> #include <ostream> #include <type_traits> #include "absl/random/internal/fast_uniform_bits.h" #include "absl/random/internal/fastmath.h" #include "absl/random/internal/generate_real.h" #include "absl/random/internal/iostream_state_saver.h" namespace absl { ABSL_NAMESPACE_BEGIN // absl::poisson_distribution: // Generates discrete variates conforming to a Poisson distribution. // p(n) = (mean^n / n!) exp(-mean) // // Depending on the parameter, the distribution selects one of the following // algorithms: // * The standard algorithm, attributed to Knuth, extended using a split method // for larger values // * The "Ratio of Uniforms as a convenient method for sampling from classical // discrete distributions", Stadlober, 1989. // http://www.sciencedirect.com/science/article/pii/0377042790903495 // // NOTE: param_type.mean() is a double, which permits values larger than // poisson_distribution<IntType>::max(), however this should be avoided and // the distribution results are limited to the max() value. // // The goals of this implementation are to provide good performance while still // beig thread-safe: This limits the implementation to not using lgamma provided // by <math.h>. // template <typename IntType = int> class poisson_distribution { public: using result_type = IntType; class param_type { public: using distribution_type = poisson_distribution; explicit param_type(double mean = 1.0); double mean() const { return mean_; } friend bool operator==(const param_type& a, const param_type& b) { return a.mean_ == b.mean_; } friend bool operator!=(const param_type& a, const param_type& b) { return !(a == b); } private: friend class poisson_distribution; double mean_; double emu_; // e ^ -mean_ double lmu_; // ln(mean_) double s_; double log_k_; int split_; static_assert(std::is_integral<IntType>::value, "Class-template absl::poisson_distribution<> must be " "parameterized using an integral type."); }; poisson_distribution() : poisson_distribution(1.0) {} explicit poisson_distribution(double mean) : param_(mean) {} explicit poisson_distribution(const param_type& p) : param_(p) {} void reset() {} // generating functions template <typename URBG> result_type operator()(URBG& g) { // NOLINT(runtime/references) return (*this)(g, param_); } template <typename URBG> result_type operator()(URBG& g, // NOLINT(runtime/references) const param_type& p); param_type param() const { return param_; } void param(const param_type& p) { param_ = p; } result_type(min)() const { return 0; } result_type(max)() const { return (std::numeric_limits<result_type>::max)(); } double mean() const { return param_.mean(); } friend bool operator==(const poisson_distribution& a, const poisson_distribution& b) { return a.param_ == b.param_; } friend bool operator!=(const poisson_distribution& a, const poisson_distribution& b) { return a.param_ != b.param_; } private: param_type param_; random_internal::FastUniformBits<uint64_t> fast_u64_; }; // ----------------------------------------------------------------------------- // Implementation details follow // ----------------------------------------------------------------------------- template <typename IntType> poisson_distribution<IntType>::param_type::param_type(double mean) : mean_(mean), split_(0) { assert(mean >= 0); assert(mean <= (std::numeric_limits<result_type>::max)()); // As a defensive measure, avoid large values of the mean. The rejection // algorithm used does not support very large values well. It my be worth // changing algorithms to better deal with these cases. assert(mean <= 1e10); if (mean_ < 10) { // For small lambda, use the knuth method. split_ = 1; emu_ = std::exp(-mean_); } else if (mean_ <= 50) { // Use split-knuth method. split_ = 1 + static_cast<int>(mean_ / 10.0); emu_ = std::exp(-mean_ / static_cast<double>(split_)); } else { // Use ratio of uniforms method. constexpr double k2E = 0.7357588823428846; constexpr double kSA = 0.4494580810294493; lmu_ = std::log(mean_); double a = mean_ + 0.5; s_ = kSA + std::sqrt(k2E * a); const double mode = std::ceil(mean_) - 1; log_k_ = lmu_ * mode - absl::random_internal::StirlingLogFactorial(mode); } } template <typename IntType> template <typename URBG> typename poisson_distribution<IntType>::result_type poisson_distribution<IntType>::operator()( URBG& g, // NOLINT(runtime/references) const param_type& p) { using random_internal::GeneratePositiveTag; using random_internal::GenerateRealFromBits; using random_internal::GenerateSignedTag; if (p.split_ != 0) { // Use Knuth's algorithm with range splitting to avoid floating-point // errors. Knuth's algorithm is: Ui is a sequence of uniform variates on // (0,1); return the number of variates required for product(Ui) < // exp(-lambda). // // The expected number of variates required for Knuth's method can be // computed as follows: // The expected value of U is 0.5, so solving for 0.5^n < exp(-lambda) gives // the expected number of uniform variates // required for a given lambda, which is: // lambda = [2, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17] // n = [3, 8, 13, 15, 16, 18, 19, 21, 22, 24, 25] // result_type n = 0; for (int split = p.split_; split > 0; --split) { double r = 1.0; do { r *= GenerateRealFromBits<double, GeneratePositiveTag, true>( fast_u64_(g)); // U(-1, 0) ++n; } while (r > p.emu_); --n; } return n; } // Use ratio of uniforms method. // // Let u ~ Uniform(0, 1), v ~ Uniform(-1, 1), // a = lambda + 1/2, // s = 1.5 - sqrt(3/e) + sqrt(2(lambda + 1/2)/e), // x = s * v/u + a. // P(floor(x) = k | u^2 < f(floor(x))/k), where // f(m) = lambda^m exp(-lambda)/ m!, for 0 <= m, and f(m) = 0 otherwise, // and k = max(f). const double a = p.mean_ + 0.5; for (;;) { const double u = GenerateRealFromBits<double, GeneratePositiveTag, false>( fast_u64_(g)); // U(0, 1) const double v = GenerateRealFromBits<double, GenerateSignedTag, false>( fast_u64_(g)); // U(-1, 1) const double x = std::floor(p.s_ * v / u + a); if (x < 0) continue; // f(negative) = 0 const double rhs = x * p.lmu_; // clang-format off double s = (x <= 1.0) ? 0.0 : (x == 2.0) ? 0.693147180559945 : absl::random_internal::StirlingLogFactorial(x); // clang-format on const double lhs = 2.0 * std::log(u) + p.log_k_ + s; if (lhs < rhs) { return x > (max)() ? (max)() : static_cast<result_type>(x); // f(x)/k >= u^2 } } } template <typename CharT, typename Traits, typename IntType> std::basic_ostream<CharT, Traits>& operator<<( std::basic_ostream<CharT, Traits>& os, // NOLINT(runtime/references) const poisson_distribution<IntType>& x) { auto saver = random_internal::make_ostream_state_saver(os); os.precision(random_internal::stream_precision_helper<double>::kPrecision); os << x.mean(); return os; } template <typename CharT, typename Traits, typename IntType> std::basic_istream<CharT, Traits>& operator>>( std::basic_istream<CharT, Traits>& is, // NOLINT(runtime/references) poisson_distribution<IntType>& x) { // NOLINT(runtime/references) using param_type = typename poisson_distribution<IntType>::param_type; auto saver = random_internal::make_istream_state_saver(is); double mean = random_internal::read_floating_point<double>(is); if (!is.fail()) { x.param(param_type(mean)); } return is; } ABSL_NAMESPACE_END } // namespace absl #endif // ABSL_RANDOM_POISSON_DISTRIBUTION_H_