1 // Copyright 2017 The Abseil Authors.
2 //
3 // Licensed under the Apache License, Version 2.0 (the "License");
4 // you may not use this file except in compliance with the License.
5 // You may obtain a copy of the License at
6 //
7 // https://www.apache.org/licenses/LICENSE-2.0
8 //
9 // Unless required by applicable law or agreed to in writing, software
10 // distributed under the License is distributed on an "AS IS" BASIS,
11 // WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
12 // See the License for the specific language governing permissions and
13 // limitations under the License.
14
15 #ifndef ABSL_RANDOM_POISSON_DISTRIBUTION_H_
16 #define ABSL_RANDOM_POISSON_DISTRIBUTION_H_
17
18 #include <cassert>
19 #include <cmath>
20 #include <istream>
21 #include <limits>
22 #include <ostream>
23 #include <type_traits>
24
25 #include "absl/random/internal/fast_uniform_bits.h"
26 #include "absl/random/internal/fastmath.h"
27 #include "absl/random/internal/generate_real.h"
28 #include "absl/random/internal/iostream_state_saver.h"
29 #include "absl/random/internal/traits.h"
30
31 namespace absl {
32 ABSL_NAMESPACE_BEGIN
33
34 // absl::poisson_distribution:
35 // Generates discrete variates conforming to a Poisson distribution.
36 // p(n) = (mean^n / n!) exp(-mean)
37 //
38 // Depending on the parameter, the distribution selects one of the following
39 // algorithms:
40 // * The standard algorithm, attributed to Knuth, extended using a split method
41 // for larger values
42 // * The "Ratio of Uniforms as a convenient method for sampling from classical
43 // discrete distributions", Stadlober, 1989.
44 // http://www.sciencedirect.com/science/article/pii/0377042790903495
45 //
46 // NOTE: param_type.mean() is a double, which permits values larger than
47 // poisson_distribution<IntType>::max(), however this should be avoided and
48 // the distribution results are limited to the max() value.
49 //
50 // The goals of this implementation are to provide good performance while still
51 // beig thread-safe: This limits the implementation to not using lgamma provided
52 // by <math.h>.
53 //
54 template <typename IntType = int>
55 class poisson_distribution {
56 public:
57 using result_type = IntType;
58
59 class param_type {
60 public:
61 using distribution_type = poisson_distribution;
62 explicit param_type(double mean = 1.0);
63
mean()64 double mean() const { return mean_; }
65
66 friend bool operator==(const param_type& a, const param_type& b) {
67 return a.mean_ == b.mean_;
68 }
69
70 friend bool operator!=(const param_type& a, const param_type& b) {
71 return !(a == b);
72 }
73
74 private:
75 friend class poisson_distribution;
76
77 double mean_;
78 double emu_; // e ^ -mean_
79 double lmu_; // ln(mean_)
80 double s_;
81 double log_k_;
82 int split_;
83
84 static_assert(random_internal::IsIntegral<IntType>::value,
85 "Class-template absl::poisson_distribution<> must be "
86 "parameterized using an integral type.");
87 };
88
poisson_distribution()89 poisson_distribution() : poisson_distribution(1.0) {}
90
poisson_distribution(double mean)91 explicit poisson_distribution(double mean) : param_(mean) {}
92
poisson_distribution(const param_type & p)93 explicit poisson_distribution(const param_type& p) : param_(p) {}
94
reset()95 void reset() {}
96
97 // generating functions
98 template <typename URBG>
operator()99 result_type operator()(URBG& g) { // NOLINT(runtime/references)
100 return (*this)(g, param_);
101 }
102
103 template <typename URBG>
104 result_type operator()(URBG& g, // NOLINT(runtime/references)
105 const param_type& p);
106
param()107 param_type param() const { return param_; }
param(const param_type & p)108 void param(const param_type& p) { param_ = p; }
109
result_type(min)110 result_type(min)() const { return 0; }
result_type(max)111 result_type(max)() const { return (std::numeric_limits<result_type>::max)(); }
112
mean()113 double mean() const { return param_.mean(); }
114
115 friend bool operator==(const poisson_distribution& a,
116 const poisson_distribution& b) {
117 return a.param_ == b.param_;
118 }
119 friend bool operator!=(const poisson_distribution& a,
120 const poisson_distribution& b) {
121 return a.param_ != b.param_;
122 }
123
124 private:
125 param_type param_;
126 random_internal::FastUniformBits<uint64_t> fast_u64_;
127 };
128
129 // -----------------------------------------------------------------------------
130 // Implementation details follow
131 // -----------------------------------------------------------------------------
132
133 template <typename IntType>
param_type(double mean)134 poisson_distribution<IntType>::param_type::param_type(double mean)
135 : mean_(mean), split_(0) {
136 assert(mean >= 0);
137 assert(mean <=
138 static_cast<double>((std::numeric_limits<result_type>::max)()));
139 // As a defensive measure, avoid large values of the mean. The rejection
140 // algorithm used does not support very large values well. It my be worth
141 // changing algorithms to better deal with these cases.
142 assert(mean <= 1e10);
143 if (mean_ < 10) {
144 // For small lambda, use the knuth method.
145 split_ = 1;
146 emu_ = std::exp(-mean_);
147 } else if (mean_ <= 50) {
148 // Use split-knuth method.
149 split_ = 1 + static_cast<int>(mean_ / 10.0);
150 emu_ = std::exp(-mean_ / static_cast<double>(split_));
151 } else {
152 // Use ratio of uniforms method.
153 constexpr double k2E = 0.7357588823428846;
154 constexpr double kSA = 0.4494580810294493;
155
156 lmu_ = std::log(mean_);
157 double a = mean_ + 0.5;
158 s_ = kSA + std::sqrt(k2E * a);
159 const double mode = std::ceil(mean_) - 1;
160 log_k_ = lmu_ * mode - absl::random_internal::StirlingLogFactorial(mode);
161 }
162 }
163
164 template <typename IntType>
165 template <typename URBG>
166 typename poisson_distribution<IntType>::result_type
operator()167 poisson_distribution<IntType>::operator()(
168 URBG& g, // NOLINT(runtime/references)
169 const param_type& p) {
170 using random_internal::GeneratePositiveTag;
171 using random_internal::GenerateRealFromBits;
172 using random_internal::GenerateSignedTag;
173
174 if (p.split_ != 0) {
175 // Use Knuth's algorithm with range splitting to avoid floating-point
176 // errors. Knuth's algorithm is: Ui is a sequence of uniform variates on
177 // (0,1); return the number of variates required for product(Ui) <
178 // exp(-lambda).
179 //
180 // The expected number of variates required for Knuth's method can be
181 // computed as follows:
182 // The expected value of U is 0.5, so solving for 0.5^n < exp(-lambda) gives
183 // the expected number of uniform variates
184 // required for a given lambda, which is:
185 // lambda = [2, 5, 9, 10, 11, 12, 13, 14, 15, 16, 17]
186 // n = [3, 8, 13, 15, 16, 18, 19, 21, 22, 24, 25]
187 //
188 result_type n = 0;
189 for (int split = p.split_; split > 0; --split) {
190 double r = 1.0;
191 do {
192 r *= GenerateRealFromBits<double, GeneratePositiveTag, true>(
193 fast_u64_(g)); // U(-1, 0)
194 ++n;
195 } while (r > p.emu_);
196 --n;
197 }
198 return n;
199 }
200
201 // Use ratio of uniforms method.
202 //
203 // Let u ~ Uniform(0, 1), v ~ Uniform(-1, 1),
204 // a = lambda + 1/2,
205 // s = 1.5 - sqrt(3/e) + sqrt(2(lambda + 1/2)/e),
206 // x = s * v/u + a.
207 // P(floor(x) = k | u^2 < f(floor(x))/k), where
208 // f(m) = lambda^m exp(-lambda)/ m!, for 0 <= m, and f(m) = 0 otherwise,
209 // and k = max(f).
210 const double a = p.mean_ + 0.5;
211 for (;;) {
212 const double u = GenerateRealFromBits<double, GeneratePositiveTag, false>(
213 fast_u64_(g)); // U(0, 1)
214 const double v = GenerateRealFromBits<double, GenerateSignedTag, false>(
215 fast_u64_(g)); // U(-1, 1)
216
217 const double x = std::floor(p.s_ * v / u + a);
218 if (x < 0) continue; // f(negative) = 0
219 const double rhs = x * p.lmu_;
220 // clang-format off
221 double s = (x <= 1.0) ? 0.0
222 : (x == 2.0) ? 0.693147180559945
223 : absl::random_internal::StirlingLogFactorial(x);
224 // clang-format on
225 const double lhs = 2.0 * std::log(u) + p.log_k_ + s;
226 if (lhs < rhs) {
227 return x > static_cast<double>((max)())
228 ? (max)()
229 : static_cast<result_type>(x); // f(x)/k >= u^2
230 }
231 }
232 }
233
234 template <typename CharT, typename Traits, typename IntType>
235 std::basic_ostream<CharT, Traits>& operator<<(
236 std::basic_ostream<CharT, Traits>& os, // NOLINT(runtime/references)
237 const poisson_distribution<IntType>& x) {
238 auto saver = random_internal::make_ostream_state_saver(os);
239 os.precision(random_internal::stream_precision_helper<double>::kPrecision);
240 os << x.mean();
241 return os;
242 }
243
244 template <typename CharT, typename Traits, typename IntType>
245 std::basic_istream<CharT, Traits>& operator>>(
246 std::basic_istream<CharT, Traits>& is, // NOLINT(runtime/references)
247 poisson_distribution<IntType>& x) { // NOLINT(runtime/references)
248 using param_type = typename poisson_distribution<IntType>::param_type;
249
250 auto saver = random_internal::make_istream_state_saver(is);
251 double mean = random_internal::read_floating_point<double>(is);
252 if (!is.fail()) {
253 x.param(param_type(mean));
254 }
255 return is;
256 }
257
258 ABSL_NAMESPACE_END
259 } // namespace absl
260
261 #endif // ABSL_RANDOM_POISSON_DISTRIBUTION_H_
262