428 lines
14 KiB
C++
428 lines
14 KiB
C++
// Copyright 2017 The Abseil Authors.
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//
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// Licensed under the Apache License, Version 2.0 (the "License");
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// you may not use this file except in compliance with the License.
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// You may obtain a copy of the License at
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//
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// https://www.apache.org/licenses/LICENSE-2.0
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//
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// Unless required by applicable law or agreed to in writing, software
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// distributed under the License is distributed on an "AS IS" BASIS,
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// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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// See the License for the specific language governing permissions and
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// limitations under the License.
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#ifndef ABSL_RANDOM_BETA_DISTRIBUTION_H_
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#define ABSL_RANDOM_BETA_DISTRIBUTION_H_
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#include <cassert>
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#include <cmath>
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#include <istream>
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#include <limits>
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#include <ostream>
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#include <type_traits>
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#include "absl/meta/type_traits.h"
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#include "absl/random/internal/fast_uniform_bits.h"
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#include "absl/random/internal/fastmath.h"
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#include "absl/random/internal/generate_real.h"
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#include "absl/random/internal/iostream_state_saver.h"
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namespace absl {
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ABSL_NAMESPACE_BEGIN
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// absl::beta_distribution:
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// Generate a floating-point variate conforming to a Beta distribution:
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// pdf(x) \propto x^(alpha-1) * (1-x)^(beta-1),
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// where the params alpha and beta are both strictly positive real values.
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//
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// The support is the open interval (0, 1), but the return value might be equal
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// to 0 or 1, due to numerical errors when alpha and beta are very different.
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//
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// Usage note: One usage is that alpha and beta are counts of number of
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// successes and failures. When the total number of trials are large, consider
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// approximating a beta distribution with a Gaussian distribution with the same
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// mean and variance. One could use the skewness, which depends only on the
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// smaller of alpha and beta when the number of trials are sufficiently large,
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// to quantify how far a beta distribution is from the normal distribution.
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template <typename RealType = double>
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class beta_distribution {
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public:
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using result_type = RealType;
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class param_type {
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public:
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using distribution_type = beta_distribution;
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explicit param_type(result_type alpha, result_type beta)
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: alpha_(alpha), beta_(beta) {
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assert(alpha >= 0);
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assert(beta >= 0);
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assert(alpha <= (std::numeric_limits<result_type>::max)());
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assert(beta <= (std::numeric_limits<result_type>::max)());
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if (alpha == 0 || beta == 0) {
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method_ = DEGENERATE_SMALL;
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x_ = (alpha >= beta) ? 1 : 0;
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return;
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}
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// a_ = min(beta, alpha), b_ = max(beta, alpha).
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if (beta < alpha) {
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inverted_ = true;
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a_ = beta;
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b_ = alpha;
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} else {
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inverted_ = false;
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a_ = alpha;
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b_ = beta;
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}
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if (a_ <= 1 && b_ >= ThresholdForLargeA()) {
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method_ = DEGENERATE_SMALL;
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x_ = inverted_ ? result_type(1) : result_type(0);
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return;
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}
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// For threshold values, see also:
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// Evaluation of Beta Generation Algorithms, Ying-Chao Hung, et. al.
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// February, 2009.
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if ((b_ < 1.0 && a_ + b_ <= 1.2) || a_ <= ThresholdForSmallA()) {
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// Choose Joehnk over Cheng when it's faster or when Cheng encounters
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// numerical issues.
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method_ = JOEHNK;
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a_ = result_type(1) / alpha_;
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b_ = result_type(1) / beta_;
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if (std::isinf(a_) || std::isinf(b_)) {
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method_ = DEGENERATE_SMALL;
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x_ = inverted_ ? result_type(1) : result_type(0);
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}
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return;
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}
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if (a_ >= ThresholdForLargeA()) {
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method_ = DEGENERATE_LARGE;
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// Note: on PPC for long double, evaluating
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// `std::numeric_limits::max() / ThresholdForLargeA` results in NaN.
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result_type r = a_ / b_;
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x_ = (inverted_ ? result_type(1) : r) / (1 + r);
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return;
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}
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x_ = a_ + b_;
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log_x_ = std::log(x_);
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if (a_ <= 1) {
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method_ = CHENG_BA;
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y_ = result_type(1) / a_;
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gamma_ = a_ + a_;
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return;
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}
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method_ = CHENG_BB;
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result_type r = (a_ - 1) / (b_ - 1);
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y_ = std::sqrt((1 + r) / (b_ * r * 2 - r + 1));
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gamma_ = a_ + result_type(1) / y_;
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}
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result_type alpha() const { return alpha_; }
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result_type beta() const { return beta_; }
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friend bool operator==(const param_type& a, const param_type& b) {
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return a.alpha_ == b.alpha_ && a.beta_ == b.beta_;
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}
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friend bool operator!=(const param_type& a, const param_type& b) {
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return !(a == b);
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}
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private:
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friend class beta_distribution;
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#ifdef _MSC_VER
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// MSVC does not have constexpr implementations for std::log and std::exp
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// so they are computed at runtime.
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#define ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR
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#else
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#define ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR constexpr
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#endif
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// The threshold for whether std::exp(1/a) is finite.
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// Note that this value is quite large, and a smaller a_ is NOT abnormal.
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static ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR result_type
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ThresholdForSmallA() {
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return result_type(1) /
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std::log((std::numeric_limits<result_type>::max)());
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}
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// The threshold for whether a * std::log(a) is finite.
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static ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR result_type
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ThresholdForLargeA() {
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return std::exp(
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std::log((std::numeric_limits<result_type>::max)()) -
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std::log(std::log((std::numeric_limits<result_type>::max)())) -
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ThresholdPadding());
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}
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#undef ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR
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// Pad the threshold for large A for long double on PPC. This is done via a
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// template specialization below.
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static constexpr result_type ThresholdPadding() { return 0; }
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enum Method {
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JOEHNK, // Uses algorithm Joehnk
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CHENG_BA, // Uses algorithm BA in Cheng
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CHENG_BB, // Uses algorithm BB in Cheng
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// Note: See also:
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// Hung et al. Evaluation of beta generation algorithms. Communications
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// in Statistics-Simulation and Computation 38.4 (2009): 750-770.
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// especially:
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// Zechner, Heinz, and Ernst Stadlober. Generating beta variates via
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// patchwork rejection. Computing 50.1 (1993): 1-18.
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DEGENERATE_SMALL, // a_ is abnormally small.
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DEGENERATE_LARGE, // a_ is abnormally large.
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};
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result_type alpha_;
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result_type beta_;
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result_type a_; // the smaller of {alpha, beta}, or 1.0/alpha_ in JOEHNK
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result_type b_; // the larger of {alpha, beta}, or 1.0/beta_ in JOEHNK
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result_type x_; // alpha + beta, or the result in degenerate cases
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result_type log_x_; // log(x_)
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result_type y_; // "beta" in Cheng
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result_type gamma_; // "gamma" in Cheng
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Method method_;
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// Placing this last for optimal alignment.
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// Whether alpha_ != a_, i.e. true iff alpha_ > beta_.
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bool inverted_;
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static_assert(std::is_floating_point<RealType>::value,
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"Class-template absl::beta_distribution<> must be "
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"parameterized using a floating-point type.");
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};
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beta_distribution() : beta_distribution(1) {}
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explicit beta_distribution(result_type alpha, result_type beta = 1)
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: param_(alpha, beta) {}
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explicit beta_distribution(const param_type& p) : param_(p) {}
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void reset() {}
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// Generating functions
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template <typename URBG>
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result_type operator()(URBG& g) { // NOLINT(runtime/references)
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return (*this)(g, param_);
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}
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template <typename URBG>
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result_type operator()(URBG& g, // NOLINT(runtime/references)
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const param_type& p);
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param_type param() const { return param_; }
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void param(const param_type& p) { param_ = p; }
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result_type(min)() const { return 0; }
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result_type(max)() const { return 1; }
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result_type alpha() const { return param_.alpha(); }
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result_type beta() const { return param_.beta(); }
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friend bool operator==(const beta_distribution& a,
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const beta_distribution& b) {
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return a.param_ == b.param_;
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}
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friend bool operator!=(const beta_distribution& a,
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const beta_distribution& b) {
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return a.param_ != b.param_;
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}
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private:
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template <typename URBG>
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result_type AlgorithmJoehnk(URBG& g, // NOLINT(runtime/references)
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const param_type& p);
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template <typename URBG>
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result_type AlgorithmCheng(URBG& g, // NOLINT(runtime/references)
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const param_type& p);
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template <typename URBG>
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result_type DegenerateCase(URBG& g, // NOLINT(runtime/references)
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const param_type& p) {
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if (p.method_ == param_type::DEGENERATE_SMALL && p.alpha_ == p.beta_) {
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// Returns 0 or 1 with equal probability.
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random_internal::FastUniformBits<uint8_t> fast_u8;
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return static_cast<result_type>((fast_u8(g) & 0x10) !=
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0); // pick any single bit.
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}
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return p.x_;
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}
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param_type param_;
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random_internal::FastUniformBits<uint64_t> fast_u64_;
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};
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#if defined(__powerpc64__) || defined(__PPC64__) || defined(__powerpc__) || \
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defined(__ppc__) || defined(__PPC__)
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// PPC needs a more stringent boundary for long double.
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template <>
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constexpr long double
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beta_distribution<long double>::param_type::ThresholdPadding() {
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return 10;
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}
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#endif
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template <typename RealType>
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template <typename URBG>
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typename beta_distribution<RealType>::result_type
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beta_distribution<RealType>::AlgorithmJoehnk(
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URBG& g, // NOLINT(runtime/references)
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const param_type& p) {
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using random_internal::GeneratePositiveTag;
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using random_internal::GenerateRealFromBits;
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using real_type =
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absl::conditional_t<std::is_same<RealType, float>::value, float, double>;
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// Based on Joehnk, M. D. Erzeugung von betaverteilten und gammaverteilten
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// Zufallszahlen. Metrika 8.1 (1964): 5-15.
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// This method is described in Knuth, Vol 2 (Third Edition), pp 134.
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result_type u, v, x, y, z;
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for (;;) {
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u = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
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fast_u64_(g));
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v = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
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fast_u64_(g));
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// Direct method. std::pow is slow for float, so rely on the optimizer to
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// remove the std::pow() path for that case.
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if (!std::is_same<float, result_type>::value) {
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x = std::pow(u, p.a_);
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y = std::pow(v, p.b_);
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z = x + y;
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if (z > 1) {
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// Reject if and only if `x + y > 1.0`
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continue;
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}
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if (z > 0) {
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// When both alpha and beta are small, x and y are both close to 0, so
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// divide by (x+y) directly may result in nan.
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return x / z;
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}
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}
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// Log transform.
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// x = log( pow(u, p.a_) ), y = log( pow(v, p.b_) )
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// since u, v <= 1.0, x, y < 0.
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x = std::log(u) * p.a_;
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y = std::log(v) * p.b_;
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if (!std::isfinite(x) || !std::isfinite(y)) {
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continue;
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}
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// z = log( pow(u, a) + pow(v, b) )
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z = x > y ? (x + std::log(1 + std::exp(y - x)))
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: (y + std::log(1 + std::exp(x - y)));
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// Reject iff log(x+y) > 0.
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if (z > 0) {
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continue;
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}
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return std::exp(x - z);
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}
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}
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template <typename RealType>
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template <typename URBG>
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typename beta_distribution<RealType>::result_type
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beta_distribution<RealType>::AlgorithmCheng(
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URBG& g, // NOLINT(runtime/references)
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const param_type& p) {
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using random_internal::GeneratePositiveTag;
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using random_internal::GenerateRealFromBits;
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using real_type =
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absl::conditional_t<std::is_same<RealType, float>::value, float, double>;
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// Based on Cheng, Russell CH. Generating beta variates with nonintegral
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// shape parameters. Communications of the ACM 21.4 (1978): 317-322.
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// (https://dl.acm.org/citation.cfm?id=359482).
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static constexpr result_type kLogFour =
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result_type(1.3862943611198906188344642429163531361); // log(4)
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static constexpr result_type kS =
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result_type(2.6094379124341003746007593332261876); // 1+log(5)
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const bool use_algorithm_ba = (p.method_ == param_type::CHENG_BA);
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result_type u1, u2, v, w, z, r, s, t, bw_inv, lhs;
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for (;;) {
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u1 = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
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fast_u64_(g));
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u2 = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
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fast_u64_(g));
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v = p.y_ * std::log(u1 / (1 - u1));
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w = p.a_ * std::exp(v);
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bw_inv = result_type(1) / (p.b_ + w);
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r = p.gamma_ * v - kLogFour;
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s = p.a_ + r - w;
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z = u1 * u1 * u2;
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if (!use_algorithm_ba && s + kS >= 5 * z) {
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break;
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}
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t = std::log(z);
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if (!use_algorithm_ba && s >= t) {
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break;
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}
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lhs = p.x_ * (p.log_x_ + std::log(bw_inv)) + r;
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if (lhs >= t) {
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break;
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}
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}
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return p.inverted_ ? (1 - w * bw_inv) : w * bw_inv;
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}
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template <typename RealType>
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template <typename URBG>
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typename beta_distribution<RealType>::result_type
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beta_distribution<RealType>::operator()(URBG& g, // NOLINT(runtime/references)
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const param_type& p) {
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switch (p.method_) {
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case param_type::JOEHNK:
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return AlgorithmJoehnk(g, p);
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case param_type::CHENG_BA:
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ABSL_FALLTHROUGH_INTENDED;
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case param_type::CHENG_BB:
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return AlgorithmCheng(g, p);
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default:
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return DegenerateCase(g, p);
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}
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}
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template <typename CharT, typename Traits, typename RealType>
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std::basic_ostream<CharT, Traits>& operator<<(
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std::basic_ostream<CharT, Traits>& os, // NOLINT(runtime/references)
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const beta_distribution<RealType>& x) {
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auto saver = random_internal::make_ostream_state_saver(os);
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os.precision(random_internal::stream_precision_helper<RealType>::kPrecision);
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os << x.alpha() << os.fill() << x.beta();
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return os;
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}
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template <typename CharT, typename Traits, typename RealType>
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std::basic_istream<CharT, Traits>& operator>>(
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std::basic_istream<CharT, Traits>& is, // NOLINT(runtime/references)
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beta_distribution<RealType>& x) { // NOLINT(runtime/references)
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using result_type = typename beta_distribution<RealType>::result_type;
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using param_type = typename beta_distribution<RealType>::param_type;
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result_type alpha, beta;
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auto saver = random_internal::make_istream_state_saver(is);
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alpha = random_internal::read_floating_point<result_type>(is);
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if (is.fail()) return is;
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beta = random_internal::read_floating_point<result_type>(is);
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if (!is.fail()) {
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x.param(param_type(alpha, beta));
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}
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return is;
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}
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ABSL_NAMESPACE_END
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} // namespace absl
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#endif // ABSL_RANDOM_BETA_DISTRIBUTION_H_
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