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Diffstat (limited to 'absl/random/internal/chi_square.cc')
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1 files changed, 230 insertions, 0 deletions
diff --git a/absl/random/internal/chi_square.cc b/absl/random/internal/chi_square.cc new file mode 100644 index 00000000..c0acc947 --- /dev/null +++ b/absl/random/internal/chi_square.cc @@ -0,0 +1,230 @@ +// 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/internal/chi_square.h" + +#include <cmath> + +#include "absl/random/internal/distribution_test_util.h" + +namespace absl { +namespace random_internal { +namespace { + +#if defined(__EMSCRIPTEN__) +// Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found. +inline double fma(double x, double y, double z) { + return (x * y) + z; +} +#endif + +// Use Horner's method to evaluate a polynomial. +template <typename T, unsigned N> +inline T EvaluatePolynomial(T x, const T (&poly)[N]) { +#if !defined(__EMSCRIPTEN__) + using std::fma; +#endif + T p = poly[N - 1]; + for (unsigned i = 2; i <= N; i++) { + p = fma(p, x, poly[N - i]); + } + return p; +} + +static constexpr int kLargeDOF = 150; + +// Returns the probability of a normal z-value. +// +// Adapted from the POZ function in: +// Ibbetson D, Algorithm 209 +// Collected Algorithms of the CACM 1963 p. 616 +// +double POZ(double z) { + static constexpr double kP1[] = { + 0.797884560593, -0.531923007300, 0.319152932694, + -0.151968751364, 0.059054035642, -0.019198292004, + 0.005198775019, -0.001075204047, 0.000124818987, + }; + static constexpr double kP2[] = { + 0.999936657524, 0.000535310849, -0.002141268741, 0.005353579108, + -0.009279453341, 0.011630447319, -0.010557625006, 0.006549791214, + -0.002034254874, -0.000794620820, 0.001390604284, -0.000676904986, + -0.000019538132, 0.000152529290, -0.000045255659, + }; + + const double kZMax = 6.0; // Maximum meaningful z-value. + if (z == 0.0) { + return 0.5; + } + double x; + double y = 0.5 * std::fabs(z); + if (y >= (kZMax * 0.5)) { + x = 1.0; + } else if (y < 1.0) { + double w = y * y; + x = EvaluatePolynomial(w, kP1) * y * 2.0; + } else { + y -= 2.0; + x = EvaluatePolynomial(y, kP2); + } + return z > 0.0 ? ((x + 1.0) * 0.5) : ((1.0 - x) * 0.5); +} + +// Approximates the survival function of the normal distribution. +// +// Algorithm 26.2.18, from: +// [Abramowitz and Stegun, Handbook of Mathematical Functions,p.932] +// http://people.math.sfu.ca/~cbm/aands/abramowitz_and_stegun.pdf +// +double normal_survival(double z) { + // Maybe replace with the alternate formulation. + // 0.5 * erfc((x - mean)/(sqrt(2) * sigma)) + static constexpr double kR[] = { + 1.0, 0.196854, 0.115194, 0.000344, 0.019527, + }; + double r = EvaluatePolynomial(z, kR); + r *= r; + return 0.5 / (r * r); +} + +} // namespace + +// Calculates the critical chi-square value given degrees-of-freedom and a +// p-value, usually using bisection. Also known by the name CRITCHI. +double ChiSquareValue(int dof, double p) { + static constexpr double kChiEpsilon = + 0.000001; // Accuracy of the approximation. + static constexpr double kChiMax = + 99999.0; // Maximum chi-squared value. + + const double p_value = 1.0 - p; + if (dof < 1 || p_value > 1.0) { + return 0.0; + } + + if (dof > kLargeDOF) { + // For large degrees of freedom, use the normal approximation by + // Wilson, E. B. and Hilferty, M. M. (1931) + // chi^2 - mean + // Z = -------------- + // stddev + const double z = InverseNormalSurvival(p_value); + const double mean = 1 - 2.0 / (9 * dof); + const double variance = 2.0 / (9 * dof); + // Cannot use this method if the variance is 0. + if (variance != 0) { + return std::pow(z * std::sqrt(variance) + mean, 3.0) * dof; + } + } + + if (p_value <= 0.0) return kChiMax; + + // Otherwise search for the p value by bisection + double min_chisq = 0.0; + double max_chisq = kChiMax; + double current = dof / std::sqrt(p_value); + while ((max_chisq - min_chisq) > kChiEpsilon) { + if (ChiSquarePValue(current, dof) < p_value) { + max_chisq = current; + } else { + min_chisq = current; + } + current = (max_chisq + min_chisq) * 0.5; + } + return current; +} + +// Calculates the p-value (probability) of a given chi-square value +// and degrees of freedom. +// +// Adapted from the POCHISQ function from: +// Hill, I. D. and Pike, M. C. Algorithm 299 +// Collected Algorithms of the CACM 1963 p. 243 +// +double ChiSquarePValue(double chi_square, int dof) { + static constexpr double kLogSqrtPi = + 0.5723649429247000870717135; // Log[Sqrt[Pi]] + static constexpr double kInverseSqrtPi = + 0.5641895835477562869480795; // 1/(Sqrt[Pi]) + + // For large degrees of freedom, use the normal approximation by + // Wilson, E. B. and Hilferty, M. M. (1931) + // Via Wikipedia: + // By the Central Limit Theorem, because the chi-square distribution is the + // sum of k independent random variables with finite mean and variance, it + // converges to a normal distribution for large k. + if (dof > kLargeDOF) { + // Re-scale everything. + const double chi_square_scaled = std::pow(chi_square / dof, 1.0 / 3); + const double mean = 1 - 2.0 / (9 * dof); + const double variance = 2.0 / (9 * dof); + // If variance is 0, this method cannot be used. + if (variance != 0) { + const double z = (chi_square_scaled - mean) / std::sqrt(variance); + if (z > 0) { + return normal_survival(z); + } else if (z < 0) { + return 1.0 - normal_survival(-z); + } else { + return 0.5; + } + } + } + + // The chi square function is >= 0 for any degrees of freedom. + // In other words, probability that the chi square function >= 0 is 1. + if (chi_square <= 0.0) return 1.0; + + // If the degrees of freedom is zero, the chi square function is always 0 by + // definition. In other words, the probability that the chi square function + // is > 0 is zero (chi square values <= 0 have been filtered above). + if (dof < 1) return 0; + + auto capped_exp = [](double x) { return x < -20 ? 0.0 : std::exp(x); }; + static constexpr double kBigX = 20; + + double a = 0.5 * chi_square; + const bool even = !(dof & 1); // True if dof is an even number. + const double y = capped_exp(-a); + double s = even ? y : (2.0 * POZ(-std::sqrt(chi_square))); + + if (dof <= 2) { + return s; + } + + chi_square = 0.5 * (dof - 1.0); + double z = (even ? 1.0 : 0.5); + if (a > kBigX) { + double e = (even ? 0.0 : kLogSqrtPi); + double c = std::log(a); + while (z <= chi_square) { + e = std::log(z) + e; + s += capped_exp(c * z - a - e); + z += 1.0; + } + return s; + } + + double e = (even ? 1.0 : (kInverseSqrtPi / std::sqrt(a))); + double c = 0.0; + while (z <= chi_square) { + e = e * (a / z); + c = c + e; + z += 1.0; + } + return c * y + s; +} + +} // namespace random_internal +} // namespace absl |