// 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 {
ABSL_NAMESPACE_BEGIN
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
ABSL_NAMESPACE_END
} // namespace absl
