about summary refs log tree commit diff
path: root/absl/random/internal/chi_square.cc
blob: 640d48cea6f47fbf47e4d5aadd916973853e1615 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
// 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