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distribution_test_util.cc (13373B)
      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 #include "absl/random/internal/distribution_test_util.h"
     16 
     17 #include <cassert>
     18 #include <cmath>
     19 #include <string>
     20 #include <vector>
     21 
     22 #include "absl/base/internal/raw_logging.h"
     23 #include "absl/base/macros.h"
     24 #include "absl/strings/str_cat.h"
     25 #include "absl/strings/str_format.h"
     26 
     27 namespace absl {
     28 ABSL_NAMESPACE_BEGIN
     29 namespace random_internal {
     30 namespace {
     31 
     32 #if defined(__EMSCRIPTEN__)
     33 // Workaround __EMSCRIPTEN__ error: llvm_fma_f64 not found.
     34 inline double fma(double x, double y, double z) { return (x * y) + z; }
     35 #endif
     36 
     37 }  // namespace
     38 
     39 DistributionMoments ComputeDistributionMoments(
     40    absl::Span<const double> data_points) {
     41  DistributionMoments result;
     42 
     43  // Compute m1
     44  for (double x : data_points) {
     45    result.n++;
     46    result.mean += x;
     47  }
     48  result.mean /= static_cast<double>(result.n);
     49 
     50  // Compute m2, m3, m4
     51  for (double x : data_points) {
     52    double v = x - result.mean;
     53    result.variance += v * v;
     54    result.skewness += v * v * v;
     55    result.kurtosis += v * v * v * v;
     56  }
     57  result.variance /= static_cast<double>(result.n - 1);
     58 
     59  result.skewness /= static_cast<double>(result.n);
     60  result.skewness /= std::pow(result.variance, 1.5);
     61 
     62  result.kurtosis /= static_cast<double>(result.n);
     63  result.kurtosis /= std::pow(result.variance, 2.0);
     64  return result;
     65 
     66  // When validating the min/max count, the following confidence intervals may
     67  // be of use:
     68  // 3.291 * stddev = 99.9% CI
     69  // 2.576 * stddev = 99% CI
     70  // 1.96 * stddev  = 95% CI
     71  // 1.65 * stddev  = 90% CI
     72 }
     73 
     74 std::ostream& operator<<(std::ostream& os, const DistributionMoments& moments) {
     75  return os << absl::StrFormat("mean=%f, stddev=%f, skewness=%f, kurtosis=%f",
     76                               moments.mean, std::sqrt(moments.variance),
     77                               moments.skewness, moments.kurtosis);
     78 }
     79 
     80 double InverseNormalSurvival(double x) {
     81  // inv_sf(u) = -sqrt(2) * erfinv(2u-1)
     82  static constexpr double kSqrt2 = 1.4142135623730950488;
     83  return -kSqrt2 * absl::random_internal::erfinv(2 * x - 1.0);
     84 }
     85 
     86 bool Near(absl::string_view msg, double actual, double expected, double bound) {
     87  assert(bound > 0.0);
     88  double delta = fabs(expected - actual);
     89  if (delta < bound) {
     90    return true;
     91  }
     92 
     93  std::string formatted = absl::StrCat(
     94      msg, " actual=", actual, " expected=", expected, " err=", delta / bound);
     95  ABSL_RAW_LOG(INFO, "%s", formatted.c_str());
     96  return false;
     97 }
     98 
     99 // TODO(absl-team): Replace with an "ABSL_HAVE_SPECIAL_MATH" and try
    100 // to use std::beta().  As of this writing P0226R1 is not implemented
    101 // in libc++: http://libcxx.llvm.org/cxx1z_status.html
    102 double beta(double p, double q) {
    103  // Beta(x, y) = Gamma(x) * Gamma(y) / Gamma(x+y)
    104  double lbeta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    105  return std::exp(lbeta);
    106 }
    107 
    108 // Approximation to inverse of the Error Function in double precision.
    109 // (http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf)
    110 double erfinv(double x) {
    111 #if !defined(__EMSCRIPTEN__)
    112  using std::fma;
    113 #endif
    114 
    115  double w = 0.0;
    116  double p = 0.0;
    117  w = -std::log((1.0 - x) * (1.0 + x));
    118  if (w < 6.250000) {
    119    w = w - 3.125000;
    120    p = -3.6444120640178196996e-21;
    121    p = fma(p, w, -1.685059138182016589e-19);
    122    p = fma(p, w, 1.2858480715256400167e-18);
    123    p = fma(p, w, 1.115787767802518096e-17);
    124    p = fma(p, w, -1.333171662854620906e-16);
    125    p = fma(p, w, 2.0972767875968561637e-17);
    126    p = fma(p, w, 6.6376381343583238325e-15);
    127    p = fma(p, w, -4.0545662729752068639e-14);
    128    p = fma(p, w, -8.1519341976054721522e-14);
    129    p = fma(p, w, 2.6335093153082322977e-12);
    130    p = fma(p, w, -1.2975133253453532498e-11);
    131    p = fma(p, w, -5.4154120542946279317e-11);
    132    p = fma(p, w, 1.051212273321532285e-09);
    133    p = fma(p, w, -4.1126339803469836976e-09);
    134    p = fma(p, w, -2.9070369957882005086e-08);
    135    p = fma(p, w, 4.2347877827932403518e-07);
    136    p = fma(p, w, -1.3654692000834678645e-06);
    137    p = fma(p, w, -1.3882523362786468719e-05);
    138    p = fma(p, w, 0.0001867342080340571352);
    139    p = fma(p, w, -0.00074070253416626697512);
    140    p = fma(p, w, -0.0060336708714301490533);
    141    p = fma(p, w, 0.24015818242558961693);
    142    p = fma(p, w, 1.6536545626831027356);
    143  } else if (w < 16.000000) {
    144    w = std::sqrt(w) - 3.250000;
    145    p = 2.2137376921775787049e-09;
    146    p = fma(p, w, 9.0756561938885390979e-08);
    147    p = fma(p, w, -2.7517406297064545428e-07);
    148    p = fma(p, w, 1.8239629214389227755e-08);
    149    p = fma(p, w, 1.5027403968909827627e-06);
    150    p = fma(p, w, -4.013867526981545969e-06);
    151    p = fma(p, w, 2.9234449089955446044e-06);
    152    p = fma(p, w, 1.2475304481671778723e-05);
    153    p = fma(p, w, -4.7318229009055733981e-05);
    154    p = fma(p, w, 6.8284851459573175448e-05);
    155    p = fma(p, w, 2.4031110387097893999e-05);
    156    p = fma(p, w, -0.0003550375203628474796);
    157    p = fma(p, w, 0.00095328937973738049703);
    158    p = fma(p, w, -0.0016882755560235047313);
    159    p = fma(p, w, 0.0024914420961078508066);
    160    p = fma(p, w, -0.0037512085075692412107);
    161    p = fma(p, w, 0.005370914553590063617);
    162    p = fma(p, w, 1.0052589676941592334);
    163    p = fma(p, w, 3.0838856104922207635);
    164  } else {
    165    w = std::sqrt(w) - 5.000000;
    166    p = -2.7109920616438573243e-11;
    167    p = fma(p, w, -2.5556418169965252055e-10);
    168    p = fma(p, w, 1.5076572693500548083e-09);
    169    p = fma(p, w, -3.7894654401267369937e-09);
    170    p = fma(p, w, 7.6157012080783393804e-09);
    171    p = fma(p, w, -1.4960026627149240478e-08);
    172    p = fma(p, w, 2.9147953450901080826e-08);
    173    p = fma(p, w, -6.7711997758452339498e-08);
    174    p = fma(p, w, 2.2900482228026654717e-07);
    175    p = fma(p, w, -9.9298272942317002539e-07);
    176    p = fma(p, w, 4.5260625972231537039e-06);
    177    p = fma(p, w, -1.9681778105531670567e-05);
    178    p = fma(p, w, 7.5995277030017761139e-05);
    179    p = fma(p, w, -0.00021503011930044477347);
    180    p = fma(p, w, -0.00013871931833623122026);
    181    p = fma(p, w, 1.0103004648645343977);
    182    p = fma(p, w, 4.8499064014085844221);
    183  }
    184  return p * x;
    185 }
    186 
    187 namespace {
    188 
    189 // Direct implementation of AS63, BETAIN()
    190 // https://www.jstor.org/stable/2346797?seq=3#page_scan_tab_contents.
    191 //
    192 // BETAIN(x, p, q, beta)
    193 //  x:     the value of the upper limit x.
    194 //  p:     the value of the parameter p.
    195 //  q:     the value of the parameter q.
    196 //  beta:  the value of ln B(p, q)
    197 //
    198 double BetaIncompleteImpl(const double x, const double p, const double q,
    199                          const double beta) {
    200  if (p < (p + q) * x) {
    201    // Incomplete beta function is symmetrical, so return the complement.
    202    return 1. - BetaIncompleteImpl(1.0 - x, q, p, beta);
    203  }
    204 
    205  double psq = p + q;
    206  const double kErr = 1e-14;
    207  const double xc = 1. - x;
    208  const double pre =
    209      std::exp(p * std::log(x) + (q - 1.) * std::log(xc) - beta) / p;
    210 
    211  double term = 1.;
    212  double ai = 1.;
    213  double result = 1.;
    214  int ns = static_cast<int>(q + xc * psq);
    215 
    216  // Use the soper reduction formula.
    217  double rx = (ns == 0) ? x : x / xc;
    218  double temp = q - ai;
    219  for (;;) {
    220    term = term * temp * rx / (p + ai);
    221    result = result + term;
    222    temp = std::fabs(term);
    223    if (temp < kErr && temp < kErr * result) {
    224      return result * pre;
    225    }
    226    ai = ai + 1.;
    227    --ns;
    228    if (ns >= 0) {
    229      temp = q - ai;
    230      if (ns == 0) {
    231        rx = x;
    232      }
    233    } else {
    234      temp = psq;
    235      psq = psq + 1.;
    236    }
    237  }
    238 
    239  // NOTE: See also TOMS Algorithm 708.
    240  // http://www.netlib.org/toms/index.html
    241  //
    242  // NOTE: The NWSC library also includes BRATIO / ISUBX (p87)
    243  // https://archive.org/details/DTIC_ADA261511/page/n75
    244 }
    245 
    246 // Direct implementation of AS109, XINBTA(p, q, beta, alpha)
    247 // https://www.jstor.org/stable/2346798?read-now=1&seq=4#page_scan_tab_contents
    248 // https://www.jstor.org/stable/2346887?seq=1#page_scan_tab_contents
    249 //
    250 // XINBTA(p, q, beta, alpha)
    251 //  p:     the value of the parameter p.
    252 //  q:     the value of the parameter q.
    253 //  beta:  the value of ln B(p, q)
    254 //  alpha: the value of the lower tail area.
    255 //
    256 double BetaIncompleteInvImpl(const double p, const double q, const double beta,
    257                             const double alpha) {
    258  if (alpha < 0.5) {
    259    // Inverse Incomplete beta function is symmetrical, return the complement.
    260    return 1. - BetaIncompleteInvImpl(q, p, beta, 1. - alpha);
    261  }
    262  const double kErr = 1e-14;
    263  double value = kErr;
    264 
    265  // Compute the initial estimate.
    266  {
    267    double r = std::sqrt(-std::log(alpha * alpha));
    268    double y =
    269        r - fma(r, 0.27061, 2.30753) / fma(r, fma(r, 0.04481, 0.99229), 1.0);
    270    if (p > 1. && q > 1.) {
    271      r = (y * y - 3.) / 6.;
    272      double s = 1. / (p + p - 1.);
    273      double t = 1. / (q + q - 1.);
    274      double h = 2. / s + t;
    275      double w =
    276          y * std::sqrt(h + r) / h - (t - s) * (r + 5. / 6. - t / (3. * h));
    277      value = p / (p + q * std::exp(w + w));
    278    } else {
    279      r = q + q;
    280      double t = 1.0 / (9. * q);
    281      double u = 1.0 - t + y * std::sqrt(t);
    282      t = r * (u * u * u);
    283      if (t <= 0) {
    284        value = 1.0 - std::exp((std::log((1.0 - alpha) * q) + beta) / q);
    285      } else {
    286        t = (4.0 * p + r - 2.0) / t;
    287        if (t <= 1) {
    288          value = std::exp((std::log(alpha * p) + beta) / p);
    289        } else {
    290          value = 1.0 - 2.0 / (t + 1.0);
    291        }
    292      }
    293    }
    294  }
    295 
    296  // Solve for x using a modified newton-raphson method using the function
    297  // BetaIncomplete.
    298  {
    299    value = std::max(value, kErr);
    300    value = std::min(value, 1.0 - kErr);
    301 
    302    const double r = 1.0 - p;
    303    const double t = 1.0 - q;
    304    double y;
    305    double yprev = 0;
    306    double sq = 1;
    307    double prev = 1;
    308    for (;;) {
    309      if (value < 0 || value > 1.0) {
    310        // Error case; value went infinite.
    311        return std::numeric_limits<double>::infinity();
    312      } else if (value == 0 || value == 1) {
    313        y = value;
    314      } else {
    315        y = BetaIncompleteImpl(value, p, q, beta);
    316        if (!std::isfinite(y)) {
    317          return y;
    318        }
    319      }
    320      y = (y - alpha) *
    321          std::exp(beta + r * std::log(value) + t * std::log(1.0 - value));
    322      if (y * yprev <= 0) {
    323        prev = std::max(sq, std::numeric_limits<double>::min());
    324      }
    325      double g = 1.0;
    326      for (;;) {
    327        const double adj = g * y;
    328        const double adj_sq = adj * adj;
    329        if (adj_sq >= prev) {
    330          g = g / 3.0;
    331          continue;
    332        }
    333        const double tx = value - adj;
    334        if (tx < 0 || tx > 1) {
    335          g = g / 3.0;
    336          continue;
    337        }
    338        if (prev < kErr) {
    339          return value;
    340        }
    341        if (y * y < kErr) {
    342          return value;
    343        }
    344        if (tx == value) {
    345          return value;
    346        }
    347        if (tx == 0 || tx == 1) {
    348          g = g / 3.0;
    349          continue;
    350        }
    351        value = tx;
    352        yprev = y;
    353        break;
    354      }
    355    }
    356  }
    357 
    358  // NOTES: See also: Asymptotic inversion of the incomplete beta function.
    359  // https://core.ac.uk/download/pdf/82140723.pdf
    360  //
    361  // NOTE: See the Boost library documentation as well:
    362  // https://www.boost.org/doc/libs/1_52_0/libs/math/doc/sf_and_dist/html/math_toolkit/special/sf_beta/ibeta_function.html
    363 }
    364 
    365 }  // namespace
    366 
    367 double BetaIncomplete(const double x, const double p, const double q) {
    368  // Error cases.
    369  if (p < 0 || q < 0 || x < 0 || x > 1.0) {
    370    return std::numeric_limits<double>::infinity();
    371  }
    372  if (x == 0 || x == 1) {
    373    return x;
    374  }
    375  // ln(Beta(p, q))
    376  double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    377  return BetaIncompleteImpl(x, p, q, beta);
    378 }
    379 
    380 double BetaIncompleteInv(const double p, const double q, const double alpha) {
    381  // Error cases.
    382  if (p < 0 || q < 0 || alpha < 0 || alpha > 1.0) {
    383    return std::numeric_limits<double>::infinity();
    384  }
    385  if (alpha == 0 || alpha == 1) {
    386    return alpha;
    387  }
    388  // ln(Beta(p, q))
    389  double beta = std::lgamma(p) + std::lgamma(q) - std::lgamma(p + q);
    390  return BetaIncompleteInvImpl(p, q, beta, alpha);
    391 }
    392 
    393 // Given `num_trials` trials each with probability `p` of success, the
    394 // probability of no failures is `p^k`. To ensure the probability of a failure
    395 // is no more than `p_fail`, it must be that `p^k == 1 - p_fail`. This function
    396 // computes `p` from that equation.
    397 double RequiredSuccessProbability(const double p_fail, const int num_trials) {
    398  double p = std::exp(std::log(1.0 - p_fail) / static_cast<double>(num_trials));
    399  ABSL_ASSERT(p > 0);
    400  return p;
    401 }
    402 
    403 double ZScore(double expected_mean, const DistributionMoments& moments) {
    404  return (moments.mean - expected_mean) /
    405         (std::sqrt(moments.variance) /
    406          std::sqrt(static_cast<double>(moments.n)));
    407 }
    408 
    409 double MaxErrorTolerance(double acceptance_probability) {
    410  double one_sided_pvalue = 0.5 * (1.0 - acceptance_probability);
    411  const double max_err = InverseNormalSurvival(one_sided_pvalue);
    412  ABSL_ASSERT(max_err > 0);
    413  return max_err;
    414 }
    415 
    416 }  // namespace random_internal
    417 ABSL_NAMESPACE_END
    418 }  // namespace absl
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