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-// SPDX-License-Identifier: GPL-3.0
-
-/********************************************************************
- *
- * File: KolmogorovSmirnovDist.c
- * Environment: ISO C99 or ANSI C89
- * Author: Richard Simard
- * Organization: DIRO, Université de Montréal
- * Date: 1 February 2012
- * Version 1.1
-
- * Copyright 1 march 2010 by Université de Montréal,
- Richard Simard and Pierre L'Ecuyer
- =====================================================================
-
- This program is free software: you can redistribute it and/or modify
- it under the terms of the GNU General Public License as published by
- the Free Software Foundation, version 3 of the License.
-
- This program is distributed in the hope that it will be useful,
- but WITHOUT ANY WARRANTY; without even the implied warranty of
- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- GNU General Public License for more details.
-
- You should have received a copy of the GNU General Public License
- along with this program. If not, see <http://www.gnu.org/licenses/>.
-
- =====================================================================*/
-
-#include "KolmogorovSmirnovDist.h"
-#include <math.h>
-#include <stdlib.h>
-
-#define num_Pi 3.14159265358979323846 /* PI */
-#define num_Ln2 0.69314718055994530941 /* log(2) */
-
-/* For x close to 0 or 1, we use the exact formulae of Ruben-Gambino in all
- cases. For n <= NEXACT, we use exact algorithms: the Durbin matrix and
- the Pomeranz algorithms. For n > NEXACT, we use asymptotic methods
- except for x close to 0 where we still use the method of Durbin
- for n <= NKOLMO. For n > NKOLMO, we use asymptotic methods only and
- so the precision is less for x close to 0.
- We could increase the limit NKOLMO to 10^6 to get better precision
- for x close to 0, but at the price of a slower speed. */
-#define NEXACT 500
-#define NKOLMO 100000
-
-/* The Durbin matrix algorithm for the Kolmogorov-Smirnov distribution */
-static double DurbinMatrix (int n, double d);
-
-
-/*========================================================================*/
-#if 0
-
-/* For ANSI C89 only, not for ISO C99 */
-#define MAXI 50
-#define EPSILON 1.0e-15
-
-double log1p (double x)
-{
- /* returns a value equivalent to log(1 + x) accurate also for small x. */
- if (fabs (x) > 0.1) {
- return log (1.0 + x);
- } else {
- double term = x;
- double sum = x;
- int s = 2;
- while ((fabs (term) > EPSILON * fabs (sum)) && (s < MAXI)) {
- term *= -x;
- sum += term / s;
- s++;
- }
- return sum;
- }
-}
-
-#undef MAXI
-#undef EPSILON
-
-#endif
-
-/*========================================================================*/
-#define MFACT 30
-
-/* The natural logarithm of factorial n! for 0 <= n <= MFACT */
-static double LnFactorial[MFACT + 1] = {
- 0.,
- 0.,
- 0.6931471805599453,
- 1.791759469228055,
- 3.178053830347946,
- 4.787491742782046,
- 6.579251212010101,
- 8.525161361065415,
- 10.60460290274525,
- 12.80182748008147,
- 15.10441257307552,
- 17.50230784587389,
- 19.98721449566188,
- 22.55216385312342,
- 25.19122118273868,
- 27.89927138384088,
- 30.67186010608066,
- 33.50507345013688,
- 36.39544520803305,
- 39.33988418719949,
- 42.33561646075348,
- 45.3801388984769,
- 48.47118135183522,
- 51.60667556776437,
- 54.7847293981123,
- 58.00360522298051,
- 61.26170176100199,
- 64.55753862700632,
- 67.88974313718154,
- 71.257038967168,
- 74.65823634883016
-};
-
-/*------------------------------------------------------------------------*/
-
-static double getLogFactorial (int n)
-{
- /* Returns the natural logarithm of factorial n! */
- if (n <= MFACT) {
- return LnFactorial[n];
-
- } else {
- double x = (double) (n + 1);
- double y = 1.0 / (x * x);
- double z = ((-(5.95238095238E-4 * y) + 7.936500793651E-4) * y -
- 2.7777777777778E-3) * y + 8.3333333333333E-2;
- z = ((x - 0.5) * log (x) - x) + 9.1893853320467E-1 + z / x;
- return z;
- }
-}
-
-/*------------------------------------------------------------------------*/
-
-static double rapfac (int n)
-{
- /* Computes n! / n^n */
- int i;
- double res = 1.0 / n;
- for (i = 2; i <= n; i++) {
- res *= (double) i / n;
- }
- return res;
-}
-
-
-/*========================================================================*/
-
-static double **CreateMatrixD (int N, int M)
-{
- int i;
- double **T2;
-
- T2 = (double **) malloc (N * sizeof (double *));
- T2[0] = (double *) malloc ((size_t) N * M * sizeof (double));
- for (i = 1; i < N; i++)
- T2[i] = T2[0] + i * M;
- return T2;
-}
-
-
-static void DeleteMatrixD (double **T)
-{
- free (T[0]);
- free (T);
-}
-
-
-/*========================================================================*/
-
-static double KSPlusbarAsymp (int n, double x)
-{
- /* Compute the probability of the KS+ distribution using an asymptotic
- formula */
- double t = (6.0 * n * x + 1);
- double z = t * t / (18.0 * n);
- double v = 1.0 - (2.0 * z * z - 4.0 * z - 1.0) / (18.0 * n);
- if (v <= 0.0)
- return 0.0;
- v = v * exp (-z);
- if (v >= 1.0)
- return 1.0;
- return v;
-}
-
-
-/*-------------------------------------------------------------------------*/
-
-static double KSPlusbarUpper (int n, double x)
-{
- /* Compute the probability of the KS+ distribution in the upper tail using
- Smirnov's stable formula */
- const double EPSILON = 1.0E-12;
- double q;
- double Sum = 0.0;
- double term;
- double t;
- double LogCom;
- double LOGJMAX;
- int j;
- int jdiv;
- int jmax = (int) (n * (1.0 - x));
-
- if (n > 200000)
- return KSPlusbarAsymp (n, x);
-
- /* Avoid log(0) for j = jmax and q ~ 1.0 */
- if ((1.0 - x - (double) jmax / n) <= 0.0)
- jmax--;
-
- if (n > 3000)
- jdiv = 2;
- else
- jdiv = 3;
-
- j = jmax / jdiv + 1;
- LogCom = getLogFactorial (n) - getLogFactorial (j) -
- getLogFactorial (n - j);
- LOGJMAX = LogCom;
-
- while (j <= jmax) {
- q = (double) j / n + x;
- term = LogCom + (j - 1) * log (q) + (n - j) * log1p (-q);
- t = exp (term);
- Sum += t;
- LogCom += log ((double) (n - j) / (j + 1));
- if (t <= Sum * EPSILON)
- break;
- j++;
- }
-
- j = jmax / jdiv;
- LogCom = LOGJMAX + log ((double) (j + 1) / (n - j));
-
- while (j > 0) {
- q = (double) j / n + x;
- term = LogCom + (j - 1) * log (q) + (n - j) * log1p (-q);
- t = exp (term);
- Sum += t;
- LogCom += log ((double) j / (n - j + 1));
- if (t <= Sum * EPSILON)
- break;
- j--;
- }
-
- Sum *= x;
- /* add the term j = 0 */
- Sum += exp (n * log1p (-x));
- return Sum;
-}
-
-
-/*========================================================================*/
-
-static double Pelz (int n, double x)
-{
- /* Approximating the Lower Tail-Areas of the Kolmogorov-Smirnov One-Sample
- Statistic,
- Wolfgang Pelz and I. J. Good,
- Journal of the Royal Statistical Society, Series B.
- Vol. 38, No. 2 (1976), pp. 152-156
- */
-
- const int JMAX = 20;
- const double EPS = 1.0e-10;
- const double C = 2.506628274631001; /* sqrt(2*Pi) */
- const double C2 = 1.2533141373155001; /* sqrt(Pi/2) */
- const double PI2 = num_Pi * num_Pi;
- const double PI4 = PI2 * PI2;
- const double RACN = sqrt ((double) n);
- const double z = RACN * x;
- const double z2 = z * z;
- const double z4 = z2 * z2;
- const double z6 = z4 * z2;
- const double w = PI2 / (2.0 * z * z);
- double ti, term, tom;
- double sum;
- int j;
-
- term = 1;
- j = 0;
- sum = 0;
- while (j <= JMAX && term > EPS * sum) {
- ti = j + 0.5;
- term = exp (-ti * ti * w);
- sum += term;
- j++;
- }
- sum *= C / z;
-
- term = 1;
- tom = 0;
- j = 0;
- while (j <= JMAX && fabs (term) > EPS * fabs (tom)) {
- ti = j + 0.5;
- term = (PI2 * ti * ti - z2) * exp (-ti * ti * w);
- tom += term;
- j++;
- }
- sum += tom * C2 / (RACN * 3.0 * z4);
-
- term = 1;
- tom = 0;
- j = 0;
- while (j <= JMAX && fabs (term) > EPS * fabs (tom)) {
- ti = j + 0.5;
- term = 6 * z6 + 2 * z4 + PI2 * (2 * z4 - 5 * z2) * ti * ti +
- PI4 * (1 - 2 * z2) * ti * ti * ti * ti;
- term *= exp (-ti * ti * w);
- tom += term;
- j++;
- }
- sum += tom * C2 / (n * 36.0 * z * z6);
-
- term = 1;
- tom = 0;
- j = 1;
- while (j <= JMAX && term > EPS * tom) {
- ti = j;
- term = PI2 * ti * ti * exp (-ti * ti * w);
- tom += term;
- j++;
- }
- sum -= tom * C2 / (n * 18.0 * z * z2);
-
- term = 1;
- tom = 0;
- j = 0;
- while (j <= JMAX && fabs (term) > EPS * fabs (tom)) {
- ti = j + 0.5;
- ti = ti * ti;
- term = -30 * z6 - 90 * z6 * z2 + PI2 * (135 * z4 - 96 * z6) * ti +
- PI4 * (212 * z4 - 60 * z2) * ti * ti + PI2 * PI4 * ti * ti * ti * (5 -
- 30 * z2);
- term *= exp (-ti * w);
- tom += term;
- j++;
- }
- sum += tom * C2 / (RACN * n * 3240.0 * z4 * z6);
-
- term = 1;
- tom = 0;
- j = 1;
- while (j <= JMAX && fabs (term) > EPS * fabs (tom)) {
- ti = j * j;
- term = (3 * PI2 * ti * z2 - PI4 * ti * ti) * exp (-ti * w);
- tom += term;
- j++;
- }
- sum += tom * C2 / (RACN * n * 108.0 * z6);
-
- return sum;
-}
-
-
-/*=========================================================================*/
-
-static void CalcFloorCeil (
- int n, /* sample size */
- double t, /* = nx */
- double *A, /* A_i */
- double *Atflo, /* floor (A_i - t) */
- double *Atcei /* ceiling (A_i + t) */
- )
-{
- /* Precompute A_i, floors, and ceilings for limits of sums in the Pomeranz
- algorithm */
- int i;
- int ell = (int) t; /* floor (t) */
- double z = t - ell; /* t - floor (t) */
- double w = ceil (t) - t;
-
- if (z > 0.5) {
- for (i = 2; i <= 2 * n + 2; i += 2)
- Atflo[i] = i / 2 - 2 - ell;
- for (i = 1; i <= 2 * n + 2; i += 2)
- Atflo[i] = i / 2 - 1 - ell;
-
- for (i = 2; i <= 2 * n + 2; i += 2)
- Atcei[i] = i / 2 + ell;
- for (i = 1; i <= 2 * n + 2; i += 2)
- Atcei[i] = i / 2 + 1 + ell;
-
- } else if (z > 0.0) {
- for (i = 1; i <= 2 * n + 2; i++)
- Atflo[i] = i / 2 - 1 - ell;
-
- for (i = 2; i <= 2 * n + 2; i++)
- Atcei[i] = i / 2 + ell;
- Atcei[1] = 1 + ell;
-
- } else { /* z == 0 */
- for (i = 2; i <= 2 * n + 2; i += 2)
- Atflo[i] = i / 2 - 1 - ell;
- for (i = 1; i <= 2 * n + 2; i += 2)
- Atflo[i] = i / 2 - ell;
-
- for (i = 2; i <= 2 * n + 2; i += 2)
- Atcei[i] = i / 2 - 1 + ell;
- for (i = 1; i <= 2 * n + 2; i += 2)
- Atcei[i] = i / 2 + ell;
- }
-
- if (w < z)
- z = w;
- A[0] = A[1] = 0;
- A[2] = z;
- A[3] = 1 - A[2];
- for (i = 4; i <= 2 * n + 1; i++)
- A[i] = A[i - 2] + 1;
- A[2 * n + 2] = n;
-}
-
-
-/*========================================================================*/
-
-static double Pomeranz (int n, double x)
-{
- /* The Pomeranz algorithm to compute the KS distribution */
- const double EPS = 1.0e-15;
- const int ENO = 350;
- const double RENO = ldexp (1.0, ENO); /* for renormalization of V */
- int coreno; /* counter: how many renormalizations */
- const double t = n * x;
- double w, sum, minsum;
- int i, j, k, s;
- int r1, r2; /* Indices i and i-1 for V[i][] */
- int jlow, jup, klow, kup, kup0;
- double *A;
- double *Atflo;
- double *Atcei;
- double **V;
- double **H; /* = pow(w, j) / Factorial(j) */
-
- A = (double *) calloc ((size_t) (2 * n + 3), sizeof (double));
- Atflo = (double *) calloc ((size_t) (2 * n + 3), sizeof (double));
- Atcei = (double *) calloc ((size_t) (2 * n + 3), sizeof (double));
- V = (double **) CreateMatrixD (2, n + 2);
- H = (double **) CreateMatrixD (4, n + 2);
-
- CalcFloorCeil (n, t, A, Atflo, Atcei);
-
- for (j = 1; j <= n + 1; j++)
- V[0][j] = 0;
- for (j = 2; j <= n + 1; j++)
- V[1][j] = 0;
- V[1][1] = RENO;
- coreno = 1;
-
- /* Precompute H[][] = (A[j] - A[j-1]^k / k! for speed */
- H[0][0] = 1;
- w = 2.0 * A[2] / n;
- for (j = 1; j <= n + 1; j++)
- H[0][j] = w * H[0][j - 1] / j;
-
- H[1][0] = 1;
- w = (1.0 - 2.0 * A[2]) / n;
- for (j = 1; j <= n + 1; j++)
- H[1][j] = w * H[1][j - 1] / j;
-
- H[2][0] = 1;
- w = A[2] / n;
- for (j = 1; j <= n + 1; j++)
- H[2][j] = w * H[2][j - 1] / j;
-
- H[3][0] = 1;
- for (j = 1; j <= n + 1; j++)
- H[3][j] = 0;
-
- r1 = 0;
- r2 = 1;
- for (i = 2; i <= 2 * n + 2; i++) {
- jlow = 2 + (int) Atflo[i];
- if (jlow < 1)
- jlow = 1;
- jup = (int) Atcei[i];
- if (jup > n + 1)
- jup = n + 1;
-
- klow = 2 + (int) Atflo[i - 1];
- if (klow < 1)
- klow = 1;
- kup0 = (int) Atcei[i - 1];
-
- /* Find to which case it corresponds */
- w = (A[i] - A[i - 1]) / n;
- s = -1;
- for (j = 0; j < 4; j++) {
- if (fabs (w - H[j][1]) <= EPS) {
- s = j;
- break;
- }
- }
- /* assert (s >= 0, "Pomeranz: s < 0"); */
-
- minsum = RENO;
- r1 = (r1 + 1) & 1; /* i - 1 */
- r2 = (r2 + 1) & 1; /* i */
-
- for (j = jlow; j <= jup; j++) {
- kup = kup0;
- if (kup > j)
- kup = j;
- sum = 0;
- for (k = kup; k >= klow; k--)
- sum += V[r1][k] * H[s][j - k];
- V[r2][j] = sum;
- if (sum < minsum)
- minsum = sum;
- }
-
- if (minsum < 1.0e-280) {
- /* V is too small: renormalize to avoid underflow of probabilities */
- for (j = jlow; j <= jup; j++)
- V[r2][j] *= RENO;
- coreno++; /* keep track of log of RENO */
- }
- }
-
- sum = V[r2][n + 1];
- free (A);
- free (Atflo);
- free (Atcei);
- DeleteMatrixD (H);
- DeleteMatrixD (V);
- w = getLogFactorial (n) - coreno * ENO * num_Ln2 + log (sum);
- if (w >= 0.)
- return 1.;
- return exp (w);
-}
-
-
-/*========================================================================*/
-
-static double cdfSpecial (int n, double x)
-{
- /* The KS distribution is known exactly for these cases */
-
- /* For nx^2 > 18, KSfbar(n, x) is smaller than 5e-16 */
- if ((n * x * x >= 18.0) || (x >= 1.0))
- return 1.0;
-
- if (x <= 0.5 / n)
- return 0.0;
-
- if (n == 1)
- return 2.0 * x - 1.0;
-
- if (x <= 1.0 / n) {
- double t = 2.0 * x * n - 1.0;
- double w;
- if (n <= NEXACT) {
- w = rapfac (n);
- return w * pow (t, (double) n);
- }
- w = getLogFactorial (n) + n * log (t / n);
- return exp (w);
- }
-
- if (x >= 1.0 - 1.0 / n) {
- return 1.0 - 2.0 * pow (1.0 - x, (double) n);
- }
-
- return -1.0;
-}
-
-
-/*========================================================================*/
-
-double KScdf (int n, double x)
-{
- const double w = n * x * x;
- double u = cdfSpecial (n, x);
- if (u >= 0.0)
- return u;
-
- if (n <= NEXACT) {
- if (w < 0.754693)
- return DurbinMatrix (n, x);
- if (w < 4.0)
- return Pomeranz (n, x);
- return 1.0 - KSfbar (n, x);
- }
-
- if ((w * x * n <= 7.0) && (n <= NKOLMO))
- return DurbinMatrix (n, x);
-
- return Pelz (n, x);
-}
-
-
-/*=========================================================================*/
-
-static double fbarSpecial (int n, double x)
-{
- const double w = n * x * x;
-
- if ((w >= 370.0) || (x >= 1.0))
- return 0.0;
- if ((w <= 0.0274) || (x <= 0.5 / n))
- return 1.0;
- if (n == 1)
- return 2.0 - 2.0 * x;
-
- if (x <= 1.0 / n) {
- double z;
- double t = 2.0 * x * n - 1.0;
- if (n <= NEXACT) {
- z = rapfac (n);
- return 1.0 - z * pow (t, (double) n);
- }
- z = getLogFactorial (n) + n * log (t / n);
- return 1.0 - exp (z);
- }
-
- if (x >= 1.0 - 1.0 / n) {
- return 2.0 * pow (1.0 - x, (double) n);
- }
- return -1.0;
-}
-
-
-/*========================================================================*/
-
-double KSfbar (int n, double x)
-{
- const double w = n * x * x;
- double v = fbarSpecial (n, x);
- if (v >= 0.0)
- return v;
-
- if (n <= NEXACT) {
- if (w < 4.0)
- return 1.0 - KScdf (n, x);
- else
- return 2.0 * KSPlusbarUpper (n, x);
- }
-
- if (w >= 2.65)
- return 2.0 * KSPlusbarUpper (n, x);
-
- return 1.0 - KScdf (n, x);
-}
-
-
-/*=========================================================================
-
-The following implements the Durbin matrix algorithm and was programmed by
-G. Marsaglia, Wai Wan Tsang and Jingbo Wong.
-
-I have made small modifications in their program. (Richard Simard)
-
-
-
-=========================================================================*/
-
-/*
- The C program to compute Kolmogorov's distribution
-
- K(n,d) = Prob(D_n < d), where
-
- D_n = max(x_1-0/n,x_2-1/n...,x_n-(n-1)/n,1/n-x_1,2/n-x_2,...,n/n-x_n)
-
- with x_1<x_2,...<x_n a purported set of n independent uniform [0,1)
- random variables sorted into increasing order.
- See G. Marsaglia, Wai Wan Tsang and Jingbo Wong,
- J.Stat.Software, 8, 18, pp 1--4, (2003).
-*/
-
-#define NORM 1.0e140
-#define INORM 1.0e-140
-#define LOGNORM 140
-
-
-/* Matrix product */
-static void mMultiply (double *A, double *B, double *C, int m);
-
-/* Matrix power */
-static void mPower (double *A, int eA, double *V, int *eV, int m, int n);
-
-
-static double DurbinMatrix (int n, double d)
-{
- int k, m, i, j, g, eH, eQ;
- double h, s, *H, *Q;
- /* OMIT NEXT TWO LINES IF YOU REQUIRE >7 DIGIT ACCURACY IN THE RIGHT TAIL */
-#if 0
- s = d * d * n;
- if (s > 7.24 || (s > 3.76 && n > 99))
- return 1 - 2 * exp (-(2.000071 + .331 / sqrt (n) + 1.409 / n) * s);
-#endif
- k = (int) (n * d) + 1;
- m = 2 * k - 1;
- h = k - n * d;
- H = (double *) malloc ((m * m) * sizeof (double));
- Q = (double *) malloc ((m * m) * sizeof (double));
- for (i = 0; i < m; i++)
- for (j = 0; j < m; j++)
- if (i - j + 1 < 0)
- H[i * m + j] = 0;
- else
- H[i * m + j] = 1;
- for (i = 0; i < m; i++) {
- H[i * m] -= pow (h, (double) (i + 1));
- H[(m - 1) * m + i] -= pow (h, (double) (m - i));
- }
- H[(m - 1) * m] += (2 * h - 1 > 0 ? pow (2 * h - 1, (double) m) : 0);
- for (i = 0; i < m; i++)
- for (j = 0; j < m; j++)
- if (i - j + 1 > 0)
- for (g = 1; g <= i - j + 1; g++)
- H[i * m + j] /= g;
- eH = 0;
- mPower (H, eH, Q, &eQ, m, n);
- s = Q[(k - 1) * m + k - 1];
-
- for (i = 1; i <= n; i++) {
- s = s * (double) i / n;
- if (s < INORM) {
- s *= NORM;
- eQ -= LOGNORM;
- }
- }
- s *= pow (10., (double) eQ);
- free (H);
- free (Q);
- return s;
-}
-
-
-static void mMultiply (double *A, double *B, double *C, int m)
-{
- int i, j, k;
- double s;
- for (i = 0; i < m; i++)
- for (j = 0; j < m; j++) {
- s = 0.;
- for (k = 0; k < m; k++)
- s += A[i * m + k] * B[k * m + j];
- C[i * m + j] = s;
- }
-}
-
-
-static void renormalize (double *V, int m, int *p)
-{
- int i;
- for (i = 0; i < m * m; i++)
- V[i] *= INORM;
- *p += LOGNORM;
-}
-
-
-static void mPower (double *A, int eA, double *V, int *eV, int m, int n)
-{
- double *B;
- int eB, i;
- if (n == 1) {
- for (i = 0; i < m * m; i++)
- V[i] = A[i];
- *eV = eA;
- return;
- }
- mPower (A, eA, V, eV, m, n / 2);
- B = (double *) malloc ((m * m) * sizeof (double));
- mMultiply (V, V, B, m);
- eB = 2 * (*eV);
- if (B[(m / 2) * m + (m / 2)] > NORM)
- renormalize (B, m, &eB);
-
- if (n % 2 == 0) {
- for (i = 0; i < m * m; i++)
- V[i] = B[i];
- *eV = eB;
- } else {
- mMultiply (A, B, V, m);
- *eV = eA + eB;
- }
-
- if (V[(m / 2) * m + (m / 2)] > NORM)
- renormalize (V, m, eV);
- free (B);
-}
generated by cgit v1.2.3 (git 2.39.1) at 2024-09-25 03:10:49 +0000