| 1 | // SPDX-License-Identifier: GPL-3.0 |
| 2 | |
| 3 | /******************************************************************** |
| 4 | * |
| 5 | * File: KolmogorovSmirnovDist.c |
| 6 | * Environment: ISO C99 or ANSI C89 |
| 7 | * Author: Richard Simard |
| 8 | * Organization: DIRO, Université de Montréal |
| 9 | * Date: 1 February 2012 |
| 10 | * Version 1.1 |
| 11 | |
| 12 | * Copyright 1 march 2010 by Université de Montréal, |
| 13 | Richard Simard and Pierre L'Ecuyer |
| 14 | ===================================================================== |
| 15 | |
| 16 | This program is free software: you can redistribute it and/or modify |
| 17 | it under the terms of the GNU General Public License as published by |
| 18 | the Free Software Foundation, version 3 of the License. |
| 19 | |
| 20 | This program is distributed in the hope that it will be useful, |
| 21 | but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 22 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 23 | GNU General Public License for more details. |
| 24 | |
| 25 | You should have received a copy of the GNU General Public License |
| 26 | along with this program. If not, see <http://www.gnu.org/licenses/>. |
| 27 | |
| 28 | =====================================================================*/ |
| 29 | |
| 30 | #include "KolmogorovSmirnovDist.h" |
| 31 | #include <math.h> |
| 32 | #include <stdlib.h> |
| 33 | |
| 34 | #define num_Pi 3.14159265358979323846 /* PI */ |
| 35 | #define num_Ln2 0.69314718055994530941 /* log(2) */ |
| 36 | |
| 37 | /* For x close to 0 or 1, we use the exact formulae of Ruben-Gambino in all |
| 38 | cases. For n <= NEXACT, we use exact algorithms: the Durbin matrix and |
| 39 | the Pomeranz algorithms. For n > NEXACT, we use asymptotic methods |
| 40 | except for x close to 0 where we still use the method of Durbin |
| 41 | for n <= NKOLMO. For n > NKOLMO, we use asymptotic methods only and |
| 42 | so the precision is less for x close to 0. |
| 43 | We could increase the limit NKOLMO to 10^6 to get better precision |
| 44 | for x close to 0, but at the price of a slower speed. */ |
| 45 | #define NEXACT 500 |
| 46 | #define NKOLMO 100000 |
| 47 | |
| 48 | /* The Durbin matrix algorithm for the Kolmogorov-Smirnov distribution */ |
| 49 | static double DurbinMatrix (int n, double d); |
| 50 | |
| 51 | |
| 52 | /*========================================================================*/ |
| 53 | #if 0 |
| 54 | |
| 55 | /* For ANSI C89 only, not for ISO C99 */ |
| 56 | #define MAXI 50 |
| 57 | #define EPSILON 1.0e-15 |
| 58 | |
| 59 | double log1p (double x) |
| 60 | { |
| 61 | /* returns a value equivalent to log(1 + x) accurate also for small x. */ |
| 62 | if (fabs (x) > 0.1) { |
| 63 | return log (1.0 + x); |
| 64 | } else { |
| 65 | double term = x; |
| 66 | double sum = x; |
| 67 | int s = 2; |
| 68 | while ((fabs (term) > EPSILON * fabs (sum)) && (s < MAXI)) { |
| 69 | term *= -x; |
| 70 | sum += term / s; |
| 71 | s++; |
| 72 | } |
| 73 | return sum; |
| 74 | } |
| 75 | } |
| 76 | |
| 77 | #undef MAXI |
| 78 | #undef EPSILON |
| 79 | |
| 80 | #endif |
| 81 | |
| 82 | /*========================================================================*/ |
| 83 | #define MFACT 30 |
| 84 | |
| 85 | /* The natural logarithm of factorial n! for 0 <= n <= MFACT */ |
| 86 | static double LnFactorial[MFACT + 1] = { |
| 87 | 0., |
| 88 | 0., |
| 89 | 0.6931471805599453, |
| 90 | 1.791759469228055, |
| 91 | 3.178053830347946, |
| 92 | 4.787491742782046, |
| 93 | 6.579251212010101, |
| 94 | 8.525161361065415, |
| 95 | 10.60460290274525, |
| 96 | 12.80182748008147, |
| 97 | 15.10441257307552, |
| 98 | 17.50230784587389, |
| 99 | 19.98721449566188, |
| 100 | 22.55216385312342, |
| 101 | 25.19122118273868, |
| 102 | 27.89927138384088, |
| 103 | 30.67186010608066, |
| 104 | 33.50507345013688, |
| 105 | 36.39544520803305, |
| 106 | 39.33988418719949, |
| 107 | 42.33561646075348, |
| 108 | 45.3801388984769, |
| 109 | 48.47118135183522, |
| 110 | 51.60667556776437, |
| 111 | 54.7847293981123, |
| 112 | 58.00360522298051, |
| 113 | 61.26170176100199, |
| 114 | 64.55753862700632, |
| 115 | 67.88974313718154, |
| 116 | 71.257038967168, |
| 117 | 74.65823634883016 |
| 118 | }; |
| 119 | |
| 120 | /*------------------------------------------------------------------------*/ |
| 121 | |
| 122 | static double getLogFactorial (int n) |
| 123 | { |
| 124 | /* Returns the natural logarithm of factorial n! */ |
| 125 | if (n <= MFACT) { |
| 126 | return LnFactorial[n]; |
| 127 | |
| 128 | } else { |
| 129 | double x = (double) (n + 1); |
| 130 | double y = 1.0 / (x * x); |
| 131 | double z = ((-(5.95238095238E-4 * y) + 7.936500793651E-4) * y - |
| 132 | 2.7777777777778E-3) * y + 8.3333333333333E-2; |
| 133 | z = ((x - 0.5) * log (x) - x) + 9.1893853320467E-1 + z / x; |
| 134 | return z; |
| 135 | } |
| 136 | } |
| 137 | |
| 138 | /*------------------------------------------------------------------------*/ |
| 139 | |
| 140 | static double rapfac (int n) |
| 141 | { |
| 142 | /* Computes n! / n^n */ |
| 143 | int i; |
| 144 | double res = 1.0 / n; |
| 145 | for (i = 2; i <= n; i++) { |
| 146 | res *= (double) i / n; |
| 147 | } |
| 148 | return res; |
| 149 | } |
| 150 | |
| 151 | |
| 152 | /*========================================================================*/ |
| 153 | |
| 154 | static double **CreateMatrixD (int N, int M) |
| 155 | { |
| 156 | int i; |
| 157 | double **T2; |
| 158 | |
| 159 | T2 = (double **) malloc (N * sizeof (double *)); |
| 160 | T2[0] = (double *) malloc ((size_t) N * M * sizeof (double)); |
| 161 | for (i = 1; i < N; i++) |
| 162 | T2[i] = T2[0] + i * M; |
| 163 | return T2; |
| 164 | } |
| 165 | |
| 166 | |
| 167 | static void DeleteMatrixD (double **T) |
| 168 | { |
| 169 | free (T[0]); |
| 170 | free (T); |
| 171 | } |
| 172 | |
| 173 | |
| 174 | /*========================================================================*/ |
| 175 | |
| 176 | static double KSPlusbarAsymp (int n, double x) |
| 177 | { |
| 178 | /* Compute the probability of the KS+ distribution using an asymptotic |
| 179 | formula */ |
| 180 | double t = (6.0 * n * x + 1); |
| 181 | double z = t * t / (18.0 * n); |
| 182 | double v = 1.0 - (2.0 * z * z - 4.0 * z - 1.0) / (18.0 * n); |
| 183 | if (v <= 0.0) |
| 184 | return 0.0; |
| 185 | v = v * exp (-z); |
| 186 | if (v >= 1.0) |
| 187 | return 1.0; |
| 188 | return v; |
| 189 | } |
| 190 | |
| 191 | |
| 192 | /*-------------------------------------------------------------------------*/ |
| 193 | |
| 194 | static double KSPlusbarUpper (int n, double x) |
| 195 | { |
| 196 | /* Compute the probability of the KS+ distribution in the upper tail using |
| 197 | Smirnov's stable formula */ |
| 198 | const double EPSILON = 1.0E-12; |
| 199 | double q; |
| 200 | double Sum = 0.0; |
| 201 | double term; |
| 202 | double t; |
| 203 | double LogCom; |
| 204 | double LOGJMAX; |
| 205 | int j; |
| 206 | int jdiv; |
| 207 | int jmax = (int) (n * (1.0 - x)); |
| 208 | |
| 209 | if (n > 200000) |
| 210 | return KSPlusbarAsymp (n, x); |
| 211 | |
| 212 | /* Avoid log(0) for j = jmax and q ~ 1.0 */ |
| 213 | if ((1.0 - x - (double) jmax / n) <= 0.0) |
| 214 | jmax--; |
| 215 | |
| 216 | if (n > 3000) |
| 217 | jdiv = 2; |
| 218 | else |
| 219 | jdiv = 3; |
| 220 | |
| 221 | j = jmax / jdiv + 1; |
| 222 | LogCom = getLogFactorial (n) - getLogFactorial (j) - |
| 223 | getLogFactorial (n - j); |
| 224 | LOGJMAX = LogCom; |
| 225 | |
| 226 | while (j <= jmax) { |
| 227 | q = (double) j / n + x; |
| 228 | term = LogCom + (j - 1) * log (q) + (n - j) * log1p (-q); |
| 229 | t = exp (term); |
| 230 | Sum += t; |
| 231 | LogCom += log ((double) (n - j) / (j + 1)); |
| 232 | if (t <= Sum * EPSILON) |
| 233 | break; |
| 234 | j++; |
| 235 | } |
| 236 | |
| 237 | j = jmax / jdiv; |
| 238 | LogCom = LOGJMAX + log ((double) (j + 1) / (n - j)); |
| 239 | |
| 240 | while (j > 0) { |
| 241 | q = (double) j / n + x; |
| 242 | term = LogCom + (j - 1) * log (q) + (n - j) * log1p (-q); |
| 243 | t = exp (term); |
| 244 | Sum += t; |
| 245 | LogCom += log ((double) j / (n - j + 1)); |
| 246 | if (t <= Sum * EPSILON) |
| 247 | break; |
| 248 | j--; |
| 249 | } |
| 250 | |
| 251 | Sum *= x; |
| 252 | /* add the term j = 0 */ |
| 253 | Sum += exp (n * log1p (-x)); |
| 254 | return Sum; |
| 255 | } |
| 256 | |
| 257 | |
| 258 | /*========================================================================*/ |
| 259 | |
| 260 | static double Pelz (int n, double x) |
| 261 | { |
| 262 | /* Approximating the Lower Tail-Areas of the Kolmogorov-Smirnov One-Sample |
| 263 | Statistic, |
| 264 | Wolfgang Pelz and I. J. Good, |
| 265 | Journal of the Royal Statistical Society, Series B. |
| 266 | Vol. 38, No. 2 (1976), pp. 152-156 |
| 267 | */ |
| 268 | |
| 269 | const int JMAX = 20; |
| 270 | const double EPS = 1.0e-10; |
| 271 | const double C = 2.506628274631001; /* sqrt(2*Pi) */ |
| 272 | const double C2 = 1.2533141373155001; /* sqrt(Pi/2) */ |
| 273 | const double PI2 = num_Pi * num_Pi; |
| 274 | const double PI4 = PI2 * PI2; |
| 275 | const double RACN = sqrt ((double) n); |
| 276 | const double z = RACN * x; |
| 277 | const double z2 = z * z; |
| 278 | const double z4 = z2 * z2; |
| 279 | const double z6 = z4 * z2; |
| 280 | const double w = PI2 / (2.0 * z * z); |
| 281 | double ti, term, tom; |
| 282 | double sum; |
| 283 | int j; |
| 284 | |
| 285 | term = 1; |
| 286 | j = 0; |
| 287 | sum = 0; |
| 288 | while (j <= JMAX && term > EPS * sum) { |
| 289 | ti = j + 0.5; |
| 290 | term = exp (-ti * ti * w); |
| 291 | sum += term; |
| 292 | j++; |
| 293 | } |
| 294 | sum *= C / z; |
| 295 | |
| 296 | term = 1; |
| 297 | tom = 0; |
| 298 | j = 0; |
| 299 | while (j <= JMAX && fabs (term) > EPS * fabs (tom)) { |
| 300 | ti = j + 0.5; |
| 301 | term = (PI2 * ti * ti - z2) * exp (-ti * ti * w); |
| 302 | tom += term; |
| 303 | j++; |
| 304 | } |
| 305 | sum += tom * C2 / (RACN * 3.0 * z4); |
| 306 | |
| 307 | term = 1; |
| 308 | tom = 0; |
| 309 | j = 0; |
| 310 | while (j <= JMAX && fabs (term) > EPS * fabs (tom)) { |
| 311 | ti = j + 0.5; |
| 312 | term = 6 * z6 + 2 * z4 + PI2 * (2 * z4 - 5 * z2) * ti * ti + |
| 313 | PI4 * (1 - 2 * z2) * ti * ti * ti * ti; |
| 314 | term *= exp (-ti * ti * w); |
| 315 | tom += term; |
| 316 | j++; |
| 317 | } |
| 318 | sum += tom * C2 / (n * 36.0 * z * z6); |
| 319 | |
| 320 | term = 1; |
| 321 | tom = 0; |
| 322 | j = 1; |
| 323 | while (j <= JMAX && term > EPS * tom) { |
| 324 | ti = j; |
| 325 | term = PI2 * ti * ti * exp (-ti * ti * w); |
| 326 | tom += term; |
| 327 | j++; |
| 328 | } |
| 329 | sum -= tom * C2 / (n * 18.0 * z * z2); |
| 330 | |
| 331 | term = 1; |
| 332 | tom = 0; |
| 333 | j = 0; |
| 334 | while (j <= JMAX && fabs (term) > EPS * fabs (tom)) { |
| 335 | ti = j + 0.5; |
| 336 | ti = ti * ti; |
| 337 | term = -30 * z6 - 90 * z6 * z2 + PI2 * (135 * z4 - 96 * z6) * ti + |
| 338 | PI4 * (212 * z4 - 60 * z2) * ti * ti + PI2 * PI4 * ti * ti * ti * (5 - |
| 339 | 30 * z2); |
| 340 | term *= exp (-ti * w); |
| 341 | tom += term; |
| 342 | j++; |
| 343 | } |
| 344 | sum += tom * C2 / (RACN * n * 3240.0 * z4 * z6); |
| 345 | |
| 346 | term = 1; |
| 347 | tom = 0; |
| 348 | j = 1; |
| 349 | while (j <= JMAX && fabs (term) > EPS * fabs (tom)) { |
| 350 | ti = j * j; |
| 351 | term = (3 * PI2 * ti * z2 - PI4 * ti * ti) * exp (-ti * w); |
| 352 | tom += term; |
| 353 | j++; |
| 354 | } |
| 355 | sum += tom * C2 / (RACN * n * 108.0 * z6); |
| 356 | |
| 357 | return sum; |
| 358 | } |
| 359 | |
| 360 | |
| 361 | /*=========================================================================*/ |
| 362 | |
| 363 | static void CalcFloorCeil ( |
| 364 | int n, /* sample size */ |
| 365 | double t, /* = nx */ |
| 366 | double *A, /* A_i */ |
| 367 | double *Atflo, /* floor (A_i - t) */ |
| 368 | double *Atcei /* ceiling (A_i + t) */ |
| 369 | ) |
| 370 | { |
| 371 | /* Precompute A_i, floors, and ceilings for limits of sums in the Pomeranz |
| 372 | algorithm */ |
| 373 | int i; |
| 374 | int ell = (int) t; /* floor (t) */ |
| 375 | double z = t - ell; /* t - floor (t) */ |
| 376 | double w = ceil (t) - t; |
| 377 | |
| 378 | if (z > 0.5) { |
| 379 | for (i = 2; i <= 2 * n + 2; i += 2) |
| 380 | Atflo[i] = i / 2 - 2 - ell; |
| 381 | for (i = 1; i <= 2 * n + 2; i += 2) |
| 382 | Atflo[i] = i / 2 - 1 - ell; |
| 383 | |
| 384 | for (i = 2; i <= 2 * n + 2; i += 2) |
| 385 | Atcei[i] = i / 2 + ell; |
| 386 | for (i = 1; i <= 2 * n + 2; i += 2) |
| 387 | Atcei[i] = i / 2 + 1 + ell; |
| 388 | |
| 389 | } else if (z > 0.0) { |
| 390 | for (i = 1; i <= 2 * n + 2; i++) |
| 391 | Atflo[i] = i / 2 - 1 - ell; |
| 392 | |
| 393 | for (i = 2; i <= 2 * n + 2; i++) |
| 394 | Atcei[i] = i / 2 + ell; |
| 395 | Atcei[1] = 1 + ell; |
| 396 | |
| 397 | } else { /* z == 0 */ |
| 398 | for (i = 2; i <= 2 * n + 2; i += 2) |
| 399 | Atflo[i] = i / 2 - 1 - ell; |
| 400 | for (i = 1; i <= 2 * n + 2; i += 2) |
| 401 | Atflo[i] = i / 2 - ell; |
| 402 | |
| 403 | for (i = 2; i <= 2 * n + 2; i += 2) |
| 404 | Atcei[i] = i / 2 - 1 + ell; |
| 405 | for (i = 1; i <= 2 * n + 2; i += 2) |
| 406 | Atcei[i] = i / 2 + ell; |
| 407 | } |
| 408 | |
| 409 | if (w < z) |
| 410 | z = w; |
| 411 | A[0] = A[1] = 0; |
| 412 | A[2] = z; |
| 413 | A[3] = 1 - A[2]; |
| 414 | for (i = 4; i <= 2 * n + 1; i++) |
| 415 | A[i] = A[i - 2] + 1; |
| 416 | A[2 * n + 2] = n; |
| 417 | } |
| 418 | |
| 419 | |
| 420 | /*========================================================================*/ |
| 421 | |
| 422 | static double Pomeranz (int n, double x) |
| 423 | { |
| 424 | /* The Pomeranz algorithm to compute the KS distribution */ |
| 425 | const double EPS = 1.0e-15; |
| 426 | const int ENO = 350; |
| 427 | const double RENO = ldexp (1.0, ENO); /* for renormalization of V */ |
| 428 | int coreno; /* counter: how many renormalizations */ |
| 429 | const double t = n * x; |
| 430 | double w, sum, minsum; |
| 431 | int i, j, k, s; |
| 432 | int r1, r2; /* Indices i and i-1 for V[i][] */ |
| 433 | int jlow, jup, klow, kup, kup0; |
| 434 | double *A; |
| 435 | double *Atflo; |
| 436 | double *Atcei; |
| 437 | double **V; |
| 438 | double **H; /* = pow(w, j) / Factorial(j) */ |
| 439 | |
| 440 | A = (double *) calloc ((size_t) (2 * n + 3), sizeof (double)); |
| 441 | Atflo = (double *) calloc ((size_t) (2 * n + 3), sizeof (double)); |
| 442 | Atcei = (double *) calloc ((size_t) (2 * n + 3), sizeof (double)); |
| 443 | V = (double **) CreateMatrixD (2, n + 2); |
| 444 | H = (double **) CreateMatrixD (4, n + 2); |
| 445 | |
| 446 | CalcFloorCeil (n, t, A, Atflo, Atcei); |
| 447 | |
| 448 | for (j = 1; j <= n + 1; j++) |
| 449 | V[0][j] = 0; |
| 450 | for (j = 2; j <= n + 1; j++) |
| 451 | V[1][j] = 0; |
| 452 | V[1][1] = RENO; |
| 453 | coreno = 1; |
| 454 | |
| 455 | /* Precompute H[][] = (A[j] - A[j-1]^k / k! for speed */ |
| 456 | H[0][0] = 1; |
| 457 | w = 2.0 * A[2] / n; |
| 458 | for (j = 1; j <= n + 1; j++) |
| 459 | H[0][j] = w * H[0][j - 1] / j; |
| 460 | |
| 461 | H[1][0] = 1; |
| 462 | w = (1.0 - 2.0 * A[2]) / n; |
| 463 | for (j = 1; j <= n + 1; j++) |
| 464 | H[1][j] = w * H[1][j - 1] / j; |
| 465 | |
| 466 | H[2][0] = 1; |
| 467 | w = A[2] / n; |
| 468 | for (j = 1; j <= n + 1; j++) |
| 469 | H[2][j] = w * H[2][j - 1] / j; |
| 470 | |
| 471 | H[3][0] = 1; |
| 472 | for (j = 1; j <= n + 1; j++) |
| 473 | H[3][j] = 0; |
| 474 | |
| 475 | r1 = 0; |
| 476 | r2 = 1; |
| 477 | for (i = 2; i <= 2 * n + 2; i++) { |
| 478 | jlow = 2 + (int) Atflo[i]; |
| 479 | if (jlow < 1) |
| 480 | jlow = 1; |
| 481 | jup = (int) Atcei[i]; |
| 482 | if (jup > n + 1) |
| 483 | jup = n + 1; |
| 484 | |
| 485 | klow = 2 + (int) Atflo[i - 1]; |
| 486 | if (klow < 1) |
| 487 | klow = 1; |
| 488 | kup0 = (int) Atcei[i - 1]; |
| 489 | |
| 490 | /* Find to which case it corresponds */ |
| 491 | w = (A[i] - A[i - 1]) / n; |
| 492 | s = -1; |
| 493 | for (j = 0; j < 4; j++) { |
| 494 | if (fabs (w - H[j][1]) <= EPS) { |
| 495 | s = j; |
| 496 | break; |
| 497 | } |
| 498 | } |
| 499 | /* assert (s >= 0, "Pomeranz: s < 0"); */ |
| 500 | |
| 501 | minsum = RENO; |
| 502 | r1 = (r1 + 1) & 1; /* i - 1 */ |
| 503 | r2 = (r2 + 1) & 1; /* i */ |
| 504 | |
| 505 | for (j = jlow; j <= jup; j++) { |
| 506 | kup = kup0; |
| 507 | if (kup > j) |
| 508 | kup = j; |
| 509 | sum = 0; |
| 510 | for (k = kup; k >= klow; k--) |
| 511 | sum += V[r1][k] * H[s][j - k]; |
| 512 | V[r2][j] = sum; |
| 513 | if (sum < minsum) |
| 514 | minsum = sum; |
| 515 | } |
| 516 | |
| 517 | if (minsum < 1.0e-280) { |
| 518 | /* V is too small: renormalize to avoid underflow of probabilities */ |
| 519 | for (j = jlow; j <= jup; j++) |
| 520 | V[r2][j] *= RENO; |
| 521 | coreno++; /* keep track of log of RENO */ |
| 522 | } |
| 523 | } |
| 524 | |
| 525 | sum = V[r2][n + 1]; |
| 526 | free (A); |
| 527 | free (Atflo); |
| 528 | free (Atcei); |
| 529 | DeleteMatrixD (H); |
| 530 | DeleteMatrixD (V); |
| 531 | w = getLogFactorial (n) - coreno * ENO * num_Ln2 + log (sum); |
| 532 | if (w >= 0.) |
| 533 | return 1.; |
| 534 | return exp (w); |
| 535 | } |
| 536 | |
| 537 | |
| 538 | /*========================================================================*/ |
| 539 | |
| 540 | static double cdfSpecial (int n, double x) |
| 541 | { |
| 542 | /* The KS distribution is known exactly for these cases */ |
| 543 | |
| 544 | /* For nx^2 > 18, KSfbar(n, x) is smaller than 5e-16 */ |
| 545 | if ((n * x * x >= 18.0) || (x >= 1.0)) |
| 546 | return 1.0; |
| 547 | |
| 548 | if (x <= 0.5 / n) |
| 549 | return 0.0; |
| 550 | |
| 551 | if (n == 1) |
| 552 | return 2.0 * x - 1.0; |
| 553 | |
| 554 | if (x <= 1.0 / n) { |
| 555 | double t = 2.0 * x * n - 1.0; |
| 556 | double w; |
| 557 | if (n <= NEXACT) { |
| 558 | w = rapfac (n); |
| 559 | return w * pow (t, (double) n); |
| 560 | } |
| 561 | w = getLogFactorial (n) + n * log (t / n); |
| 562 | return exp (w); |
| 563 | } |
| 564 | |
| 565 | if (x >= 1.0 - 1.0 / n) { |
| 566 | return 1.0 - 2.0 * pow (1.0 - x, (double) n); |
| 567 | } |
| 568 | |
| 569 | return -1.0; |
| 570 | } |
| 571 | |
| 572 | |
| 573 | /*========================================================================*/ |
| 574 | |
| 575 | double KScdf (int n, double x) |
| 576 | { |
| 577 | const double w = n * x * x; |
| 578 | double u = cdfSpecial (n, x); |
| 579 | if (u >= 0.0) |
| 580 | return u; |
| 581 | |
| 582 | if (n <= NEXACT) { |
| 583 | if (w < 0.754693) |
| 584 | return DurbinMatrix (n, x); |
| 585 | if (w < 4.0) |
| 586 | return Pomeranz (n, x); |
| 587 | return 1.0 - KSfbar (n, x); |
| 588 | } |
| 589 | |
| 590 | if ((w * x * n <= 7.0) && (n <= NKOLMO)) |
| 591 | return DurbinMatrix (n, x); |
| 592 | |
| 593 | return Pelz (n, x); |
| 594 | } |
| 595 | |
| 596 | |
| 597 | /*=========================================================================*/ |
| 598 | |
| 599 | static double fbarSpecial (int n, double x) |
| 600 | { |
| 601 | const double w = n * x * x; |
| 602 | |
| 603 | if ((w >= 370.0) || (x >= 1.0)) |
| 604 | return 0.0; |
| 605 | if ((w <= 0.0274) || (x <= 0.5 / n)) |
| 606 | return 1.0; |
| 607 | if (n == 1) |
| 608 | return 2.0 - 2.0 * x; |
| 609 | |
| 610 | if (x <= 1.0 / n) { |
| 611 | double z; |
| 612 | double t = 2.0 * x * n - 1.0; |
| 613 | if (n <= NEXACT) { |
| 614 | z = rapfac (n); |
| 615 | return 1.0 - z * pow (t, (double) n); |
| 616 | } |
| 617 | z = getLogFactorial (n) + n * log (t / n); |
| 618 | return 1.0 - exp (z); |
| 619 | } |
| 620 | |
| 621 | if (x >= 1.0 - 1.0 / n) { |
| 622 | return 2.0 * pow (1.0 - x, (double) n); |
| 623 | } |
| 624 | return -1.0; |
| 625 | } |
| 626 | |
| 627 | |
| 628 | /*========================================================================*/ |
| 629 | |
| 630 | double KSfbar (int n, double x) |
| 631 | { |
| 632 | const double w = n * x * x; |
| 633 | double v = fbarSpecial (n, x); |
| 634 | if (v >= 0.0) |
| 635 | return v; |
| 636 | |
| 637 | if (n <= NEXACT) { |
| 638 | if (w < 4.0) |
| 639 | return 1.0 - KScdf (n, x); |
| 640 | else |
| 641 | return 2.0 * KSPlusbarUpper (n, x); |
| 642 | } |
| 643 | |
| 644 | if (w >= 2.65) |
| 645 | return 2.0 * KSPlusbarUpper (n, x); |
| 646 | |
| 647 | return 1.0 - KScdf (n, x); |
| 648 | } |
| 649 | |
| 650 | |
| 651 | /*========================================================================= |
| 652 | |
| 653 | The following implements the Durbin matrix algorithm and was programmed by |
| 654 | G. Marsaglia, Wai Wan Tsang and Jingbo Wong. |
| 655 | |
| 656 | I have made small modifications in their program. (Richard Simard) |
| 657 | |
| 658 | |
| 659 | |
| 660 | =========================================================================*/ |
| 661 | |
| 662 | /* |
| 663 | The C program to compute Kolmogorov's distribution |
| 664 | |
| 665 | K(n,d) = Prob(D_n < d), where |
| 666 | |
| 667 | 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) |
| 668 | |
| 669 | with x_1<x_2,...<x_n a purported set of n independent uniform [0,1) |
| 670 | random variables sorted into increasing order. |
| 671 | See G. Marsaglia, Wai Wan Tsang and Jingbo Wong, |
| 672 | J.Stat.Software, 8, 18, pp 1--4, (2003). |
| 673 | */ |
| 674 | |
| 675 | #define NORM 1.0e140 |
| 676 | #define INORM 1.0e-140 |
| 677 | #define LOGNORM 140 |
| 678 | |
| 679 | |
| 680 | /* Matrix product */ |
| 681 | static void mMultiply (double *A, double *B, double *C, int m); |
| 682 | |
| 683 | /* Matrix power */ |
| 684 | static void mPower (double *A, int eA, double *V, int *eV, int m, int n); |
| 685 | |
| 686 | |
| 687 | static double DurbinMatrix (int n, double d) |
| 688 | { |
| 689 | int k, m, i, j, g, eH, eQ; |
| 690 | double h, s, *H, *Q; |
| 691 | /* OMIT NEXT TWO LINES IF YOU REQUIRE >7 DIGIT ACCURACY IN THE RIGHT TAIL */ |
| 692 | #if 0 |
| 693 | s = d * d * n; |
| 694 | if (s > 7.24 || (s > 3.76 && n > 99)) |
| 695 | return 1 - 2 * exp (-(2.000071 + .331 / sqrt (n) + 1.409 / n) * s); |
| 696 | #endif |
| 697 | k = (int) (n * d) + 1; |
| 698 | m = 2 * k - 1; |
| 699 | h = k - n * d; |
| 700 | H = (double *) calloc ((m * m), sizeof (double)); |
| 701 | Q = (double *) calloc ((m * m), sizeof (double)); |
| 702 | for (i = 0; i < m; i++) |
| 703 | for (j = 0; j < m; j++) |
| 704 | if (i - j + 1 < 0) |
| 705 | H[i * m + j] = 0; |
| 706 | else |
| 707 | H[i * m + j] = 1; |
| 708 | for (i = 0; i < m; i++) { |
| 709 | H[i * m] -= pow (h, (double) (i + 1)); |
| 710 | H[(m - 1) * m + i] -= pow (h, (double) (m - i)); |
| 711 | } |
| 712 | H[(m - 1) * m] += (2 * h - 1 > 0 ? pow (2 * h - 1, (double) m) : 0); |
| 713 | for (i = 0; i < m; i++) |
| 714 | for (j = 0; j < m; j++) |
| 715 | if (i - j + 1 > 0) |
| 716 | for (g = 1; g <= i - j + 1; g++) |
| 717 | H[i * m + j] /= g; |
| 718 | eH = 0; |
| 719 | mPower (H, eH, Q, &eQ, m, n); |
| 720 | s = Q[(k - 1) * m + k - 1]; |
| 721 | |
| 722 | for (i = 1; i <= n; i++) { |
| 723 | s = s * (double) i / n; |
| 724 | if (s < INORM) { |
| 725 | s *= NORM; |
| 726 | eQ -= LOGNORM; |
| 727 | } |
| 728 | } |
| 729 | s *= pow (10., (double) eQ); |
| 730 | free (H); |
| 731 | free (Q); |
| 732 | return s; |
| 733 | } |
| 734 | |
| 735 | |
| 736 | static void mMultiply (double *A, double *B, double *C, int m) |
| 737 | { |
| 738 | int i, j, k; |
| 739 | double s; |
| 740 | for (i = 0; i < m; i++) |
| 741 | for (j = 0; j < m; j++) { |
| 742 | s = 0.; |
| 743 | for (k = 0; k < m; k++) |
| 744 | s += A[i * m + k] * B[k * m + j]; |
| 745 | C[i * m + j] = s; |
| 746 | } |
| 747 | } |
| 748 | |
| 749 | |
| 750 | static void renormalize (double *V, int m, int *p) |
| 751 | { |
| 752 | int i; |
| 753 | for (i = 0; i < m * m; i++) |
| 754 | V[i] *= INORM; |
| 755 | *p += LOGNORM; |
| 756 | } |
| 757 | |
| 758 | |
| 759 | static void mPower (double *A, int eA, double *V, int *eV, int m, int n) |
| 760 | { |
| 761 | double *B; |
| 762 | int eB, i; |
| 763 | if (n == 1) { |
| 764 | for (i = 0; i < m * m; i++) |
| 765 | V[i] = A[i]; |
| 766 | *eV = eA; |
| 767 | return; |
| 768 | } |
| 769 | mPower (A, eA, V, eV, m, n / 2); |
| 770 | B = (double *) malloc ((m * m) * sizeof (double)); |
| 771 | mMultiply (V, V, B, m); |
| 772 | eB = 2 * (*eV); |
| 773 | if (B[(m / 2) * m + (m / 2)] > NORM) |
| 774 | renormalize (B, m, &eB); |
| 775 | |
| 776 | if (n % 2 == 0) { |
| 777 | for (i = 0; i < m * m; i++) |
| 778 | V[i] = B[i]; |
| 779 | *eV = eB; |
| 780 | } else { |
| 781 | mMultiply (A, B, V, m); |
| 782 | *eV = eA + eB; |
| 783 | } |
| 784 | |
| 785 | if (V[(m / 2) * m + (m / 2)] > NORM) |
| 786 | renormalize (V, m, eV); |
| 787 | free (B); |
| 788 | } |