Mercurial > hg > octave-nkf
annotate liboctave/randpoisson.c @ 7948:af10baa63915 ss-3-1-50
3.1.50 snapshot
| author | John W. Eaton <jwe@octave.org> |
|---|---|
| date | Fri, 18 Jul 2008 17:42:48 -0400 |
| parents | 8a939b217863 |
| children | eb63fbe60fab |
| rev | line source |
|---|---|
| 7019 | 1 /* |
| 2 | |
| 3 Copyright (C) 2006, 2007 John W. Eaton | |
| 4 | |
| 5 This file is part of Octave. | |
| 6 | |
| 7 Octave is free software; you can redistribute it and/or modify it | |
| 8 under the terms of the GNU General Public License as published by the | |
| 9 Free Software Foundation; either version 3 of the License, or (at your | |
| 10 option) any later version. | |
| 11 | |
| 12 Octave is distributed in the hope that it will be useful, but WITHOUT | |
| 13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
| 14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
| 15 for more details. | |
| 16 | |
| 17 You should have received a copy of the GNU General Public License | |
| 18 along with Octave; see the file COPYING. If not, see | |
| 19 <http://www.gnu.org/licenses/>. | |
| 20 | |
| 21 */ | |
| 22 | |
| 23 /* Original version written by Paul Kienzle distributed as free | |
| 24 software in the in the public domain. */ | |
| 5742 | 25 |
| 26 /* Needs the following defines: | |
| 27 * NAN: value to return for Not-A-Number | |
| 28 * RUNI: uniform generator on (0,1) | |
| 29 * RNOR: normal generator | |
| 30 * LGAMMA: log gamma function | |
| 31 * INFINITE: function to test whether a value is infinite | |
| 32 */ | |
| 33 | |
| 34 #if defined (HAVE_CONFIG_H) | |
| 35 #include <config.h> | |
| 36 #endif | |
| 37 | |
| 38 #include <stdio.h> | |
| 39 | |
| 40 #include "f77-fcn.h" | |
| 7231 | 41 #include "lo-error.h" |
| 5742 | 42 #include "lo-ieee.h" |
| 7231 | 43 #include "lo-math.h" |
| 5742 | 44 #include "randmtzig.h" |
| 45 #include "randpoisson.h" | |
| 46 | |
| 47 #undef NAN | |
| 48 #define NAN octave_NaN | |
| 6096 | 49 #undef INFINITE |
| 5742 | 50 #define INFINITE lo_ieee_isinf |
| 51 #define RUNI oct_randu() | |
| 52 #define RNOR oct_randn() | |
| 53 #define LGAMMA xlgamma | |
| 54 | |
| 55 F77_RET_T | |
| 56 F77_FUNC (dlgams, DLGAMS) (const double *, double *, double *); | |
| 57 | |
| 58 static double | |
| 59 xlgamma (double x) | |
| 60 { | |
| 61 double result; | |
|
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62 #ifdef HAVE_LGAMMA |
|
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63 result = lgamma (x); |
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64 #else |
| 5742 | 65 double sgngam; |
| 66 | |
| 67 if (lo_ieee_isnan (x)) | |
| 68 result = x; | |
| 69 else if (x <= 0 || lo_ieee_isinf (x)) | |
| 70 result = octave_Inf; | |
| 71 else | |
| 72 F77_XFCN (dlgams, DLGAMS, (&x, &result, &sgngam)); | |
|
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73 #endif |
| 5742 | 74 return result; |
| 75 } | |
| 76 | |
| 77 /* ---- pprsc.c from Stadloeber's winrand --- */ | |
| 78 | |
| 79 /* flogfak(k) = ln(k!) */ | |
| 80 static double | |
| 81 flogfak (double k) | |
| 82 { | |
| 83 #define C0 9.18938533204672742e-01 | |
| 84 #define C1 8.33333333333333333e-02 | |
| 85 #define C3 -2.77777777777777778e-03 | |
| 86 #define C5 7.93650793650793651e-04 | |
| 87 #define C7 -5.95238095238095238e-04 | |
| 88 | |
| 89 static double logfak[30L] = { | |
| 90 0.00000000000000000, 0.00000000000000000, 0.69314718055994531, | |
| 91 1.79175946922805500, 3.17805383034794562, 4.78749174278204599, | |
| 92 6.57925121201010100, 8.52516136106541430, 10.60460290274525023, | |
| 93 12.80182748008146961, 15.10441257307551530, 17.50230784587388584, | |
| 94 19.98721449566188615, 22.55216385312342289, 25.19122118273868150, | |
| 95 27.89927138384089157, 30.67186010608067280, 33.50507345013688888, | |
| 96 36.39544520803305358, 39.33988418719949404, 42.33561646075348503, | |
| 97 45.38013889847690803, 48.47118135183522388, 51.60667556776437357, | |
| 98 54.78472939811231919, 58.00360522298051994, 61.26170176100200198, | |
| 99 64.55753862700633106, 67.88974313718153498, 71.25703896716800901 | |
| 100 }; | |
| 101 | |
| 102 double r, rr; | |
| 103 | |
| 104 if (k >= 30.0) | |
| 105 { | |
| 106 r = 1.0 / k; | |
| 107 rr = r * r; | |
| 108 return ((k + 0.5)*log(k) - k + C0 + r*(C1 + rr*(C3 + rr*(C5 + rr*C7)))); | |
| 109 } | |
| 110 else | |
| 111 return (logfak[(int)k]); | |
| 112 } | |
| 113 | |
| 114 | |
| 115 /****************************************************************** | |
| 116 * * | |
| 117 * Poisson Distribution - Patchwork Rejection/Inversion * | |
| 118 * * | |
| 119 ****************************************************************** | |
| 120 * * | |
| 121 * For parameter my < 10 Tabulated Inversion is applied. * | |
| 122 * For my >= 10 Patchwork Rejection is employed: * | |
| 123 * The area below the histogram function f(x) is rearranged in * | |
| 124 * its body by certain point reflections. Within a large center * | |
| 125 * interval variates are sampled efficiently by rejection from * | |
| 126 * uniform hats. Rectangular immediate acceptance regions speed * | |
| 127 * up the generation. The remaining tails are covered by * | |
| 128 * exponential functions. * | |
| 129 * * | |
| 130 ****************************************************************** | |
| 131 * * | |
| 132 * FUNCTION : - pprsc samples a random number from the Poisson * | |
| 133 * distribution with parameter my > 0. * | |
| 134 * REFERENCE : - H. Zechner (1994): Efficient sampling from * | |
| 135 * continuous and discrete unimodal distributions, * | |
| 136 * Doctoral Dissertation, 156 pp., Technical * | |
| 137 * University Graz, Austria. * | |
| 138 * SUBPROGRAM : - drand(seed) ... (0,1)-Uniform generator with * | |
| 139 * unsigned long integer *seed. * | |
| 140 * * | |
| 141 * Implemented by H. Zechner, January 1994 * | |
| 142 * Revised by F. Niederl, July 1994 * | |
| 143 * * | |
| 144 ******************************************************************/ | |
| 145 | |
| 146 static double | |
| 147 f (double k, double l_nu, double c_pm) | |
| 148 { | |
| 149 return exp(k * l_nu - flogfak(k) - c_pm); | |
| 150 } | |
| 151 | |
| 152 static double | |
| 153 pprsc (double my) | |
| 154 { | |
| 155 static double my_last = -1.0; | |
| 156 static double m, k2, k4, k1, k5; | |
| 157 static double dl, dr, r1, r2, r4, r5, ll, lr, l_my, c_pm, | |
| 158 f1, f2, f4, f5, p1, p2, p3, p4, p5, p6; | |
| 159 double Dk, X, Y; | |
| 160 double Ds, U, V, W; | |
| 161 | |
| 162 if (my != my_last) | |
| 163 { /* set-up */ | |
| 164 my_last = my; | |
| 165 /* approximate deviation of reflection points k2, k4 from my - 1/2 */ | |
| 166 Ds = sqrt(my + 0.25); | |
| 167 | |
| 168 /* mode m, reflection points k2 and k4, and points k1 and k5, */ | |
| 169 /* which delimit the centre region of h(x) */ | |
| 170 m = floor(my); | |
| 171 k2 = ceil(my - 0.5 - Ds); | |
| 172 k4 = floor(my - 0.5 + Ds); | |
| 173 k1 = k2 + k2 - m + 1L; | |
| 174 k5 = k4 + k4 - m; | |
| 175 | |
| 176 /* range width of the critical left and right centre region */ | |
| 177 dl = (k2 - k1); | |
| 178 dr = (k5 - k4); | |
| 179 | |
| 180 /* recurrence constants r(k)=p(k)/p(k-1) at k = k1, k2, k4+1, k5+1 */ | |
| 181 r1 = my / k1; | |
| 182 r2 = my / k2; | |
| 183 r4 = my / (k4 + 1.0); | |
| 184 r5 = my / (k5 + 1.0); | |
| 185 | |
| 186 /* reciprocal values of the scale parameters of exp. tail envelope */ | |
| 187 ll = log(r1); /* expon. tail left */ | |
| 188 lr = -log(r5); /* expon. tail right*/ | |
| 189 | |
| 190 /* Poisson constants, necessary for computing function values f(k) */ | |
| 191 l_my = log(my); | |
| 192 c_pm = m * l_my - flogfak(m); | |
| 193 | |
| 194 /* function values f(k) = p(k)/p(m) at k = k2, k4, k1, k5 */ | |
| 195 f2 = f(k2, l_my, c_pm); | |
| 196 f4 = f(k4, l_my, c_pm); | |
| 197 f1 = f(k1, l_my, c_pm); | |
| 198 f5 = f(k5, l_my, c_pm); | |
| 199 | |
| 200 /* area of the two centre and the two exponential tail regions */ | |
| 201 /* area of the two immediate acceptance regions between k2, k4 */ | |
| 202 p1 = f2 * (dl + 1.0); /* immed. left */ | |
| 203 p2 = f2 * dl + p1; /* centre left */ | |
| 204 p3 = f4 * (dr + 1.0) + p2; /* immed. right */ | |
| 205 p4 = f4 * dr + p3; /* centre right */ | |
| 206 p5 = f1 / ll + p4; /* exp. tail left */ | |
| 207 p6 = f5 / lr + p5; /* exp. tail right*/ | |
| 208 } | |
| 209 | |
| 210 for (;;) | |
| 211 { | |
| 212 /* generate uniform number U -- U(0, p6) */ | |
| 213 /* case distinction corresponding to U */ | |
| 214 if ((U = RUNI * p6) < p2) | |
| 215 { /* centre left */ | |
| 216 | |
| 217 /* immediate acceptance region | |
| 218 R2 = [k2, m) *[0, f2), X = k2, ... m -1 */ | |
| 219 if ((V = U - p1) < 0.0) return(k2 + floor(U/f2)); | |
| 220 /* immediate acceptance region | |
| 221 R1 = [k1, k2)*[0, f1), X = k1, ... k2-1 */ | |
| 222 if ((W = V / dl) < f1 ) return(k1 + floor(V/f1)); | |
| 223 | |
| 224 /* computation of candidate X < k2, and its counterpart Y > k2 */ | |
| 225 /* either squeeze-acceptance of X or acceptance-rejection of Y */ | |
| 226 Dk = floor(dl * RUNI) + 1.0; | |
| 227 if (W <= f2 - Dk * (f2 - f2/r2)) | |
| 228 { /* quick accept of */ | |
| 229 return(k2 - Dk); /* X = k2 - Dk */ | |
| 230 } | |
| 231 if ((V = f2 + f2 - W) < 1.0) | |
| 232 { /* quick reject of Y*/ | |
| 233 Y = k2 + Dk; | |
| 234 if (V <= f2 + Dk * (1.0 - f2)/(dl + 1.0)) | |
| 235 { /* quick accept of */ | |
| 236 return(Y); /* Y = k2 + Dk */ | |
| 237 } | |
| 238 if (V <= f(Y, l_my, c_pm)) return(Y); /* final accept of Y*/ | |
| 239 } | |
| 240 X = k2 - Dk; | |
| 241 } | |
| 242 else if (U < p4) | |
| 243 { /* centre right */ | |
| 244 /* immediate acceptance region | |
| 245 R3 = [m, k4+1)*[0, f4), X = m, ... k4 */ | |
| 246 if ((V = U - p3) < 0.0) return(k4 - floor((U - p2)/f4)); | |
| 247 /* immediate acceptance region | |
| 248 R4 = [k4+1, k5+1)*[0, f5) */ | |
| 249 if ((W = V / dr) < f5 ) return(k5 - floor(V/f5)); | |
| 250 | |
| 251 /* computation of candidate X > k4, and its counterpart Y < k4 */ | |
| 252 /* either squeeze-acceptance of X or acceptance-rejection of Y */ | |
| 253 Dk = floor(dr * RUNI) + 1.0; | |
| 254 if (W <= f4 - Dk * (f4 - f4*r4)) | |
| 255 { /* quick accept of */ | |
| 256 return(k4 + Dk); /* X = k4 + Dk */ | |
| 257 } | |
| 258 if ((V = f4 + f4 - W) < 1.0) | |
| 259 { /* quick reject of Y*/ | |
| 260 Y = k4 - Dk; | |
| 261 if (V <= f4 + Dk * (1.0 - f4)/ dr) | |
| 262 { /* quick accept of */ | |
| 263 return(Y); /* Y = k4 - Dk */ | |
| 264 } | |
| 265 if (V <= f(Y, l_my, c_pm)) return(Y); /* final accept of Y*/ | |
| 266 } | |
| 267 X = k4 + Dk; | |
| 268 } | |
| 269 else | |
| 270 { | |
| 271 W = RUNI; | |
| 272 if (U < p5) | |
| 273 { /* expon. tail left */ | |
| 274 Dk = floor(1.0 - log(W)/ll); | |
| 275 if ((X = k1 - Dk) < 0L) continue; /* 0 <= X <= k1 - 1 */ | |
| 276 W *= (U - p4) * ll; /* W -- U(0, h(x)) */ | |
| 277 if (W <= f1 - Dk * (f1 - f1/r1)) | |
| 278 return(X); /* quick accept of X*/ | |
| 279 } | |
| 280 else | |
| 281 { /* expon. tail right*/ | |
| 282 Dk = floor(1.0 - log(W)/lr); | |
| 283 X = k5 + Dk; /* X >= k5 + 1 */ | |
| 284 W *= (U - p5) * lr; /* W -- U(0, h(x)) */ | |
| 285 if (W <= f5 - Dk * (f5 - f5*r5)) | |
| 286 return(X); /* quick accept of X*/ | |
| 287 } | |
| 288 } | |
| 289 | |
| 290 /* acceptance-rejection test of candidate X from the original area */ | |
| 291 /* test, whether W <= f(k), with W = U*h(x) and U -- U(0, 1)*/ | |
| 292 /* log f(X) = (X - m)*log(my) - log X! + log m! */ | |
| 293 if (log(W) <= X * l_my - flogfak(X) - c_pm) return(X); | |
| 294 } | |
| 295 } | |
| 296 /* ---- pprsc.c end ------ */ | |
| 297 | |
| 298 | |
| 299 /* The remainder of the file is by Paul Kienzle */ | |
| 300 | |
| 301 /* Given uniform u, find x such that CDF(L,x)==u. Return x. */ | |
| 302 static void | |
| 303 poisson_cdf_lookup(double lambda, double *p, size_t n) | |
| 304 { | |
| 305 /* Table size is predicated on the maximum value of lambda | |
| 306 * we want to store in the table, and the maximum value of | |
| 307 * returned by the uniform random number generator on [0,1). | |
| 308 * With lambda==10 and u_max = 1 - 1/(2^32+1), we | |
| 309 * have poisson_pdf(lambda,36) < 1-u_max. If instead our | |
| 310 * generator uses more bits of mantissa or returns a value | |
| 311 * in the range [0,1], then for lambda==10 we need a table | |
| 312 * size of 46 instead. For long doubles, the table size | |
| 313 * will need to be longer still. */ | |
| 314 #define TABLESIZE 46 | |
| 315 double t[TABLESIZE]; | |
| 316 | |
| 317 /* Precompute the table for the u up to and including 0.458. | |
| 318 * We will almost certainly need it. */ | |
| 319 int intlambda = (int)floor(lambda); | |
| 320 double P; | |
| 321 int tableidx; | |
| 322 size_t i = n; | |
| 323 | |
| 324 t[0] = P = exp(-lambda); | |
| 325 for (tableidx = 1; tableidx <= intlambda; tableidx++) { | |
| 326 P = P*lambda/(double)tableidx; | |
| 327 t[tableidx] = t[tableidx-1] + P; | |
| 328 } | |
| 329 | |
| 330 while (i-- > 0) { | |
| 331 double u = RUNI; | |
| 332 | |
| 333 /* If u > 0.458 we know we can jump to floor(lambda) before | |
| 334 * comparing (this observation is based on Stadlober's winrand | |
| 335 * code). For lambda >= 1, this will be a win. Lambda < 1 | |
| 336 * is already fast, so adding an extra comparison is not a | |
| 337 * problem. */ | |
| 338 int k = (u > 0.458 ? intlambda : 0); | |
| 339 | |
| 340 /* We aren't using a for loop here because when we find the | |
| 341 * right k we want to jump to the next iteration of the | |
| 342 * outer loop, and the continue statement will only work for | |
| 343 * the inner loop. */ | |
| 344 nextk: | |
| 345 if ( u <= t[k] ) { | |
| 346 p[i] = (double) k; | |
| 347 continue; | |
| 348 } | |
| 349 if (++k < tableidx) goto nextk; | |
| 350 | |
| 351 /* We only need high values of the table very rarely so we | |
| 352 * don't automatically compute the entire table. */ | |
| 353 while (tableidx < TABLESIZE) { | |
| 354 P = P*lambda/(double)tableidx; | |
| 355 t[tableidx] = t[tableidx-1] + P; | |
| 356 /* Make sure we converge to 1.0 just in case u is uniform | |
| 357 * on [0,1] rather than [0,1). */ | |
| 358 if (t[tableidx] == t[tableidx-1]) t[tableidx] = 1.0; | |
| 359 tableidx++; | |
| 360 if (u <= t[tableidx-1]) break; | |
| 361 } | |
| 362 | |
| 363 /* We are assuming that the table size is big enough here. | |
| 364 * This should be true even if RUNI is returning values in | |
| 365 * the range [0,1] rather than [0,1). | |
| 366 */ | |
| 367 p[i] = (double)(tableidx-1); | |
| 368 } | |
| 369 } | |
| 370 | |
| 371 /* From Press, et al., Numerical Recipes */ | |
| 372 static void | |
| 373 poisson_rejection (double lambda, double *p, size_t n) | |
| 374 { | |
| 375 double sq = sqrt(2.0*lambda); | |
| 376 double alxm = log(lambda); | |
| 377 double g = lambda*alxm - LGAMMA(lambda+1.0); | |
| 378 size_t i; | |
| 379 | |
| 380 for (i = 0; i < n; i++) | |
| 381 { | |
| 382 double y, em, t; | |
| 383 do { | |
| 384 do { | |
| 385 y = tan(M_PI*RUNI); | |
| 386 em = sq * y + lambda; | |
| 387 } while (em < 0.0); | |
| 388 em = floor(em); | |
| 389 t = 0.9*(1.0+y*y)*exp(em*alxm-flogfak(em)-g); | |
| 390 } while (RUNI > t); | |
| 391 p[i] = em; | |
| 392 } | |
| 393 } | |
| 394 | |
| 395 /* The cutoff of L <= 1e8 in the following two functions before using | |
| 396 * the normal approximation is based on: | |
| 397 * > L=1e8; x=floor(linspace(0,2*L,1000)); | |
| 398 * > max(abs(normal_pdf(x,L,L)-poisson_pdf(x,L))) | |
| 399 * ans = 1.1376e-28 | |
| 400 * For L=1e7, the max is around 1e-9, which is within the step size of RUNI. | |
| 401 * For L>1e10 the pprsc function breaks down, as I saw from the histogram | |
| 402 * of a large sample, so 1e8 is both small enough and large enough. */ | |
| 403 | |
| 404 /* Generate a set of poisson numbers with the same distribution */ | |
| 405 void | |
| 406 oct_fill_randp (double L, octave_idx_type n, double *p) | |
| 407 { | |
| 408 octave_idx_type i; | |
| 409 if (L < 0.0 || INFINITE(L)) | |
| 410 { | |
| 411 for (i=0; i<n; i++) | |
| 412 p[i] = NAN; | |
| 413 } | |
| 414 else if (L <= 10.0) | |
| 415 { | |
| 416 poisson_cdf_lookup(L, p, n); | |
| 417 } | |
| 418 else if (L <= 1e8) | |
| 419 { | |
| 420 for (i=0; i<n; i++) | |
| 421 p[i] = pprsc(L); | |
| 422 } | |
| 423 else | |
| 424 { | |
| 425 /* normal approximation: from Phys. Rev. D (1994) v50 p1284 */ | |
| 426 const double sqrtL = sqrt(L); | |
| 6198 | 427 for (i = 0; i < n; i++) |
| 5742 | 428 { |
| 429 p[i] = floor(RNOR*sqrtL + L + 0.5); | |
| 430 if (p[i] < 0.0) | |
| 431 p[i] = 0.0; /* will probably never happen */ | |
| 432 } | |
| 433 } | |
| 434 } | |
| 435 | |
| 436 /* Generate one poisson variate */ | |
| 437 double | |
| 438 oct_randp (double L) | |
| 439 { | |
| 440 double ret; | |
| 441 if (L < 0.0) ret = NAN; | |
| 442 else if (L <= 12.0) { | |
| 443 /* From Press, et al. Numerical recipes */ | |
| 444 double g = exp(-L); | |
| 445 int em = -1; | |
| 446 double t = 1.0; | |
| 447 do { | |
| 448 ++em; | |
| 449 t *= RUNI; | |
| 450 } while (t > g); | |
| 451 ret = em; | |
| 452 } else if (L <= 1e8) { | |
| 453 /* numerical recipes */ | |
| 454 poisson_rejection(L, &ret, 1); | |
| 455 } else if (INFINITE(L)) { | |
| 5775 | 456 /* FIXME R uses NaN, but the normal approx. suggests that as |
| 5742 | 457 * limit should be inf. Which is correct? */ |
| 458 ret = NAN; | |
| 459 } else { | |
| 460 /* normal approximation: from Phys. Rev. D (1994) v50 p1284 */ | |
| 461 ret = floor(RNOR*sqrt(L) + L + 0.5); | |
| 462 if (ret < 0.0) ret = 0.0; /* will probably never happen */ | |
| 463 } | |
| 464 return ret; | |
| 465 } | |
| 466 | |
| 467 /* | |
| 468 ;;; Local Variables: *** | |
| 469 ;;; mode: C *** | |
| 470 ;;; End: *** | |
| 471 */ |