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- /* randist/binomial_tpe.c
- *
- * Copyright (C) 1996, 2003, 2007 James Theiler, Brian Gough
- *
- * 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; either version 3 of the License, or (at
- * your option) any later version.
- *
- * 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, write to the Free Software
- * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
- */
- #include "gsl__config.h"
- #include <math.h>
- #include "gsl_rng.h"
- #include "gsl_randist.h"
- #include "gsl_pow_int.h"
- #include "gsl_sf_gamma.h"
- /* The binomial distribution has the form,
- f(x) = n!/(x!(n-x)!) * p^x (1-p)^(n-x) for integer 0 <= x <= n
- = 0 otherwise
- This implementation follows the public domain ranlib function
- "ignbin", the bulk of which is the BTPE (Binomial Triangle
- Parallelogram Exponential) algorithm introduced in
- Kachitvichyanukul and Schmeiser[1]. It has been translated to use
- modern C coding standards.
- If n is small and/or p is near 0 or near 1 (specifically, if
- n*min(p,1-p) < SMALL_MEAN), then a different algorithm, called
- BINV, is used which has an average runtime that scales linearly
- with n*min(p,1-p).
- But for larger problems, the BTPE algorithm takes the form of two
- functions b(x) and t(x) -- "bottom" and "top" -- for which b(x) <
- f(x)/f(M) < t(x), with M = floor(n*p+p). b(x) defines a triangular
- region, and t(x) includes a parallelogram and two tails. Details
- (including a nice drawing) are in the paper.
- [1] Kachitvichyanukul, V. and Schmeiser, B. W. Binomial Random
- Variate Generation. Communications of the ACM, 31, 2 (February,
- 1988) 216.
- Note, Bruce Schmeiser (personal communication) points out that if
- you want very fast binomial deviates, and you are happy with
- approximate results, and/or n and n*p are both large, then you can
- just use gaussian estimates: mean=n*p, variance=n*p*(1-p).
- This implementation by James Theiler, April 2003, after obtaining
- permission -- and some good advice -- from Drs. Kachitvichyanukul
- and Schmeiser to use their code as a starting point, and then doing
- a little bit of tweaking.
- Additional polishing for GSL coding standards by Brian Gough. */
- #define SMALL_MEAN 14 /* If n*p < SMALL_MEAN then use BINV
- algorithm. The ranlib
- implementation used cutoff=30; but
- on my computer 14 works better */
- #define BINV_CUTOFF 110 /* In BINV, do not permit ix too large */
- #define FAR_FROM_MEAN 20 /* If ix-n*p is larger than this, then
- use the "squeeze" algorithm.
- Ranlib used 20, and this seems to
- be the best choice on my machine as
- well */
- #define LNFACT(x) gsl_sf_lnfact(x)
- inline static double
- Stirling (double y1)
- {
- double y2 = y1 * y1;
- double s =
- (13860.0 -
- (462.0 - (132.0 - (99.0 - 140.0 / y2) / y2) / y2) / y2) / y1 / 166320.0;
- return s;
- }
- unsigned int
- gsl_ran_binomial_tpe (const gsl_rng * rng, double p, unsigned int n)
- {
- return gsl_ran_binomial (rng, p, n);
- }
- unsigned int
- gsl_ran_binomial (const gsl_rng * rng, double p, unsigned int n)
- {
- int ix; /* return value */
- int flipped = 0;
- double q, s, np;
- if (n == 0)
- return 0;
- if (p > 0.5)
- {
- p = 1.0 - p; /* work with small p */
- flipped = 1;
- }
- q = 1 - p;
- s = p / q;
- np = n * p;
- /* Inverse cdf logic for small mean (BINV in K+S) */
- if (np < SMALL_MEAN)
- {
- double f0 = gsl_pow_int (q, n); /* f(x), starting with x=0 */
- while (1)
- {
- /* This while(1) loop will almost certainly only loop once; but
- * if u=1 to within a few epsilons of machine precision, then it
- * is possible for roundoff to prevent the main loop over ix to
- * achieve its proper value. following the ranlib implementation,
- * we introduce a check for that situation, and when it occurs,
- * we just try again.
- */
- double f = f0;
- double u = gsl_rng_uniform (rng);
- for (ix = 0; ix <= BINV_CUTOFF; ++ix)
- {
- if (u < f)
- goto Finish;
- u -= f;
- /* Use recursion f(x+1) = f(x)*[(n-x)/(x+1)]*[p/(1-p)] */
- f *= s * (n - ix) / (ix + 1);
- }
- /* It should be the case that the 'goto Finish' was encountered
- * before this point was ever reached. But if we have reached
- * this point, then roundoff has prevented u from decreasing
- * all the way to zero. This can happen only if the initial u
- * was very nearly equal to 1, which is a rare situation. In
- * that rare situation, we just try again.
- *
- * Note, following the ranlib implementation, we loop ix only to
- * a hardcoded value of SMALL_MEAN_LARGE_N=110; we could have
- * looped to n, and 99.99...% of the time it won't matter. This
- * choice, I think is a little more robust against the rare
- * roundoff error. If n>LARGE_N, then it is technically
- * possible for ix>LARGE_N, but it is astronomically rare, and
- * if ix is that large, it is more likely due to roundoff than
- * probability, so better to nip it at LARGE_N than to take a
- * chance that roundoff will somehow conspire to produce an even
- * larger (and more improbable) ix. If n<LARGE_N, then once
- * ix=n, f=0, and the loop will continue until ix=LARGE_N.
- */
- }
- }
- else
- {
- /* For n >= SMALL_MEAN, we invoke the BTPE algorithm */
- int k;
- double ffm = np + p; /* ffm = n*p+p */
- int m = (int) ffm; /* m = int floor[n*p+p] */
- double fm = m; /* fm = double m; */
- double xm = fm + 0.5; /* xm = half integer mean (tip of triangle) */
- double npq = np * q; /* npq = n*p*q */
- /* Compute cumulative area of tri, para, exp tails */
- /* p1: radius of triangle region; since height=1, also: area of region */
- /* p2: p1 + area of parallelogram region */
- /* p3: p2 + area of left tail */
- /* p4: p3 + area of right tail */
- /* pi/p4: probability of i'th area (i=1,2,3,4) */
- /* Note: magic numbers 2.195, 4.6, 0.134, 20.5, 15.3 */
- /* These magic numbers are not adjustable...at least not easily! */
- double p1 = floor (2.195 * sqrt (npq) - 4.6 * q) + 0.5;
- /* xl, xr: left and right edges of triangle */
- double xl = xm - p1;
- double xr = xm + p1;
- /* Parameter of exponential tails */
- /* Left tail: t(x) = c*exp(-lambda_l*[xl - (x+0.5)]) */
- /* Right tail: t(x) = c*exp(-lambda_r*[(x+0.5) - xr]) */
- double c = 0.134 + 20.5 / (15.3 + fm);
- double p2 = p1 * (1.0 + c + c);
- double al = (ffm - xl) / (ffm - xl * p);
- double lambda_l = al * (1.0 + 0.5 * al);
- double ar = (xr - ffm) / (xr * q);
- double lambda_r = ar * (1.0 + 0.5 * ar);
- double p3 = p2 + c / lambda_l;
- double p4 = p3 + c / lambda_r;
- double var, accept;
- double u, v; /* random variates */
- TryAgain:
- /* generate random variates, u specifies which region: Tri, Par, Tail */
- u = gsl_rng_uniform (rng) * p4;
- v = gsl_rng_uniform (rng);
- if (u <= p1)
- {
- /* Triangular region */
- ix = (int) (xm - p1 * v + u);
- goto Finish;
- }
- else if (u <= p2)
- {
- /* Parallelogram region */
- double x = xl + (u - p1) / c;
- v = v * c + 1.0 - fabs (x - xm) / p1;
- if (v > 1.0 || v <= 0.0)
- goto TryAgain;
- ix = (int) x;
- }
- else if (u <= p3)
- {
- /* Left tail */
- ix = (int) (xl + log (v) / lambda_l);
- if (ix < 0)
- goto TryAgain;
- v *= ((u - p2) * lambda_l);
- }
- else
- {
- /* Right tail */
- ix = (int) (xr - log (v) / lambda_r);
- if (ix > (double) n)
- goto TryAgain;
- v *= ((u - p3) * lambda_r);
- }
- /* At this point, the goal is to test whether v <= f(x)/f(m)
- *
- * v <= f(x)/f(m) = (m!(n-m)! / (x!(n-x)!)) * (p/q)^{x-m}
- *
- */
- /* Here is a direct test using logarithms. It is a little
- * slower than the various "squeezing" computations below, but
- * if things are working, it should give exactly the same answer
- * (given the same random number seed). */
- #ifdef DIRECT
- var = log (v);
- accept =
- LNFACT (m) + LNFACT (n - m) - LNFACT (ix) - LNFACT (n - ix)
- + (ix - m) * log (p / q);
- #else /* SQUEEZE METHOD */
- /* More efficient determination of whether v < f(x)/f(M) */
- k = abs (ix - m);
- if (k <= FAR_FROM_MEAN)
- {
- /*
- * If ix near m (ie, |ix-m|<FAR_FROM_MEAN), then do
- * explicit evaluation using recursion relation for f(x)
- */
- double g = (n + 1) * s;
- double f = 1.0;
- var = v;
- if (m < ix)
- {
- int i;
- for (i = m + 1; i <= ix; i++)
- {
- f *= (g / i - s);
- }
- }
- else if (m > ix)
- {
- int i;
- for (i = ix + 1; i <= m; i++)
- {
- f /= (g / i - s);
- }
- }
- accept = f;
- }
- else
- {
- /* If ix is far from the mean m: k=ABS(ix-m) large */
- var = log (v);
- if (k < npq / 2 - 1)
- {
- /* "Squeeze" using upper and lower bounds on
- * log(f(x)) The squeeze condition was derived
- * under the condition k < npq/2-1 */
- double amaxp =
- k / npq * ((k * (k / 3.0 + 0.625) + (1.0 / 6.0)) / npq + 0.5);
- double ynorm = -(k * k / (2.0 * npq));
- if (var < ynorm - amaxp)
- goto Finish;
- if (var > ynorm + amaxp)
- goto TryAgain;
- }
- /* Now, again: do the test log(v) vs. log f(x)/f(M) */
- #if USE_EXACT
- /* This is equivalent to the above, but is a little (~20%) slower */
- /* There are five log's vs three above, maybe that's it? */
- accept = LNFACT (m) + LNFACT (n - m)
- - LNFACT (ix) - LNFACT (n - ix) + (ix - m) * log (p / q);
- #else /* USE STIRLING */
- /* The "#define Stirling" above corresponds to the first five
- * terms in asymptoic formula for
- * log Gamma (y) - (y-0.5)log(y) + y - 0.5 log(2*pi);
- * See Abramowitz and Stegun, eq 6.1.40
- */
- /* Note below: two Stirling's are added, and two are
- * subtracted. In both K+S, and in the ranlib
- * implementation, all four are added. I (jt) believe that
- * is a mistake -- this has been confirmed by personal
- * correspondence w/ Dr. Kachitvichyanukul. Note, however,
- * the corrections are so small, that I couldn't find an
- * example where it made a difference that could be
- * observed, let alone tested. In fact, define'ing Stirling
- * to be zero gave identical results!! In practice, alv is
- * O(1), ranging 0 to -10 or so, while the Stirling
- * correction is typically O(10^{-5}) ...setting the
- * correction to zero gives about a 2% performance boost;
- * might as well keep it just to be pendantic. */
- {
- double x1 = ix + 1.0;
- double w1 = n - ix + 1.0;
- double f1 = fm + 1.0;
- double z1 = n + 1.0 - fm;
- accept = xm * log (f1 / x1) + (n - m + 0.5) * log (z1 / w1)
- + (ix - m) * log (w1 * p / (x1 * q))
- + Stirling (f1) + Stirling (z1) - Stirling (x1) - Stirling (w1);
- }
- #endif
- #endif
- }
- if (var <= accept)
- {
- goto Finish;
- }
- else
- {
- goto TryAgain;
- }
- }
- Finish:
- return (flipped) ? (n - ix) : (unsigned int)ix;
- }
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