/* gammadist.c -- computes probability of samples under / produces samples from a Gamma distribution * * Mark Johnson, 22nd March 2008 * * WARNING: you need to set the flag -std=c99 to compile * * gammavariate() was translated from random.py in Python library * * The Gamma distribution is: * * Gamma(x | alpha, beta) = pow(x/beta, alpha-1) * exp(-x/beta) / (gamma(alpha)*beta) * * shape parameter alpha > 0 (also called c), scale parameter beta > 0 (also called s); * mean is alpha*beta, variance is alpha*beta**2 * * Note that many parameterizations of the Gamma function are in terms of an _inverse_ * scale parameter beta, which is the inverse of the beta given here. * * To define a main() that tests the routines, uncomment the following #define: */ /* #define GAMMATEST */ #include <assert.h> #include <math.h> #include "gammadist.h" #include "mt19937ar.h" /* gammadist() returns the probability density of x under a Gamma(alpha,beta) * distribution */ long double gammadist(long double x, long double alpha, long double beta) { assert(alpha > 0); assert(beta > 0); return pow(x/beta, alpha-1) * exp(-x/beta) / (tgamma(alpha)*beta); } /* lgammadist() returns the log probability density of x under a Gamma(alpha,beta) * distribution */ long double lgammadist(long double x, long double alpha, long double beta) { assert(alpha > 0); assert(beta > 0); return (alpha-1)*log(x) - alpha*log(beta) - x/beta - lgamma(alpha); } /* This definition of gammavariate is from Python code in * the Python random module. */ long double gammavariate(long double alpha, long double beta) { assert(alpha > 0); assert(beta > 0); if (alpha > 1.0) { /* Uses R.C.H. Cheng, "The generation of Gamma variables with non-integral shape parameters", Applied Statistics, (1977), 26, No. 1, p71-74 */ long double ainv = sqrt(2.0 * alpha - 1.0); long double bbb = alpha - log(4.0); long double ccc = alpha + ainv; while (1) { long double u1 = mt_genrand_real3(); if (u1 > 1e-7 || u1 < 0.9999999) { long double u2 = 1.0 - mt_genrand_real3(); long double v = log(u1/(1.0-u1))/ainv; long double x = alpha*exp(v); long double z = u1*u1*u2; long double r = bbb+ccc*v-x; if (r + (1.0+log(4.5)) - 4.5*z >= 0.0 || r >= log(z)) return x * beta; } } } else if (alpha == 1.0) { long double u = mt_genrand_real3(); while (u <= 1e-7) u = mt_genrand_real3(); return -log(u) * beta; } else { /* alpha is between 0 and 1 (exclusive) Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle */ while (1) { long double u = mt_genrand_real3(); long double b = (exp(1) + alpha)/exp(1); long double p = b*u; long double x = (p <= 1.0) ? pow(p, 1.0/alpha) : -log((b-p)/alpha); long double u1 = mt_genrand_real3(); if (! (((p <= 1.0) && (u1 > exp(-x))) || ((p > 1.0) && (u1 > pow(x, alpha - 1.0))))) return x * beta; } } } /* betadist() returns the probability density of x under a Beta(alpha,beta) * distribution. */ long double betadist(long double x, long double alpha, long double beta) { assert(x >= 0); assert(x <= 1); assert(alpha > 0); assert(beta > 0); return pow(x,alpha-1)*pow(1-x,beta-1)*tgamma(alpha+beta)/(tgamma(alpha)*tgamma(beta)); } /* lbetadist() returns the log probability density of x under a Beta(alpha,beta) * distribution. */ long double lbetadist(long double x, long double alpha, long double beta) { assert(x > 0); assert(x < 1); assert(alpha > 0); assert(beta > 0); return (alpha-1)*log(x)+(beta-1)*log(1-x)+lgamma(alpha+beta)-lgamma(alpha)-lgamma(beta); } /* betavariate() generates a sample from a Beta distribution with * parameters alpha and beta. * * 0 < alpha < 1, 0 < beta < 1, mean is alpha/(alpha+beta) */ long double betavariate(long double alpha, long double beta) { long double x = gammavariate(alpha, 1); long double y = gammavariate(beta, 1); return x/(x+y); } #ifdef GAMMATEST #include <stdio.h> int main(int argc, char **argv) { int iteration, niterations = 1000; for (iteration = 0; iteration < niterations; ++iteration) { long double alpha = 100*mt_genrand_real3(); long double gv = gammavariate(alpha, 1); long double pgv = gammadist(gv, alpha, 1); long double pgvl = exp(lgammadist(gv, alpha, 1)); fprintf(stderr, "iteration = %d, gammavariate(%lg,1) = %lg, gammadist(%lg,%lg,1) = %lg, exp(lgammadist(%lg,%lg,1) = %lg\n", iteration, alpha, gv, gv, alpha, pgv, gv, alpha, pgvl); } return 0; } #endif /* GAMMATEST */ /* Other routines I tried, but which weren't as good as the ones above */ #if 0 /*! gammavariate() returns samples from a Gamma distribution *! where alpha is the shape parameter and beta is the scale *! parameter, using the algorithm described on p. 94 of *! Gentle (1998) Random Number Generation and Monte Carlo Methods, *! Springer. */ long double gammavariate(long double alpha) { assert(alpha > 0); if (alpha > 1.0) { while (1) { long double u1 = mt_genrand_real3(); long double u2 = mt_genrand_real3(); long double v = (alpha - 1/(6*alpha))*u1/(alpha-1)*u2; if (2*(u2-1)/(alpha-1) + v + 1/v <= 2 || 2*log(u2)/(alpha-1) - log(v) + v <= 1) return (alpha-1)*v; } } else if (alpha < 1.0) { while (1) { long double t = 0.07 + 0.75*sqrt(1-alpha); long double b = alpha + exp(-t)*alpha/t; long double u1 = mt_genrand_real3(); long double u2 = mt_genrand_real3(); long double v = b*u1; if (v <= 1) { long double x = t*pow(v, 1/alpha); if (u2 <= (2 - x)/(2 + x)) return x; if (u2 <= exp(-x)) return x; } else { long double x = log(t*(b-v)/alpha); long double y = x/t; if (u2*(alpha + y*(1-alpha)) <= 1) return x; if (u2 <= pow(y,alpha-1)) return x; } } } else return -log(mt_genrand_real3()); } /*! gammavariate() returns a deviate distributed as a gamma *! distribution of order alpha, beta, i.e., a waiting time to the alpha'th *! event in a Poisson process of unit mean. *! *! Code from Numerical Recipes */ long double nr_gammavariate(long double ia) { int j; long double am,e,s,v1,v2,x,y; assert(ia > 0); if (ia < 10) { x=1.0; for (j=1;j<=ia;j++) x *= mt_genrand_real3(); x = -log(x); } else { do { do { do { v1=mt_genrand_real3(); v2=2.0*mt_genrand_real3()-1.0; } while (v1*v1+v2*v2 > 1.0); y=v2/v1; am=ia-1; s=sqrt(2.0*am+1.0); x=s*y+am; } while (x <= 0.0); e=(1.0+y*y)*exp(am*log(x/am)-s*y); } while (mt_genrand_real3() > e); } return x; } #endif