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/* 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
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