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/* randist/dirichlet.c
*
* Copyright (C) 2007 Brian Gough
* Copyright (C) 2002 Gavin E. Crooks <gec@compbio.berkeley.edu>
*
* 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 <config.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_rng.h>
#include <gsl/gsl_randist.h>
#include <gsl/gsl_sf_gamma.h>
/* The Dirichlet probability distribution of order K-1 is
p(\theta_1,...,\theta_K) d\theta_1 ... d\theta_K =
(1/Z) \prod_i=1,K \theta_i^{alpha_i - 1} \delta(1 -\sum_i=1,K \theta_i)
The normalization factor Z can be expressed in terms of gamma functions:
Z = {\prod_i=1,K \Gamma(\alpha_i)} / {\Gamma( \sum_i=1,K \alpha_i)}
The K constants, \alpha_1,...,\alpha_K, must be positive. The K parameters,
\theta_1,...,\theta_K are nonnegative and sum to 1.
The random variates are generated by sampling K values from gamma
distributions with parameters a=\alpha_i, b=1, and renormalizing.
See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).
Gavin E. Crooks <gec@compbio.berkeley.edu> (2002)
*/
static void ran_dirichlet_small (const gsl_rng * r, const size_t K, const double alpha[], double theta[]);
void
gsl_ran_dirichlet (const gsl_rng * r, const size_t K,
const double alpha[], double theta[])
{
size_t i;
double norm = 0.0;
for (i = 0; i < K; i++)
{
theta[i] = gsl_ran_gamma (r, alpha[i], 1.0);
}
for (i = 0; i < K; i++)
{
norm += theta[i];
}
if (norm < GSL_SQRT_DBL_MIN) /* Handle underflow */
{
ran_dirichlet_small (r, K, alpha, theta);
return;
}
for (i = 0; i < K; i++)
{
theta[i] /= norm;
}
}
/* When the values of alpha[] are small, scale the variates to avoid
underflow so that the result is not 0/0. Note that the Dirichlet
distribution is defined by a ratio of gamma functions so we can
take out an arbitrary factor to keep the values in the range of
double precision. */
static void
ran_dirichlet_small (const gsl_rng * r, const size_t K,
const double alpha[], double theta[])
{
size_t i;
double norm = 0.0, umax = 0;
for (i = 0; i < K; i++)
{
double u = log(gsl_rng_uniform_pos (r)) / alpha[i];
theta[i] = u;
if (u > umax || i == 0) {
umax = u;
}
}
for (i = 0; i < K; i++)
{
theta[i] = exp(theta[i] - umax);
}
for (i = 0; i < K; i++)
{
theta[i] = theta[i] * gsl_ran_gamma (r, alpha[i] + 1.0, 1.0);
}
for (i = 0; i < K; i++)
{
norm += theta[i];
}
for (i = 0; i < K; i++)
{
theta[i] /= norm;
}
}
double
gsl_ran_dirichlet_pdf (const size_t K,
const double alpha[], const double theta[])
{
return exp (gsl_ran_dirichlet_lnpdf (K, alpha, theta));
}
double
gsl_ran_dirichlet_lnpdf (const size_t K,
const double alpha[], const double theta[])
{
/*We calculate the log of the pdf to minimize the possibility of overflow */
size_t i;
double log_p = 0.0;
double sum_alpha = 0.0;
for (i = 0; i < K; i++)
{
log_p += (alpha[i] - 1.0) * log (theta[i]);
}
for (i = 0; i < K; i++)
{
sum_alpha += alpha[i];
}
log_p += gsl_sf_lngamma (sum_alpha);
for (i = 0; i < K; i++)
{
log_p -= gsl_sf_lngamma (alpha[i]);
}
return log_p;
}
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