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/* randist/dirichlet.c
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*
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* Copyright (C) 2007 Brian Gough
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* Copyright (C) 2002 Gavin E. Crooks <gec@compbio.berkeley.edu>
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*
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* This program is free software; you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation; either version 3 of the License, or (at
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* your option) any later version.
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*
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* This program is distributed in the hope that it will be useful, but
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* WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with this program; if not, write to the Free Software
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* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
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*/
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#include <config.h>
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#include <math.h>
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#include <gsl/gsl_math.h>
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#include <gsl/gsl_rng.h>
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#include <gsl/gsl_randist.h>
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#include <gsl/gsl_sf_gamma.h>
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/* The Dirichlet probability distribution of order K-1 is
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p(\theta_1,...,\theta_K) d\theta_1 ... d\theta_K =
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(1/Z) \prod_i=1,K \theta_i^{alpha_i - 1} \delta(1 -\sum_i=1,K \theta_i)
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The normalization factor Z can be expressed in terms of gamma functions:
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Z = {\prod_i=1,K \Gamma(\alpha_i)} / {\Gamma( \sum_i=1,K \alpha_i)}
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The K constants, \alpha_1,...,\alpha_K, must be positive. The K parameters,
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\theta_1,...,\theta_K are nonnegative and sum to 1.
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The random variates are generated by sampling K values from gamma
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distributions with parameters a=\alpha_i, b=1, and renormalizing.
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See A.M. Law, W.D. Kelton, Simulation Modeling and Analysis (1991).
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Gavin E. Crooks <gec@compbio.berkeley.edu> (2002)
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*/
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static void ran_dirichlet_small (const gsl_rng * r, const size_t K, const double alpha[], double theta[]);
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void
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gsl_ran_dirichlet (const gsl_rng * r, const size_t K,
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const double alpha[], double theta[])
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{
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size_t i;
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double norm = 0.0;
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for (i = 0; i < K; i++)
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{
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theta[i] = gsl_ran_gamma (r, alpha[i], 1.0);
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}
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for (i = 0; i < K; i++)
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{
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norm += theta[i];
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}
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if (norm < GSL_SQRT_DBL_MIN) /* Handle underflow */
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{
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ran_dirichlet_small (r, K, alpha, theta);
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return;
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}
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for (i = 0; i < K; i++)
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{
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theta[i] /= norm;
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}
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}
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/* When the values of alpha[] are small, scale the variates to avoid
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underflow so that the result is not 0/0. Note that the Dirichlet
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distribution is defined by a ratio of gamma functions so we can
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take out an arbitrary factor to keep the values in the range of
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double precision. */
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static void
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ran_dirichlet_small (const gsl_rng * r, const size_t K,
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const double alpha[], double theta[])
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{
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size_t i;
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double norm = 0.0, umax = 0;
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for (i = 0; i < K; i++)
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{
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double u = log(gsl_rng_uniform_pos (r)) / alpha[i];
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theta[i] = u;
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if (u > umax || i == 0) {
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umax = u;
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}
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}
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for (i = 0; i < K; i++)
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{
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theta[i] = exp(theta[i] - umax);
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}
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for (i = 0; i < K; i++)
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{
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theta[i] = theta[i] * gsl_ran_gamma (r, alpha[i] + 1.0, 1.0);
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}
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for (i = 0; i < K; i++)
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{
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norm += theta[i];
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}
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for (i = 0; i < K; i++)
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{
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theta[i] /= norm;
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}
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}
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double
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gsl_ran_dirichlet_pdf (const size_t K,
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const double alpha[], const double theta[])
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{
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return exp (gsl_ran_dirichlet_lnpdf (K, alpha, theta));
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}
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double
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gsl_ran_dirichlet_lnpdf (const size_t K,
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const double alpha[], const double theta[])
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{
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/*We calculate the log of the pdf to minimize the possibility of overflow */
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size_t i;
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double log_p = 0.0;
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double sum_alpha = 0.0;
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for (i = 0; i < K; i++)
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{
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log_p += (alpha[i] - 1.0) * log (theta[i]);
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}
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for (i = 0; i < K; i++)
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{
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sum_alpha += alpha[i];
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}
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log_p += gsl_sf_lngamma (sum_alpha);
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for (i = 0; i < K; i++)
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{
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log_p -= gsl_sf_lngamma (alpha[i]);
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}
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return log_p;
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}
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