diff hmm/hmm.c @ 241:a98dd8ec96f8

* Move dsp/maths to maths ; bring PCA and HMM across from Soundbite

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/hmm/hmm.c	Wed Jan 09 10:31:29 2008 +0000
@@ -0,0 +1,800 @@
+/*
+ *  hmm.c
+ *  soundbite
+ *
+ *  Created by Mark Levy on 12/02/2006.
+ *  Copyright 2006 Centre for Digital Music, Queen Mary, University of London. All rights reserved.
+ *
+ */
+
+#include <stdio.h>
+#include <math.h>
+#include <stdlib.h>
+#include <float.h>
+#include <time.h>				/* to seed random number generator */
+#include "clapack.h"		/* LAPACK for matrix inversion */
+#ifdef _MAC_OS_X
+#include <vecLib/cblas.h>
+#else
+#include <cblas.h>		/* BLAS for matrix multiplication */
+#endif
+
+#include "hmm.h"
+
+model_t* hmm_init(double** x, int T, int L, int N)
+{
+	int i, j, d, e, t;
+	double s, ss;
+	
+	model_t* model;
+	model = (model_t*) malloc(sizeof(model_t));
+	model->N = N;
+	model->L = L;	
+	model->p0 = (double*) malloc(N*sizeof(double));
+	model->a = (double**) malloc(N*sizeof(double*));
+	model->mu = (double**) malloc(N*sizeof(double*));
+	for (i = 0; i < N; i++)
+	{
+		model->a[i] = (double*) malloc(N*sizeof(double));
+		model->mu[i] = (double*) malloc(L*sizeof(double));
+	}
+	model->cov = (double**) malloc(L*sizeof(double*));
+	for (i = 0; i < L; i++)
+		model->cov[i] = (double*) malloc(L*sizeof(double));
+	
+	srand(time(0));
+	double* global_mean = (double*) malloc(L*sizeof(double));
+	
+	/* find global mean */
+	for (d = 0; d < L; d++)
+	{
+		global_mean[d] = 0;
+		for (t = 0; t < T; t++)
+			global_mean[d] += x[t][d];
+		global_mean[d] /= T;
+	}
+	
+	/* calculate global diagonal covariance */
+	for (d = 0; d < L; d++)
+	{
+		for (e = 0; e < L; e++)
+			model->cov[d][e] = 0;
+		for (t = 0; t < T; t++)
+			model->cov[d][d] += (x[t][d] - global_mean[d]) * (x[t][d] - global_mean[d]);
+		model->cov[d][d] /= T-1;
+	}
+	
+	/* set all means close to global mean */
+	for (i = 0; i < N; i++)
+	{
+		for (d = 0; d < L; d++)
+		{
+			/* add some random noise related to covariance */
+			/* ideally the random number would be Gaussian(0,1), as a hack we make it uniform on [-0.25,0.25] */
+			model->mu[i][d] = global_mean[d] + (0.5 * rand() / (double) RAND_MAX - 0.25) * sqrt(model->cov[d][d]);
+		}
+	}	
+	
+	/* random intial and transition probs */
+	s = 0;
+	for (i = 0; i < N; i++)
+	{
+		model->p0[i] = 1 + rand() / (double) RAND_MAX;
+		s += model->p0[i];
+		ss = 0;
+		for (j = 0; j < N; j++)
+		{
+			model->a[i][j] = 1 + rand() / (double) RAND_MAX;
+			ss += model->a[i][j];
+		}
+		for (j = 0; j < N; j++)
+		{
+			model->a[i][j] /= ss;
+		}
+	}
+	for (i = 0; i < N; i++)
+		model->p0[i] /= s;
+	
+	free(global_mean);
+	
+	return model;
+}
+
+void hmm_close(model_t* model)
+{
+	int i;
+	
+	for (i = 0; i < model->N; i++)
+	{
+		free(model->a[i]);
+		free(model->mu[i]);
+	}
+	free(model->a);
+	free(model->mu);
+	for (i = 0; i < model->L; i++)
+		free(model->cov[i]);
+	free(model->cov);	  
+	free(model);	
+}
+
+void hmm_train(double** x, int T, model_t* model)
+{
+	int i, t;
+	double loglik;	/* overall log-likelihood at each iteration */
+	
+	int N = model->N;
+	int L = model->L;
+	double* p0 = model->p0;
+	double** a = model->a;
+	double** mu = model->mu;
+	double** cov = model->cov;
+	
+	/* allocate memory */	
+	double** gamma = (double**) malloc(T*sizeof(double*));
+	double*** xi = (double***) malloc(T*sizeof(double**));
+	for (t = 0; t < T; t++)
+	{
+		gamma[t] = (double*) malloc(N*sizeof(double));
+		xi[t] = (double**) malloc(N*sizeof(double*));
+		for (i = 0; i < N; i++)
+			xi[t][i] = (double*) malloc(N*sizeof(double));
+	}
+	
+	/* temporary memory */
+	double* gauss_y = (double*) malloc(L*sizeof(double));
+	double* gauss_z = (double*) malloc(L*sizeof(double));	
+			
+	/* obs probs P(j|{x}) */
+	double** b = (double**) malloc(T*sizeof(double*));
+	for (t = 0; t < T; t++)
+		b[t] = (double*) malloc(N*sizeof(double));
+	
+	/* inverse covariance and its determinant */
+	double** icov = (double**) malloc(L*sizeof(double*));
+	for (i = 0; i < L; i++)
+		icov[i] = (double*) malloc(L*sizeof(double));
+	double detcov;
+	
+	double thresh = 0.0001;
+	int niter = 50;	
+	int iter = 0;
+	double loglik1, loglik2;
+	while(iter < niter && !(iter > 1 && (loglik - loglik1) < thresh * (loglik1 - loglik2)))	
+	{
+		++iter;
+		
+		fprintf(stderr, "calculating obsprobs...\n");
+		fflush(stderr);
+		
+		/* precalculate obs probs */
+		invert(cov, L, icov, &detcov);
+		
+		for (t = 0; t < T; t++)
+		{
+			//int allzero = 1;
+			for (i = 0; i < N; i++)
+			{
+				b[t][i] = exp(loggauss(x[t], L, mu[i], icov, detcov, gauss_y, gauss_z));
+		
+				//if (b[t][i] != 0)
+				//	allzero = 0;
+			}
+			/*
+			if (allzero)
+			{
+				printf("all the b[t][i] were zero for t = %d, correcting...\n", t);
+				for (i = 0; i < N; i++)
+				{
+					b[t][i] = 0.00001;
+				}
+			}
+			*/
+		}
+		
+		fprintf(stderr, "forwards-backwards...\n");
+		fflush(stderr);
+		
+		forward_backwards(xi, gamma, &loglik, &loglik1, &loglik2, iter, N, T, p0, a, b);
+		
+		fprintf(stderr, "iteration %d: loglik = %f\n", iter, loglik);		
+		fprintf(stderr, "re-estimation...\n");
+		fflush(stderr);
+		
+		baum_welch(p0, a, mu, cov, N, T, L, x, xi, gamma);
+			
+		/*
+		printf("a:\n");
+		for (i = 0; i < model->N; i++)
+		{
+			for (j = 0; j < model->N; j++)
+				printf("%f ", model->a[i][j]);
+			printf("\n");
+		}
+		printf("\n\n");
+		 */
+		//hmm_print(model);
+	}
+	
+	/* deallocate memory */
+	for (t = 0; t < T; t++)
+	{
+		free(gamma[t]);
+		free(b[t]);
+		for (i = 0; i < N; i++)
+			free(xi[t][i]);
+		free(xi[t]);
+	}
+	free(gamma);
+	free(xi);
+	free(b);	
+	
+	for (i = 0; i < L; i++)
+		free(icov[i]);
+	free(icov);
+	
+	free(gauss_y);
+	free(gauss_z);
+}
+
+void mlss_reestimate(double* p0, double** a, double** mu, double** cov, int N, int T, int L, int* q, double** x)
+{
+	/* fit a single Gaussian to observations in each state */
+	
+	/* calculate the mean observation in each state */
+	
+	/* calculate the overall covariance */
+	
+	/* count transitions */
+	
+	/* estimate initial probs from transitions (???) */
+}
+
+void baum_welch(double* p0, double** a, double** mu, double** cov, int N, int T, int L, double** x, double*** xi, double** gamma)
+{
+	int i, j, t;
+	
+	double* sum_gamma = (double*) malloc(N*sizeof(double));
+	
+	/* temporary memory */
+	double* u = (double*) malloc(L*L*sizeof(double));
+	double* yy = (double*) malloc(T*L*sizeof(double));
+	double* yy2 = (double*) malloc(T*L*sizeof(double));	
+	
+	/* re-estimate transition probs */
+	for (i = 0; i < N; i++)
+	{
+		sum_gamma[i] = 0;
+		for (t = 0; t < T-1; t++)
+			sum_gamma[i] += gamma[t][i];
+	}
+	
+	for (i = 0; i < N; i++)
+	{
+		if (sum_gamma[i] == 0)
+		{
+			fprintf(stderr, "sum_gamma[%d] was zero...\n", i);
+		}
+		//double s = 0;
+		for (j = 0; j < N; j++)
+		{
+			a[i][j] = 0;
+			for (t = 0; t < T-1; t++)
+				a[i][j] += xi[t][i][j];
+			//s += a[i][j];
+			a[i][j] /= sum_gamma[i];	
+		}
+		/*
+		 for (j = 0; j < N; j++)
+		 {
+			 a[i][j] /= s;
+		 }
+		 */
+	}
+	
+	/* NB: now we need to sum gamma over all t */
+	for (i = 0; i < N; i++)
+		sum_gamma[i] += gamma[T-1][i];
+	
+	/* re-estimate initial probs */
+	for (i = 0; i < N; i++)
+		p0[i] = gamma[0][i];
+	
+	/* re-estimate covariance */
+	int d, e;
+	double sum_sum_gamma = 0;
+	for (i = 0; i < N; i++)
+		sum_sum_gamma += sum_gamma[i];		
+	
+	/*
+	 for (d = 0; d < L; d++)
+	 {
+		 for (e = d; e < L; e++)
+		 {
+			 cov[d][e] = 0;
+			 for (t = 0; t < T; t++)
+				 for (j = 0; j < N; j++)
+					 cov[d][e] += gamma[t][j] * (x[t][d] - mu[j][d]) * (x[t][e] - mu[j][e]);
+			 
+			 cov[d][e] /= sum_sum_gamma;
+			 
+			 if (isnan(cov[d][e]))
+			 {
+				 printf("cov[%d][%d] was nan\n", d, e);
+				 for (j = 0; j < N; j++)
+					 for (i = 0; i < L; i++)
+						 if (isnan(mu[j][i]))
+							 printf("mu[%d][%d] was nan\n", j, i);
+				 for (t = 0; t < T; t++)
+					 for (j = 0; j < N; j++)
+						 if (isnan(gamma[t][j]))
+							 printf("gamma[%d][%d] was nan\n", t, j);
+				 exit(-1);
+			 }
+		 }
+	 }
+	 for (d = 0; d < L; d++)
+	 for (e = 0; e < d; e++)
+	 cov[d][e] = cov[e][d];
+	 */
+	
+	/* using BLAS */
+	for (d = 0; d < L; d++)
+		for (e = 0; e < L; e++)
+			cov[d][e] = 0;
+	
+	for (j = 0; j < N; j++)
+	{
+		for (d = 0; d < L; d++)
+			for (t = 0; t < T; t++)
+			{
+				yy[d*T+t] = x[t][d] - mu[j][d];
+				yy2[d*T+t] = gamma[t][j] * (x[t][d] - mu[j][d]);
+			}
+				
+				cblas_dgemm(CblasColMajor, CblasTrans, CblasNoTrans, L, L, T, 1.0, yy, T, yy2, T, 0, u, L);
+		
+		for (e = 0; e < L; e++)
+			for (d = 0; d < L; d++)
+				cov[d][e] += u[e*L+d];
+	}
+	
+	for (d = 0; d < L; d++)
+		for (e = 0; e < L; e++)
+			cov[d][e] /= T; /* sum_sum_gamma; */			
+	
+	//printf("sum_sum_gamma = %f\n", sum_sum_gamma); /* fine, = T IS THIS ALWAYS TRUE with pooled cov?? */
+	
+	/* re-estimate means */
+	for (j = 0; j < N; j++)
+	{
+		for (d = 0; d < L; d++)
+		{
+			mu[j][d] = 0;
+			for (t = 0; t < T; t++)
+				mu[j][d] += gamma[t][j] * x[t][d];
+			mu[j][d] /= sum_gamma[j];
+		}
+	}
+	
+	/* deallocate memory */
+	free(sum_gamma);
+	free(yy);
+	free(yy2);
+	free(u);
+}
+
+void forward_backwards(double*** xi, double** gamma, double* loglik, double* loglik1, double* loglik2, int iter, int N, int T, double* p0, double** a, double** b)
+{
+	/* forwards-backwards with scaling */
+	int i, j, t;
+	
+	double** alpha = (double**) malloc(T*sizeof(double*));
+	double** beta = (double**) malloc(T*sizeof(double*));
+	for (t = 0; t < T; t++)
+	{
+		alpha[t] = (double*) malloc(N*sizeof(double));
+		beta[t] = (double*) malloc(N*sizeof(double));
+	}
+	
+	/* scaling coefficients */
+	double* c = (double*) malloc(T*sizeof(double));
+	
+	/* calculate forward probs and scale coefficients */
+	c[0] = 0;
+	for (i = 0; i < N; i++)
+	{
+		alpha[0][i] = p0[i] * b[0][i];
+		c[0] += alpha[0][i];
+		
+		//printf("p0[%d] = %f, b[0][%d] = %f\n", i, p0[i], i, b[0][i]);
+	}
+	c[0] = 1 / c[0];
+	for (i = 0; i < N; i++)
+	{
+		alpha[0][i] *= c[0];		
+		
+		//printf("alpha[0][%d] = %f\n", i, alpha[0][i]);	/* OK agrees with Matlab */
+	}
+	
+	*loglik1 = *loglik;
+	*loglik = -log(c[0]);
+	if (iter == 2)
+		*loglik2 = *loglik;
+	
+	for (t = 1; t < T; t++)
+	{			
+		c[t] = 0;
+		for (j = 0; j < N; j++)
+		{
+			alpha[t][j] = 0;
+			for (i = 0; i < N; i++)
+				alpha[t][j] += alpha[t-1][i] * a[i][j];
+			alpha[t][j] *= b[t][j];
+			
+			c[t] += alpha[t][j];
+		}
+		
+		/*
+		 if (c[t] == 0)
+		 {
+			 printf("c[%d] = 0, going to blow up so exiting\n", t);
+			 for (i = 0; i < N; i++)
+				 if (b[t][i] == 0)
+					 fprintf(stderr, "b[%d][%d] was zero\n", t, i);
+			 fprintf(stderr, "x[t] was \n");
+			 for (i = 0; i < L; i++)
+				 fprintf(stderr, "%f ", x[t][i]);
+			 fprintf(stderr, "\n\n");
+			 exit(-1);
+		 }
+		 */
+		
+		c[t] = 1 / c[t];
+		for (j = 0; j < N; j++)
+			alpha[t][j] *= c[t];
+		
+		//printf("c[%d] = %e\n", t, c[t]);
+		
+		*loglik -= log(c[t]);
+	}
+	
+	/* calculate backwards probs using same coefficients */
+	for (i = 0; i < N; i++)
+		beta[T-1][i] = 1;
+	t = T - 1;
+	while (1)
+	{
+		for (i = 0; i < N; i++)
+			beta[t][i] *= c[t];
+		
+		if (t == 0)
+			break;
+		
+		for (i = 0; i < N; i++)
+		{
+			beta[t-1][i] = 0;
+			for (j = 0; j < N; j++)
+				beta[t-1][i] += a[i][j] * b[t][j] * beta[t][j];
+		}
+		
+		t--;
+	}
+	
+	/*
+	 printf("alpha:\n");
+	 for (t = 0; t < T; t++)
+	 {
+		 for (i = 0; i < N; i++)
+			 printf("%4.4e\t\t", alpha[t][i]);
+		 printf("\n");
+	 }
+	 printf("\n\n");printf("beta:\n");
+	 for (t = 0; t < T; t++)
+	 {
+		 for (i = 0; i < N; i++)
+			 printf("%4.4e\t\t", beta[t][i]);
+		 printf("\n");
+	 }
+	 printf("\n\n");
+	 */
+	
+	/* calculate posterior probs */
+	double tot;
+	for (t = 0; t < T; t++)
+	{
+		tot = 0;
+		for (i = 0; i < N; i++)
+		{
+			gamma[t][i] = alpha[t][i] * beta[t][i];
+			tot += gamma[t][i];
+		}
+		for (i = 0; i < N; i++)
+		{
+			gamma[t][i] /= tot;				
+			
+			//printf("gamma[%d][%d] = %f\n", t, i, gamma[t][i]);				
+		}
+	}		
+	
+	for (t = 0; t < T-1; t++)
+	{
+		tot = 0;
+		for (i = 0; i < N; i++)
+		{
+			for (j = 0; j < N; j++)
+			{
+				xi[t][i][j] = alpha[t][i] * a[i][j] * b[t+1][j] * beta[t+1][j];
+				tot += xi[t][i][j];
+			}
+		}
+		for (i = 0; i < N; i++)
+			for (j = 0; j < N; j++)
+				xi[t][i][j] /= tot;
+	}
+	
+	/*
+	 // CHECK - fine
+	 // gamma[t][i] = \sum_j{xi[t][i][j]}
+	 tot = 0;
+	 for (j = 0; j < N; j++)
+	 tot += xi[3][1][j];
+	 printf("gamma[3][1] = %f, sum_j(xi[3][1][j]) = %f\n", gamma[3][1], tot);
+	 */	
+	
+	for (t = 0; t < T; t++)
+	{
+		free(alpha[t]);
+		free(beta[t]);
+	}
+	free(alpha);
+	free(beta);
+	free(c);
+}
+
+void viterbi_decode(double** x, int T, model_t* model, int* q)
+{
+	int i, j, t;
+	double max;
+	
+	int N = model->N;
+	int L = model->L;
+	double* p0 = model->p0;
+	double** a = model->a;
+	double** mu = model->mu;
+	double** cov = model->cov;
+	
+	/* inverse covariance and its determinant */
+	double** icov = (double**) malloc(L*sizeof(double*));
+	for (i = 0; i < L; i++)
+		icov[i] = (double*) malloc(L*sizeof(double));
+	double detcov;
+	
+	double** logb = (double**) malloc(T*sizeof(double*));
+	double** phi = (double**) malloc(T*sizeof(double*));
+	int** psi = (int**) malloc(T*sizeof(int*));
+	for (t = 0; t < T; t++)
+	{
+		logb[t] = (double*) malloc(N*sizeof(double));
+		phi[t] = (double*) malloc(N*sizeof(double));
+		psi[t] = (int*) malloc(N*sizeof(int));
+	}
+	
+	/* temporary memory */
+	double* gauss_y = (double*) malloc(L*sizeof(double));
+	double* gauss_z = (double*) malloc(L*sizeof(double));	
+	
+	/* calculate observation logprobs */
+	invert(cov, L, icov, &detcov);
+	for (t = 0; t < T; t++)
+		for (i = 0; i < N; i++)
+			logb[t][i] = loggauss(x[t], L, mu[i], icov, detcov, gauss_y, gauss_z);
+	
+	/* initialise */
+	for (i = 0; i < N; i++)
+	{
+		phi[0][i] = log(p0[i]) + logb[0][i];
+		psi[0][i] = 0;
+	}
+	
+	for (t = 1; t < T; t++)
+	{
+		for (j = 0; j < N; j++)
+		{
+			max = -1000000;		// TODO: what should this be??  = smallest possible sumlogprob
+			psi[t][j] = 0;
+			for (i = 0; i < N; i++)
+			{
+				if (phi[t-1][i] + log(a[i][j]) > max)
+				{
+					max = phi[t-1][i] + log(a[i][j]);
+					phi[t][j] = max + logb[t][j];
+					psi[t][j] = i;
+				}
+			}
+		}
+	}
+	
+	/* find maximising state at time T-1 */
+	max = phi[T-1][0];
+	q[T-1] = 0;
+	for (i = 1; i < N; i++)
+	{
+		if (phi[T-1][i] > max)
+		{
+			max = phi[T-1][i];
+			q[T-1] = i;
+		}
+	}
+
+	
+	/* track back */
+	t = T - 2;
+	while (t >= 0)
+	{
+		q[t] = psi[t+1][q[t+1]];
+		t--;
+	}
+	
+	/* de-allocate memory */
+	for (i = 0; i < L; i++)
+		free(icov[i]);
+	free(icov);
+	for (t = 0; t < T; t++)
+	{
+		free(logb[t]);
+		free(phi[t]);
+		free(psi[t]);
+	}
+	free(logb);
+	free(phi);
+	free(psi);
+	
+	free(gauss_y);
+	free(gauss_z);
+}
+
+/* invert matrix and calculate determinant using LAPACK */
+void invert(double** cov, int L, double** icov, double* detcov)
+{
+	/* copy square matrix into a vector in column-major order */
+	double* a = (double*) malloc(L*L*sizeof(double));
+	int i, j;
+	for(j=0; j < L; j++)
+		for (i=0; i < L; i++) 
+			a[j*L+i] = cov[i][j];
+	
+	long M = (long) L;	
+	long* ipiv = (long*) malloc(L*L*sizeof(int));
+	long ret;
+	
+	/* LU decomposition */
+	ret = dgetrf_(&M, &M, a, &M, ipiv, &ret);	/* ret should be zero, negative if cov is singular */
+	if (ret < 0)
+	{
+		fprintf(stderr, "Covariance matrix was singular, couldn't invert\n");
+		exit(-1);
+	}
+	
+	/* find determinant */
+	double det = 1;
+	for(i = 0; i < L; i++)
+		det *= a[i*L+i];
+	// TODO: get this to work!!! If detcov < 0 then cov is bad anyway...
+	/*
+	int sign = 1;
+	for (i = 0; i < L; i++)
+		if (ipiv[i] != i)
+			sign = -sign;
+	det *= sign;
+	 */
+	if (det < 0)
+		det = -det;
+	*detcov = det;
+	
+	/* allocate required working storage */
+	long lwork = -1;
+	double lwbest;
+	dgetri_(&M, a, &M, ipiv, &lwbest, &lwork, &ret);
+	lwork = (long) lwbest;	
+	double* work  = (double*) malloc(lwork*sizeof(double));
+	
+	/* find inverse */
+	dgetri_(&M, a, &M, ipiv, work, &lwork, &ret);
+	
+	for(j=0; j < L; j++)
+		for (i=0; i < L; i++) 
+			icov[i][j] = a[j*L+i];	
+	
+	free(work);
+	free(a);	
+}
+
+/* probability of multivariate Gaussian given mean, inverse and determinant of covariance */
+double gauss(double* x, int L, double* mu, double** icov, double detcov, double* y, double* z)
+{
+	int i, j;
+	double s = 0;
+	for (i = 0; i < L; i++)
+		y[i] = x[i] - mu[i];
+	for (i = 0; i < L; i++)
+	{
+		//z[i] = 0;
+		//for (j = 0; j < L; j++)
+		//	z[i] += icov[i][j] *  y[j];
+		z[i] = cblas_ddot(L, &icov[i][0], 1, y, 1);
+	}
+	s = cblas_ddot(L, z, 1, y, 1);
+	//for (i = 0; i < L; i++)
+	//	s += z[i] * y[i];	
+	
+	return exp(-s/2.0) / (pow(2*PI, L/2.0) * sqrt(detcov));
+}
+
+/* log probability of multivariate Gaussian given mean, inverse and determinant of covariance */
+double loggauss(double* x, int L, double* mu, double** icov, double detcov, double* y, double* z)
+{
+	int i, j;
+	double s = 0;
+	double ret;
+	for (i = 0; i < L; i++)
+		y[i] = x[i] - mu[i];
+	for (i = 0; i < L; i++)
+	{
+		//z[i] = 0;
+		//for (j = 0; j < L; j++)
+		//	z[i] += icov[i][j] *  y[j];
+		z[i] = cblas_ddot(L, &icov[i][0], 1, y, 1);
+	}
+	s = cblas_ddot(L, z, 1, y, 1);
+	//for (i = 0; i < L; i++)
+	//	s += z[i] * y[i];	
+	
+	ret = -0.5 * (s + L * log(2*PI) + log(detcov));
+	
+	/*
+	// TEST
+	if (isinf(ret) > 0)
+		printf("loggauss returning infinity\n");
+	if (isinf(ret) < 0)
+		printf("loggauss returning -infinity\n");
+	if (isnan(ret))
+		printf("loggauss returning nan\n");	
+	*/
+	
+	return ret;
+}
+
+void hmm_print(model_t* model)
+{
+	int i, j;
+	printf("p0:\n");
+	for (i = 0; i < model->N; i++)
+		printf("%f ", model->p0[i]);
+	printf("\n\n");
+	printf("a:\n");
+	for (i = 0; i < model->N; i++)
+	{
+		for (j = 0; j < model->N; j++)
+			printf("%f ", model->a[i][j]);
+		printf("\n");
+	}
+	printf("\n\n");
+	printf("mu:\n");
+	for (i = 0; i < model->N; i++)
+	{
+		for (j = 0; j < model->L; j++)
+			printf("%f ", model->mu[i][j]);
+		printf("\n");
+	}
+	printf("\n\n");
+	printf("cov:\n");
+	for (i = 0; i < model->L; i++)
+	{
+		for (j = 0; j < model->L; j++)
+			printf("%f ", model->cov[i][j]);
+		printf("\n");
+	}
+	printf("\n\n");
+}
+
+