/* log_weight_div_det[k] = -2*log(weights_k) + log(det(Sigma_k)))
covs[k] = cov_rotate_mats[k] * cov_eigen_values[k] * (cov_rotate_mats[k])'
cov_rotate_mats[k] are orthogonal matrices of eigenvectors and
cov_eigen_values[k] are diagonal matrices (represented by 1D vectors) of eigen values.
The <alpha_ik> is the probability of the vector x_i to belong to the k-th cluster:
<alpha_ik> ~ weights_k * exp{ -0.5[ln(det(Sigma_k)) + (x_i - mu_k)' Sigma_k^(-1) (x_i - mu_k)] }
We calculate these probabilities here by the equivalent formulae:
Denote
S_ik = -0.5(log(det(Sigma_k)) + (x_i - mu_k)' Sigma_k^(-1) (x_i - mu_k)) + log(weights_k),
M_i = max_k S_ik = S_qi, so that the q-th class is the one where maximum reaches. Then
alpha_ik = exp{ S_ik - M_i } / ( 1 + sum_j!=q exp{ S_ji - M_i })
*/
double CvEM::run_em( const CvVectors& train_data )
{
CvMat* centered_sample = 0;
CvMat* covs_item = 0;
CvMat* log_det = 0;
CvMat* log_weights = 0;
CvMat* cov_eigen_values = 0;
CvMat* samples = 0;
CvMat* sum_probs = 0;
log_likelihood = -DBL_MAX;
CV_FUNCNAME( "CvEM::run_em" );
__BEGIN__;
int nsamples = train_data.count, dims = train_data.dims, nclusters = params.nclusters;
double min_variation = FLT_EPSILON;
double min_det_value = MAX( DBL_MIN, pow( min_variation, dims ));
double likelihood_bias = -CV_LOG2PI * (double)nsamples * (double)dims / 2., _log_likelihood = -DBL_MAX;
int start_step = params.start_step;
int i, j, k, n;
int is_general = 0, is_diagonal = 0, is_spherical = 0;
double prev_log_likelihood = -DBL_MAX / 1000., det, d;
CvMat whdr, iwhdr, diag, *w, *iw;
double* w_data;
double* sp_data;
if( nclusters == 1 )
{
double log_weight;
CV_CALL( cvSet( probs, cvScalar(1.)) );
if( params.cov_mat_type == COV_MAT_SPHERICAL )
{
d = cvTrace(*covs).val[0]/dims;
d = MAX( d, FLT_EPSILON );
inv_eigen_values->data.db[0] = 1./d;
log_weight = pow( d, dims*0.5 );
}
else
{
w_data = inv_eigen_values->data.db;
if( params.cov_mat_type == COV_MAT_GENERIC )
cvSVD( *covs, inv_eigen_values, *cov_rotate_mats, 0, CV_SVD_U_T );
else
cvTranspose( cvGetDiag(*covs, &diag), inv_eigen_values );
cvMaxS( inv_eigen_values, FLT_EPSILON, inv_eigen_values );
for( j = 0, det = 1.; j < dims; j++ )
det *= w_data[j];
log_weight = sqrt(det);
cvDiv( 0, inv_eigen_values, inv_eigen_values );
}
log_weight_div_det->data.db[0] = -2*log(weights->data.db[0]/log_weight);
log_likelihood = DBL_MAX/1000.;
EXIT;
}
if( params.cov_mat_type == COV_MAT_GENERIC )
is_general = 1;
else if( params.cov_mat_type == COV_MAT_DIAGONAL )
is_diagonal = 1;
else if( params.cov_mat_type == COV_MAT_SPHERICAL )
is_spherical = 1;
/* In the case of <cov_mat_type> == COV_MAT_DIAGONAL, the k-th row of cov_eigen_values
contains the diagonal elements (variations). In the case of
<cov_mat_type> == COV_MAT_SPHERICAL - the 0-ths elements of the vectors cov_eigen_values[k]
are to be equal to the mean of the variations over all the dimensions. */
CV_CALL( log_det = cvCreateMat( 1, nclusters, CV_64FC1 ));
CV_CALL( log_weights = cvCreateMat( 1, nclusters, CV_64FC1 ));
CV_CALL( covs_item = cvCreateMat( dims, dims, CV_64FC1 ));
CV_CALL( centered_sample = cvCreateMat( 1, dims, CV_64FC1 ));
CV_CALL( cov_eigen_values = cvCreateMat( inv_eigen_values->rows, inv_eigen_values->cols, CV_64FC1 ));
CV_CALL( samples = cvCreateMat( nsamples, dims, CV_64FC1 ));
CV_CALL( sum_probs = cvCreateMat( 1, nclusters, CV_64FC1 ));
sp_data = sum_probs->data.db;
// copy the training data into double-precision matrix
for( i = 0; i < nsamples; i++ )
{
const float* src = train_data.data.fl[i];
double* dst = (double*)(samples->data.ptr + samples->step*i);
for( j = 0; j < dims; j++ )
dst[j] = src[j];
}
if( start_step != START_M_STEP )
{
for( k = 0; k < nclusters; k++ )
{
if( is_general || is_diagonal )
{
w = cvGetRow( cov_eigen_values, &whdr, k );
if( is_general )
cvSVD( covs[k], w, cov_rotate_mats[k], 0, CV_SVD_U_T );
else
cvTranspose( cvGetDiag( covs[k], &diag ), w );
w_data = w->data.db;
for( j = 0, det = 1.; j < dims; j++ )
det *= w_data[j];
if( det < min_det_value )
{
if( start_step == START_AUTO_STEP )
det = min_det_value;
else
EXIT;
}
log_det->data.db[k] = det;
}
else
{
d = cvTrace(covs[k]).val[0]/(double)dims;
if( d < min_variation )
{
if( start_step == START_AUTO_STEP )
d = min_variation;
else
EXIT;
}
cov_eigen_values->data.db[k] = d;
log_det->data.db[k] = d;
}
}
cvLog( log_det, log_det );
if( is_spherical )
cvScale( log_det, log_det, dims );
}
for( n = 0; n < params.term_crit.max_iter; n++ )
{
if( n > 0 || start_step != START_M_STEP )
{
// e-step: compute probs_ik from means_k, covs_k and weights_k.
CV_CALL(cvLog( weights, log_weights ));
// S_ik = -0.5[log(det(Sigma_k)) + (x_i - mu_k)' Sigma_k^(-1) (x_i - mu_k)] + log(weights_k)
for( k = 0; k < nclusters; k++ )
{
CvMat* u = cov_rotate_mats[k];
const double* mean = (double*)(means->data.ptr + means->step*k);
w = cvGetRow( cov_eigen_values, &whdr, k );
iw = cvGetRow( inv_eigen_values, &iwhdr, k );
cvDiv( 0, w, iw );
w_data = (double*)(inv_eigen_values->data.ptr + inv_eigen_values->step*k);
for( i = 0; i < nsamples; i++ )
{
double *csample = centered_sample->data.db, p = log_det->data.db[k];
const double* sample = (double*)(samples->data.ptr + samples->step*i);
double* pp = (double*)(probs->data.ptr + probs->step*i);
for( j = 0; j < dims; j++ )
csample[j] = sample[j] - mean[j];
if( is_general )
cvGEMM( centered_sample, u, 1, 0, 0, centered_sample, CV_GEMM_B_T );
for( j = 0; j < dims; j++ )
p += csample[j]*csample[j]*w_data[is_spherical ? 0 : j];
pp[k] = -0.5*p + log_weights->data.db[k];
// S_ik <- S_ik - max_j S_ij
if( k == nclusters - 1 )
{
double max_val = 0;
for( j = 0; j < nclusters; j++ )
max_val = MAX( max_val, pp[j] );
for( j = 0; j < nclusters; j++ )
pp[j] -= max_val;
}
}
}
CV_CALL(cvExp( probs, probs )); // exp( S_ik )
cvZero( sum_probs );
// alpha_ik = exp( S_ik ) / sum_j exp( S_ij ),
// log_likelihood = sum_i log (sum_j exp(S_ij))
for( i = 0, _log_likelihood = likelihood_bias; i < nsamples; i++ )
{
double* pp = (double*)(probs->data.ptr + probs->step*i), sum = 0;
for( j = 0; j < nclusters; j++ )
sum += pp[j];
sum = 1./MAX( sum, DBL_EPSILON );
for( j = 0; j < nclusters; j++ )
{
double p = pp[j] *= sum;
sp_data[j] += p;
}
_log_likelihood -= log( sum );
}
// check termination criteria
if( fabs( (_log_likelihood - prev_log_likelihood) / prev_log_likelihood ) < params.term_crit.epsilon )
break;
prev_log_likelihood = _log_likelihood;
}
// m-step: update means_k, covs_k and weights_k from probs_ik
cvGEMM( probs, samples, 1, 0, 0, means, CV_GEMM_A_T );
for( k = 0; k < nclusters; k++ )
{
double sum = sp_data[k], inv_sum = 1./sum;
CvMat* cov = covs[k], _mean, _sample;
w = cvGetRow( cov_eigen_values, &whdr, k );
w_data = w->data.db;
cvGetRow( means, &_mean, k );
cvGetRow( samples, &_sample, k );
// update weights_k
weights->data.db[k] = sum;
// update means_k
cvScale( &_mean, &_mean, inv_sum );
// compute covs_k
cvZero( cov );
cvZero( w );
for( i = 0; i < nsamples; i++ )
{
double p = probs->data.db[i*nclusters + k]*inv_sum;
_sample.data.db = (double*)(samples->data.ptr + samples->step*i);
if( is_general )
{
cvMulTransposed( &_sample, covs_item, 1, &_mean );
cvScaleAdd( covs_item, cvRealScalar(p), cov, cov );
}
else
for( j = 0; j < dims; j++ )
{
double val = _sample.data.db[j] - _mean.data.db[j];
w_data[is_spherical ? 0 : j] += p*val*val;
}
}
if( is_spherical )
{
d = w_data[0]/(double)dims;
d = MAX( d, min_variation );
w->data.db[0] = d;
log_det->data.db[k] = d;
}
else
{
if( is_general )
cvSVD( cov, w, cov_rotate_mats[k], 0, CV_SVD_U_T );
cvMaxS( w, min_variation, w );
for( j = 0, det = 1.; j < dims; j++ )
det *= w_data[j];
log_det->data.db[k] = det;
}
}
cvConvertScale( weights, weights, 1./(double)nsamples, 0 );
cvMaxS( weights, DBL_MIN, weights );
cvLog( log_det, log_det );
if( is_spherical )
cvScale( log_det, log_det, dims );
} // end of iteration process
//log_weight_div_det[k] = -2*log(weights_k/det(Sigma_k))^0.5) = -2*log(weights_k) + log(det(Sigma_k)))
if( log_weight_div_det )
{
cvScale( log_weights, log_weight_div_det, -2 );
cvAdd( log_weight_div_det, log_det, log_weight_div_det );
}
/* Now finalize all the covariation matrices:
1) if <cov_mat_type> == COV_MAT_DIAGONAL we used array of <w> as diagonals.
Now w[k] should be copied back to the diagonals of covs[k];
2) if <cov_mat_type> == COV_MAT_SPHERICAL we used the 0-th element of w[k]
as an average variation in each cluster. The value of the 0-th element of w[k]
should be copied to the all of the diagonal elements of covs[k]. */
if( is_spherical )
{
for( k = 0; k < nclusters; k++ )
cvSetIdentity( covs[k], cvScalar(cov_eigen_values->data.db[k]));
}
else if( is_diagonal )
{
for( k = 0; k < nclusters; k++ )
cvTranspose( cvGetRow( cov_eigen_values, &whdr, k ),
cvGetDiag( covs[k], &diag ));
}
cvDiv( 0, cov_eigen_values, inv_eigen_values );
log_likelihood = _log_likelihood;
__END__;
cvReleaseMat( &log_det );
cvReleaseMat( &log_weights );
cvReleaseMat( &covs_item );
cvReleaseMat( ¢ered_sample );
cvReleaseMat( &cov_eigen_values );
cvReleaseMat( &samples );
cvReleaseMat( &sum_probs );
return log_likelihood;
}
CvMat* cveGetDiag(CvArr* arr, CvMat* submat, int diag)
{
return cvGetDiag(arr, submat, diag);
}
