src/sources/EMGrank.c

   1 #include <stdlib.h>
   2 #include <gsl/gsl_linalg.h>
   3 #include "utils.h"
   4
   5 // Compute pseudo-inverse of a square matrix
   6 static double* pinv(const double* matrix, int dim)
   7 {
   8     gsl_matrix* U = gsl_matrix_alloc(dim,dim);
   9     gsl_matrix* V = gsl_matrix_alloc(dim,dim);
  10     gsl_vector* S = gsl_vector_alloc(dim);
  11     gsl_vector* work = gsl_vector_alloc(dim);
  12     double EPS = 1e-10; //threshold for singular value "== 0"
  13
  14     //copy matrix into U
  15     copyArray(matrix, U->data, dim*dim);
  16
  17     //U,S,V = SVD of matrix
  18     gsl_linalg_SV_decomp(U, V, S, work);
  19     gsl_vector_free(work);
  20
  21     // Obtain pseudo-inverse by V*S^{-1}*t(U)
  22     double* inverse = (double*)malloc(dim*dim*sizeof(double));
  23     for (int i=0; i<dim; i++)
  24     {
  25         for (int ii=0; ii<dim; ii++)
  26         {
  27             double dotProduct = 0.0;
  28             for (int j=0; j<dim; j++)
  29                 dotProduct += V->data[i*dim+j] * (S->data[j] > EPS ? 1.0/S->data[j] : 0.0) * U->data[ii*dim+j];
  30             inverse[i*dim+ii] = dotProduct;
  31         }
  32     }
  33
  34     gsl_matrix_free(U);
  35     gsl_matrix_free(V);
  36     gsl_vector_free(S);
  37     return inverse;
  38 }
  39
  40 // TODO: comment EMGrank purpose
  41 void EMGrank_core(
  42     // IN parameters
  43     const double* Pi, // parametre de proportion
  44     const double* Rho, // parametre initial de variance renormalisé
  45     int mini, // nombre minimal d'itérations dans l'algorithme EM
  46     int maxi, // nombre maximal d'itérations dans l'algorithme EM
  47     const double* X, // régresseurs
  48     const double* Y, // réponse
  49     double tau, // seuil pour accepter la convergence
  50     const int* rank, // vecteur des rangs possibles
  51     // OUT parameters
  52     double* phi, // parametre de moyenne renormalisé, calculé par l'EM
  53     double* LLF, // log vraisemblance associé à cet échantillon, pour les valeurs estimées des paramètres
  54     // additional size parameters
  55     int n, // taille de l'echantillon
  56     int p, // nombre de covariables
  57     int m, // taille de Y (multivarié)
  58     int k) // nombre de composantes
  59 {
  60     // Allocations, initializations
  61     double* Phi = (double*)calloc(p*m*k,sizeof(double));
  62     double* hatBetaR = (double*)malloc(p*m*sizeof(double));
  63     int signum;
  64     double invN = 1.0/n;
  65     int deltaPhiBufferSize = 20;
  66     double* deltaPhi = (double*)malloc(deltaPhiBufferSize*sizeof(double));
  67     int ite = 0;
  68     double sumDeltaPhi = 0.0;
  69     double* YiRhoR = (double*)malloc(m*sizeof(double));
  70     double* XiPhiR = (double*)malloc(m*sizeof(double));
  71     double* Xr = (double*)malloc(n*p*sizeof(double));
  72     double* Yr = (double*)malloc(n*m*sizeof(double));
  73     double* tXrXr = (double*)malloc(p*p*sizeof(double));
  74     double* tXrYr = (double*)malloc(p*m*sizeof(double));
  75     gsl_matrix* matrixM = gsl_matrix_alloc(p, m);
  76     gsl_matrix* matrixE = gsl_matrix_alloc(m, m);
  77     gsl_permutation* permutation = gsl_permutation_alloc(m);
  78     gsl_matrix* V = gsl_matrix_alloc(m,m);
  79     gsl_vector* S = gsl_vector_alloc(m);
  80     gsl_vector* work = gsl_vector_alloc(m);
  81
  82     //Initialize class memberships (all elements in class 0; TODO: randomize ?)
  83     int* Z = (int*)calloc(n, sizeof(int));
  84
  85     //Initialize phi to zero, because some M loops might exit before phi affectation
  86     zeroArray(phi, p*m*k);
  87
  88     while (ite<mini || (ite<maxi && sumDeltaPhi>tau))
  89     {
  90         /////////////
  91         // Etape M //
  92         /////////////
  93
  94         //M step: Mise à jour de Beta (et donc phi)
  95         for (int r=0; r<k; r++)
  96         {
  97             //Compute Xr = X(Z==r,:) and Yr = Y(Z==r,:)
  98             int cardClustR=0;
  99             for (int i=0; i<n; i++)
 100             {
 101                 if (Z[i] == r)
 102                 {
 103                     for (int j=0; j<p; j++)
 104                         Xr[mi(cardClustR,j,n,p)] = X[mi(i,j,n,p)];
 105                     for (int j=0; j<m; j++)
 106                         Yr[mi(cardClustR,j,n,m)] = Y[mi(i,j,n,m)];
 107                     cardClustR++;
 108                 }
 109             }
 110             if (cardClustR == 0)
 111                 continue;
 112
 113             //Compute tXrXr = t(Xr) * Xr
 114             for (int j=0; j<p; j++)
 115             {
 116                 for (int jj=0; jj<p; jj++)
 117                 {
 118                     double dotProduct = 0.0;
 119                     for (int u=0; u<cardClustR; u++)
 120                         dotProduct += Xr[mi(u,j,n,p)] * Xr[mi(u,jj,n,p)];
 121                     tXrXr[mi(j,jj,p,p)] = dotProduct;
 122                 }
 123             }
 124
 125             //Get pseudo inverse = (t(Xr)*Xr)^{-1}
 126             double* invTXrXr = pinv(tXrXr, p);
 127
 128             // Compute tXrYr = t(Xr) * Yr
 129             for (int j=0; j<p; j++)
 130             {
 131                 for (int jj=0; jj<m; jj++)
 132                 {
 133                     double dotProduct = 0.0;
 134                     for (int u=0; u<cardClustR; u++)
 135                         dotProduct += Xr[mi(u,j,n,p)] * Yr[mi(u,j,n,m)];
 136                     tXrYr[mi(j,jj,p,m)] = dotProduct;
 137                 }
 138             }
 139
 140             //Fill matrixM with inverse * tXrYr = (t(Xr)*Xr)^{-1} * t(Xr) * Yr
 141             for (int j=0; j<p; j++)
 142             {
 143                 for (int jj=0; jj<m; jj++)
 144                 {
 145                     double dotProduct = 0.0;
 146                     for (int u=0; u<p; u++)
 147                         dotProduct += invTXrXr[mi(j,u,p,p)] * tXrYr[mi(u,jj,p,m)];
 148                     matrixM->data[j*m+jj] = dotProduct;
 149                 }
 150             }
 151             free(invTXrXr);
 152
 153             //U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr
 154             gsl_linalg_SV_decomp(matrixM, V, S, work);
 155
 156             //Set m-rank(r) singular values to zero, and recompose
 157             //best rank(r) approximation of the initial product
 158             for (int j=rank[r]; j<m; j++)
 159                 S->data[j] = 0.0;
 160
 161             //[intermediate step] Compute hatBetaR = U * S * t(V)
 162             double* U = matrixM->data;
 163             for (int j=0; j<p; j++)
 164             {
 165                 for (int jj=0; jj<m; jj++)
 166                 {
 167                     double dotProduct = 0.0;
 168                     for (int u=0; u<m; u++)
 169                         dotProduct += U[j*m+u] * S->data[u] * V->data[jj*m+u];
 170                     hatBetaR[mi(j,jj,p,m)] = dotProduct;
 171                 }
 172             }
 173
 174             //Compute phi(:,:,r) = hatBetaR * Rho(:,:,r)
 175             for (int j=0; j<p; j++)
 176             {
 177                 for (int jj=0; jj<m; jj++)
 178                 {
 179                     double dotProduct=0.0;
 180                     for (int u=0; u<m; u++)
 181                         dotProduct += hatBetaR[mi(j,u,p,m)] * Rho[ai(u,jj,r,m,m,k)];
 182                     phi[ai(j,jj,r,p,m,k)] = dotProduct;
 183                 }
 184         }
 185         }
 186
 187         /////////////
 188         // Etape E //
 189         /////////////
 190
 191         double sumLogLLF2 = 0.0;
 192         for (int i=0; i<n; i++)
 193         {
 194             double sumLLF1 = 0.0;
 195             double maxLogGamIR = -INFINITY;
 196             for (int r=0; r<k; r++)
 197             {
 198                 //Compute
 199                 //Gam(i,r) = Pi(r) * det(Rho(:,:,r)) * exp( -1/2 * (Y(i,:)*Rho(:,:,r) - X(i,:)...
 200                 //*phi(:,:,r)) * transpose( Y(i,:)*Rho(:,:,r) - X(i,:)*phi(:,:,r) ) );
 201                 //split in several sub-steps
 202
 203                 //compute det(Rho(:,:,r)) [TODO: avoid re-computations]
 204                 for (int j=0; j<m; j++)
 205                 {
 206                     for (int jj=0; jj<m; jj++)
 207                         matrixE->data[j*m+jj] = Rho[ai(j,jj,r,m,m,k)];
 208                 }
 209                 gsl_linalg_LU_decomp(matrixE, permutation, &signum);
 210                 double detRhoR = gsl_linalg_LU_det(matrixE, signum);
 211
 212                 //compute Y(i,:)*Rho(:,:,r)
 213                 for (int j=0; j<m; j++)
 214                 {
 215                     YiRhoR[j] = 0.0;
 216                     for (int u=0; u<m; u++)
 217                         YiRhoR[j] += Y[mi(i,u,n,m)] * Rho[ai(u,j,r,m,m,k)];
 218                 }
 219
 220                 //compute X(i,:)*phi(:,:,r)
 221                 for (int j=0; j<m; j++)
 222                 {
 223                     XiPhiR[j] = 0.0;
 224                     for (int u=0; u<p; u++)
 225                         XiPhiR[j] += X[mi(i,u,n,p)] * phi[ai(u,j,r,p,m,k)];
 226                 }
 227
 228                 //compute dotProduct < Y(:,i)*rho(:,:,r)-X(i,:)*phi(:,:,r) . Y(:,i)*rho(:,:,r)-X(i,:)*phi(:,:,r) >
 229                 double dotProduct = 0.0;
 230                 for (int u=0; u<m; u++)
 231                     dotProduct += (YiRhoR[u]-XiPhiR[u]) * (YiRhoR[u]-XiPhiR[u]);
 232                 double logGamIR = log(Pi[r]) + log(detRhoR) - 0.5*dotProduct;
 233
 234                 //Z(i) = index of max (gam(i,:))
 235                 if (logGamIR > maxLogGamIR)
 236                 {
 237                     Z[i] = r;
 238                     maxLogGamIR = logGamIR;
 239                 }
 240                 sumLLF1 += exp(logGamIR) / pow(2*M_PI,m/2.0);
 241             }
 242
 243             sumLogLLF2 += log(sumLLF1);
 244         }
 245
 246         // Assign output variable LLF
 247         *LLF = -invN * sumLogLLF2;
 248
 249         //newDeltaPhi = max(max((abs(phi-Phi))./(1+abs(phi))));
 250         double newDeltaPhi = 0.0;
 251         for (int j=0; j<p; j++)
 252         {
 253             for (int jj=0; jj<m; jj++)
 254             {
 255                 for (int r=0; r<k; r++)
 256                 {
 257                     double tmpDist = fabs(phi[ai(j,jj,r,p,m,k)]-Phi[ai(j,jj,r,p,m,k)])
 258                         / (1.0+fabs(phi[ai(j,jj,r,p,m,k)]));
 259                     if (tmpDist > newDeltaPhi)
 260                         newDeltaPhi = tmpDist;
 261                 }
 262             }
 263         }
 264
 265         //update distance parameter to check algorithm convergence (delta(phi, Phi))
 266         //TODO: deltaPhi should be a linked list for perf.
 267         if (ite < deltaPhiBufferSize)
 268             deltaPhi[ite] = newDeltaPhi;
 269         else
 270         {
 271             sumDeltaPhi -= deltaPhi[0];
 272             for (int u=0; u<deltaPhiBufferSize-1; u++)
 273                 deltaPhi[u] = deltaPhi[u+1];
 274             deltaPhi[deltaPhiBufferSize-1] = newDeltaPhi;
 275         }
 276         sumDeltaPhi += newDeltaPhi;
 277
 278         // update other local variables
 279         for (int j=0; j<m; j++)
 280         {
 281             for (int jj=0; jj<p; jj++)
 282             {
 283                 for (int r=0; r<k; r++)
 284                     Phi[ai(j,jj,r,p,m,k)] = phi[ai(j,jj,r,p,m,k)];
 285             }
 286         }
 287         ite++;
 288     }
 289
 290     //free memory
 291     free(hatBetaR);
 292     free(deltaPhi);
 293     free(Phi);
 294     gsl_matrix_free(matrixE);
 295     gsl_matrix_free(matrixM);
 296     gsl_permutation_free(permutation);
 297     gsl_vector_free(work);
 298     gsl_matrix_free(V);
 299     gsl_vector_free(S);
 300     free(XiPhiR);
 301     free(YiRhoR);
 302     free(Xr);
 303     free(Yr);
 304     free(tXrXr);
 305     free(tXrYr);
 306     free(Z);
 307 }