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   1 #include "utils.h"
   2 #include <stdlib.h>
   3 #include <math.h>
   4 #include <gsl/gsl_linalg.h>
   5
   6 // TODO: don't recompute indexes ai(...) and mi(...) when possible
   7 void EMGLLF_core(
   8     // IN parameters
   9     const Real* phiInit, // parametre initial de moyenne renormalisé
  10     const Real* rhoInit, // parametre initial de variance renormalisé
  11     const Real* piInit,  // parametre initial des proportions
  12     const Real* gamInit, // paramètre initial des probabilités a posteriori de chaque échantillon
  13     int mini, // nombre minimal d'itérations dans l'algorithme EM
  14     int maxi, // nombre maximal d'itérations dans l'algorithme EM
  15     Real gamma, // puissance des proportions dans la pénalisation pour un Lasso adaptatif
  16     Real lambda, // valeur du paramètre de régularisation du Lasso
  17     const Real* X, // régresseurs
  18     const Real* Y, // réponse
  19     Real tau, // seuil pour accepter la convergence
  20     // OUT parameters (all pointers, to be modified)
  21     Real* phi, // parametre de moyenne renormalisé, calculé par l'EM
  22     Real* rho, // parametre de variance renormalisé, calculé par l'EM
  23     Real* pi, // parametre des proportions renormalisé, calculé par l'EM
  24     Real* llh, // (derniere) log vraisemblance associée à cet échantillon,
  25                // pour les valeurs estimées des paramètres
  26     Real* S,
  27     int* affec,
  28     // additional size parameters
  29     int n, // nombre d'echantillons
  30     int p, // nombre de covariables
  31     int m, // taille de Y (multivarié)
  32     int k) // nombre de composantes dans le mélange
  33 {
  34     //Initialize outputs
  35     copyArray(phiInit, phi, p*m*k);
  36     copyArray(rhoInit, rho, m*m*k);
  37     copyArray(piInit, pi, k);
  38     //S is already allocated, and doesn't need to be 'zeroed'
  39
  40     //Other local variables: same as in R
  41     Real* gam = (Real*)malloc(n*k*sizeof(Real));
  42     copyArray(gamInit, gam, n*k);
  43     Real* Gram2 = (Real*)malloc(p*p*k*sizeof(Real));
  44     Real* ps2 = (Real*)malloc(p*m*k*sizeof(Real));
  45     Real* b = (Real*)malloc(k*sizeof(Real));
  46     Real* X2 = (Real*)malloc(n*p*k*sizeof(Real));
  47     Real* Y2 = (Real*)malloc(n*m*k*sizeof(Real));
  48     *llh = -INFINITY;
  49     Real* pi2 = (Real*)malloc(k*sizeof(Real));
  50     const Real EPS = 1e-15;
  51     // Additional (not at this place, in R file)
  52     Real* gam2 = (Real*)malloc(k*sizeof(Real));
  53     Real* sqNorm2 = (Real*)malloc(k*sizeof(Real));
  54     Real* detRho = (Real*)malloc(k*sizeof(Real));
  55     gsl_matrix* matrix = gsl_matrix_alloc(m, m);
  56     gsl_permutation* permutation = gsl_permutation_alloc(m);
  57     Real* YiRhoR = (Real*)malloc(m*sizeof(Real));
  58     Real* XiPhiR = (Real*)malloc(m*sizeof(Real));
  59     const Real gaussConstM = pow(2.*M_PI,m/2.);
  60     Real* Phi = (Real*)malloc(p*m*k*sizeof(Real));
  61     Real* Rho = (Real*)malloc(m*m*k*sizeof(Real));
  62     Real* Pi = (Real*)malloc(k*sizeof(Real));
  63
  64     for (int ite=1; ite<=maxi; ite++)
  65     {
  66         copyArray(phi, Phi, p*m*k);
  67         copyArray(rho, Rho, m*m*k);
  68         copyArray(pi, Pi, k);
  69
  70         // Calculs associés a Y et X
  71         for (int r=0; r<k; r++)
  72         {
  73             for (int mm=0; mm<m; mm++)
  74             {
  75                 //Y2[,mm,r] = sqrt(gam[,r]) * Y[,mm]
  76                 for (int u=0; u<n; u++)
  77                     Y2[ai(u,mm,r,n,m,k)] = sqrt(gam[mi(u,r,n,k)]) * Y[mi(u,mm,n,m)];
  78             }
  79             for (int i=0; i<n; i++)
  80             {
  81                 //X2[i,,r] = sqrt(gam[i,r]) * X[i,]
  82                 for (int u=0; u<p; u++)
  83                     X2[ai(i,u,r,n,p,k)] = sqrt(gam[mi(i,r,n,k)]) * X[mi(i,u,n,p)];
  84             }
  85             for (int mm=0; mm<m; mm++)
  86             {
  87                 //ps2[,mm,r] = crossprod(X2[,,r],Y2[,mm,r])
  88                 for (int u=0; u<p; u++)
  89                 {
  90                     Real dotProduct = 0.;
  91                     for (int v=0; v<n; v++)
  92                         dotProduct += X2[ai(v,u,r,n,p,k)] * Y2[ai(v,mm,r,n,m,k)];
  93                     ps2[ai(u,mm,r,p,m,k)] = dotProduct;
  94                 }
  95             }
  96             for (int j=0; j<p; j++)
  97             {
  98                 for (int s=0; s<p; s++)
  99                 {
 100                     //Gram2[j,s,r] = crossprod(X2[,j,r], X2[,s,r])
 101                     Real dotProduct = 0.;
 102                     for (int u=0; u<n; u++)
 103                         dotProduct += X2[ai(u,j,r,n,p,k)] * X2[ai(u,s,r,n,p,k)];
 104                     Gram2[ai(j,s,r,p,p,k)] = dotProduct;
 105                 }
 106             }
 107         }
 108
 109         /////////////
 110         // Etape M //
 111         /////////////
 112
 113         // Pour pi
 114         for (int r=0; r<k; r++)
 115         {
 116             //b[r] = sum(abs(phi[,,r]))
 117             Real sumAbsPhi = 0.;
 118             for (int u=0; u<p; u++)
 119                 for (int v=0; v<m; v++)
 120                     sumAbsPhi += fabs(phi[ai(u,v,r,p,m,k)]);
 121             b[r] = sumAbsPhi;
 122         }
 123         //gam2 = colSums(gam)
 124         for (int u=0; u<k; u++)
 125         {
 126             Real sumOnColumn = 0.;
 127             for (int v=0; v<n; v++)
 128                 sumOnColumn += gam[mi(v,u,n,k)];
 129             gam2[u] = sumOnColumn;
 130         }
 131         //a = sum(gam %*% log(pi))
 132         Real a = 0.;
 133         for (int u=0; u<n; u++)
 134         {
 135             Real dotProduct = 0.;
 136             for (int v=0; v<k; v++)
 137                 dotProduct += gam[mi(u,v,n,k)] * log(pi[v]);
 138             a += dotProduct;
 139         }
 140
 141         //tant que les proportions sont negatives
 142         int kk = 0,
 143             pi2AllPositive = 0;
 144         Real invN = 1./n;
 145         while (!pi2AllPositive)
 146         {
 147             //pi2 = pi + 0.1^kk * ((1/n)*gam2 - pi)
 148             Real pow_01_kk = pow(0.1,kk);
 149             for (int r=0; r<k; r++)
 150                 pi2[r] = pi[r] + pow_01_kk * (invN*gam2[r] - pi[r]);
 151             //pi2AllPositive = all(pi2 >= 0)
 152             pi2AllPositive = 1;
 153             for (int r=0; r<k; r++)
 154             {
 155                 if (pi2[r] < 0)
 156                 {
 157                     pi2AllPositive = 0;
 158                     break;
 159                 }
 160             }
 161             kk++;
 162         }
 163
 164         //sum(pi^gamma * b)
 165         Real piPowGammaDotB = 0.;
 166         for (int v=0; v<k; v++)
 167             piPowGammaDotB += pow(pi[v],gamma) * b[v];
 168         //sum(pi2^gamma * b)
 169         Real pi2PowGammaDotB = 0.;
 170         for (int v=0; v<k; v++)
 171             pi2PowGammaDotB += pow(pi2[v],gamma) * b[v];
 172         //sum(gam2 * log(pi2))
 173         Real gam2DotLogPi2 = 0.;
 174         for (int v=0; v<k; v++)
 175             gam2DotLogPi2 += gam2[v] * log(pi2[v]);
 176
 177         //t(m) la plus grande valeur dans la grille O.1^k tel que ce soit décroissante ou constante
 178         while (-invN*a + lambda*piPowGammaDotB < -invN*gam2DotLogPi2 + lambda*pi2PowGammaDotB
 179             && kk<1000)
 180         {
 181             Real pow_01_kk = pow(0.1,kk);
 182             //pi2 = pi + 0.1^kk * (1/n*gam2 - pi)
 183             for (int v=0; v<k; v++)
 184                 pi2[v] = pi[v] + pow_01_kk * (invN*gam2[v] - pi[v]);
 185             //pi2 was updated, so we recompute pi2PowGammaDotB and gam2DotLogPi2
 186             pi2PowGammaDotB = 0.;
 187             for (int v=0; v<k; v++)
 188                 pi2PowGammaDotB += pow(pi2[v],gamma) * b[v];
 189             gam2DotLogPi2 = 0.;
 190             for (int v=0; v<k; v++)
 191                 gam2DotLogPi2 += gam2[v] * log(pi2[v]);
 192             kk++;
 193         }
 194         Real t = pow(0.1,kk);
 195         //sum(pi + t*(pi2-pi))
 196         Real sumPiPlusTbyDiff = 0.;
 197         for (int v=0; v<k; v++)
 198             sumPiPlusTbyDiff += (pi[v] + t*(pi2[v] - pi[v]));
 199         //pi = (pi + t*(pi2-pi)) / sum(pi + t*(pi2-pi))
 200         for (int v=0; v<k; v++)
 201             pi[v] = (pi[v] + t*(pi2[v] - pi[v])) / sumPiPlusTbyDiff;
 202
 203         //Pour phi et rho
 204         for (int r=0; r<k; r++)
 205         {
 206             for (int mm=0; mm<m; mm++)
 207             {
 208                 Real ps = 0.,
 209                     nY2 = 0.;
 210                 // Compute ps, and nY2 = sum(Y2[,mm,r]^2)
 211                 for (int i=0; i<n; i++)
 212                 {
 213                     //< X2[i,,r] , phi[,mm,r] >
 214                     Real dotProduct = 0.;
 215                     for (int u=0; u<p; u++)
 216                         dotProduct += X2[ai(i,u,r,n,p,k)] * phi[ai(u,mm,r,p,m,k)];
 217                     //ps = ps + Y2[i,mm,r] * sum(X2[i,,r] * phi[,mm,r])
 218                     ps += Y2[ai(i,mm,r,n,m,k)] * dotProduct;
 219                     nY2 += Y2[ai(i,mm,r,n,m,k)] * Y2[ai(i,mm,r,n,m,k)];
 220                 }
 221                 //rho[mm,mm,r] = (ps+sqrt(ps^2+4*nY2*gam2[r])) / (2*nY2)
 222                 rho[ai(mm,mm,r,m,m,k)] = (ps + sqrt(ps*ps + 4*nY2 * gam2[r])) / (2*nY2);
 223             }
 224         }
 225
 226         for (int r=0; r<k; r++)
 227         {
 228             for (int j=0; j<p; j++)
 229             {
 230                 for (int mm=0; mm<m; mm++)
 231                 {
 232                     //sum(phi[-j,mm,r] * Gram2[j,-j,r])
 233                     Real phiDotGram2 = 0.;
 234                     for (int u=0; u<p; u++)
 235                     {
 236                         if (u != j)
 237                             phiDotGram2 += phi[ai(u,mm,r,p,m,k)] * Gram2[ai(j,u,r,p,p,k)];
 238                     }
 239                     //S[j,mm,r] = -rho[mm,mm,r]*ps2[j,mm,r] + sum(phi[-j,mm,r] * Gram2[j,-j,r])
 240                     S[ai(j,mm,r,p,m,k)] = -rho[ai(mm,mm,r,m,m,k)] * ps2[ai(j,mm,r,p,m,k)]
 241                         + phiDotGram2;
 242                     Real pirPowGamma = pow(pi[r],gamma);
 243                     if (fabs(S[ai(j,mm,r,p,m,k)]) <= n*lambda*pirPowGamma)
 244                         phi[ai(j,mm,r,p,m,k)] = 0.;
 245                     else if (S[ai(j,mm,r,p,m,k)] > n*lambda*pirPowGamma)
 246                     {
 247                         phi[ai(j,mm,r,p,m,k)] = (n*lambda*pirPowGamma - S[ai(j,mm,r,p,m,k)])
 248                             / Gram2[ai(j,j,r,p,p,k)];
 249                     }
 250                     else
 251                     {
 252                         phi[ai(j,mm,r,p,m,k)] = -(n*lambda*pirPowGamma + S[ai(j,mm,r,p,m,k)])
 253                             / Gram2[ai(j,j,r,p,p,k)];
 254                     }
 255                 }
 256             }
 257         }
 258
 259         /////////////
 260         // Etape E //
 261         /////////////
 262
 263         // Precompute det(rho[,,r]) for r in 1...k
 264         int signum;
 265         for (int r=0; r<k; r++)
 266         {
 267             for (int u=0; u<m; u++)
 268             {
 269                 for (int v=0; v<m; v++)
 270                     matrix->data[u*m+v] = rho[ai(u,v,r,m,m,k)];
 271             }
 272             gsl_linalg_LU_decomp(matrix, permutation, &signum);
 273             detRho[r] = gsl_linalg_LU_det(matrix, signum);
 274         }
 275
 276         Real sumLogLLH = 0.;
 277         for (int i=0; i<n; i++)
 278         {
 279             for (int r=0; r<k; r++)
 280             {
 281                 //compute Y[i,]%*%rho[,,r]
 282                 for (int u=0; u<m; u++)
 283                 {
 284                     YiRhoR[u] = 0.;
 285                     for (int v=0; v<m; v++)
 286                         YiRhoR[u] += Y[mi(i,v,n,m)] * rho[ai(v,u,r,m,m,k)];
 287                 }
 288
 289                 //compute X[i,]%*%phi[,,r]
 290                 for (int u=0; u<m; u++)
 291                 {
 292                     XiPhiR[u] = 0.;
 293                     for (int v=0; v<p; v++)
 294                         XiPhiR[u] += X[mi(i,v,n,p)] * phi[ai(v,u,r,p,m,k)];
 295                 }
 296
 297                 //compute sq norm || Y(:,i)*rho(:,:,r)-X(i,:)*phi(:,:,r) ||_2^2
 298                 sqNorm2[r] = 0.;
 299                 for (int u=0; u<m; u++)
 300                     sqNorm2[r] += (YiRhoR[u]-XiPhiR[u]) * (YiRhoR[u]-XiPhiR[u]);
 301             }
 302
 303             Real sumGamI = 0.;
 304             for (int r=0; r<k; r++)
 305             {
 306                 gam[mi(i,r,n,k)] = pi[r] * exp(-.5*sqNorm2[r]) * detRho[r];
 307                 sumGamI += gam[mi(i,r,n,k)];
 308             }
 309
 310             sumLogLLH += log(sumGamI) - log(gaussConstM);
 311             if (sumGamI > EPS) //else: gam[i,] is already ~=0
 312             {
 313                 for (int r=0; r<k; r++)
 314                     gam[mi(i,r,n,k)] /= sumGamI;
 315             }
 316         }
 317
 318         //sumPen = sum(pi^gamma * b)
 319         Real sumPen = 0.;
 320         for (int r=0; r<k; r++)
 321             sumPen += pow(pi[r],gamma) * b[r];
 322         Real last_llh = *llh;
 323         //llh = -sumLogLLH/n + lambda*sumPen
 324         *llh = -invN * sumLogLLH + lambda * sumPen;
 325         Real dist = ite==1 ? *llh : (*llh - last_llh) / (1. + fabs(*llh));
 326
 327         //Dist1 = max( abs(phi-Phi) / (1+abs(phi)) )
 328         Real Dist1 = 0.;
 329         for (int u=0; u<p; u++)
 330         {
 331             for (int v=0; v<m; v++)
 332             {
 333                 for (int w=0; w<k; w++)
 334                 {
 335                     Real tmpDist = fabs(phi[ai(u,v,w,p,m,k)]-Phi[ai(u,v,w,p,m,k)])
 336                         / (1.+fabs(phi[ai(u,v,w,p,m,k)]));
 337                     if (tmpDist > Dist1)
 338                         Dist1 = tmpDist;
 339                 }
 340             }
 341         }
 342         //Dist2 = max( (abs(rho-Rho)) / (1+abs(rho)) )
 343         Real Dist2 = 0.;
 344         for (int u=0; u<m; u++)
 345         {
 346             for (int v=0; v<m; v++)
 347             {
 348                 for (int w=0; w<k; w++)
 349                 {
 350                     Real tmpDist = fabs(rho[ai(u,v,w,m,m,k)]-Rho[ai(u,v,w,m,m,k)])
 351                         / (1.+fabs(rho[ai(u,v,w,m,m,k)]));
 352                     if (tmpDist > Dist2)
 353                         Dist2 = tmpDist;
 354                 }
 355             }
 356         }
 357         //Dist3 = max( (abs(pi-Pi)) / (1+abs(Pi)))
 358         Real Dist3 = 0.;
 359         for (int u=0; u<n; u++)
 360         {
 361             for (int v=0; v<k; v++)
 362             {
 363                 Real tmpDist = fabs(pi[v]-Pi[v]) / (1.+fabs(pi[v]));
 364                 if (tmpDist > Dist3)
 365                     Dist3 = tmpDist;
 366             }
 367         }
 368         //dist2=max([max(Dist1),max(Dist2),max(Dist3)]);
 369         Real dist2 = Dist1;
 370         if (Dist2 > dist2)
 371             dist2 = Dist2;
 372         if (Dist3 > dist2)
 373             dist2 = Dist3;
 374
 375         if (ite >= mini && (dist >= tau || dist2 >= sqrt(tau)))
 376             break;
 377     }
 378
 379     //affec = apply(gam, 1, which.max)
 380     for (int i=0; i<n; i++)
 381     {
 382         Real rowMax = 0.;
 383         affec[i] = 0;
 384         for (int j=0; j<k; j++)
 385         {
 386             if (gam[mi(i,j,n,k)] > rowMax)
 387             {
 388                 affec[i] = j+1; //R indices start at 1
 389                 rowMax = gam[mi(i,j,n,k)];
 390             }
 391         }
 392     }
 393
 394     //free memory
 395     free(b);
 396     free(gam);
 397     free(Phi);
 398     free(Rho);
 399     free(Pi);
 400     free(Gram2);
 401     free(ps2);
 402     free(detRho);
 403     gsl_matrix_free(matrix);
 404     gsl_permutation_free(permutation);
 405     free(XiPhiR);
 406     free(YiRhoR);
 407     free(gam2);
 408     free(pi2);
 409     free(X2);
 410     free(Y2);
 411     free(sqNorm2);
 412 }