2 #include <gsl/gsl_linalg.h>
5 // Compute pseudo-inverse of a square matrix
6 static Real* pinv(const Real* matrix, int dim)
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 Real EPS = 1e-10; //threshold for singular value "== 0"
15 copyArray(matrix, U->data, dim*dim);
17 //U,S,V = SVD of matrix
18 gsl_linalg_SV_decomp(U, V, S, work);
19 gsl_vector_free(work);
21 // Obtain pseudo-inverse by V*S^{-1}*t(U)
22 Real* inverse = (Real*)malloc(dim*dim*sizeof(Real));
23 for (int i=0; i<dim; i++)
25 for (int ii=0; ii<dim; ii++)
27 Real 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;
40 // TODO: comment EMGrank purpose
43 const Real* Pi, // parametre de proportion
44 const Real* 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 Real* X, // régresseurs
48 const Real* Y, // réponse
49 Real tau, // seuil pour accepter la convergence
50 const int* rank, // vecteur des rangs possibles
52 Real* phi, // parametre de moyenne renormalisé, calculé par l'EM
53 Real* 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
60 // Allocations, initializations
61 Real* Phi = (Real*)calloc(p*m*k,sizeof(Real));
62 Real* hatBetaR = (Real*)malloc(p*m*sizeof(Real));
65 int deltaPhiBufferSize = 20;
66 Real* deltaPhi = (Real*)malloc(deltaPhiBufferSize*sizeof(Real));
68 Real sumDeltaPhi = 0.0;
69 Real* YiRhoR = (Real*)malloc(m*sizeof(Real));
70 Real* XiPhiR = (Real*)malloc(m*sizeof(Real));
71 Real* Xr = (Real*)malloc(n*p*sizeof(Real));
72 Real* Yr = (Real*)malloc(n*m*sizeof(Real));
73 Real* tXrXr = (Real*)malloc(p*p*sizeof(Real));
74 Real* tXrYr = (Real*)malloc(p*m*sizeof(Real));
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);
82 //Initialize class memberships (all elements in class 0; TODO: randomize ?)
83 int* Z = (int*)calloc(n, sizeof(int));
85 //Initialize phi to zero, because some M loops might exit before phi affectation
86 zeroArray(phi, p*m*k);
88 while (ite<mini || (ite<maxi && sumDeltaPhi>tau))
94 //M step: Mise à jour de Beta (et donc phi)
95 for (int r=0; r<k; r++)
97 //Compute Xr = X(Z==r,:) and Yr = Y(Z==r,:)
99 for (int i=0; i<n; i++)
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)];
113 //Compute tXrXr = t(Xr) * Xr
114 for (int j=0; j<p; j++)
116 for (int jj=0; jj<p; jj++)
118 Real 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;
125 //Get pseudo inverse = (t(Xr)*Xr)^{-1}
126 Real* invTXrXr = pinv(tXrXr, p);
128 // Compute tXrYr = t(Xr) * Yr
129 for (int j=0; j<p; j++)
131 for (int jj=0; jj<m; jj++)
133 Real dotProduct = 0.0;
134 for (int u=0; u<cardClustR; u++)
135 dotProduct += Xr[mi(u,j,n,p)] * Yr[mi(u,jj,n,m)];
136 tXrYr[mi(j,jj,p,m)] = dotProduct;
140 //Fill matrixM with inverse * tXrYr = (t(Xr)*Xr)^{-1} * t(Xr) * Yr
141 for (int j=0; j<p; j++)
143 for (int jj=0; jj<m; jj++)
145 Real 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;
153 //U,S,V = SVD of (t(Xr)Xr)^{-1} * t(Xr) * Yr
154 gsl_linalg_SV_decomp(matrixM, V, S, work);
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++)
161 //[intermediate step] Compute hatBetaR = U * S * t(V)
162 double* U = matrixM->data; //GSL require double precision
163 for (int j=0; j<p; j++)
165 for (int jj=0; jj<m; jj++)
167 Real 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;
174 //Compute phi(:,:,r) = hatBetaR * Rho(:,:,r)
175 for (int j=0; j<p; j++)
177 for (int jj=0; jj<m; jj++)
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;
191 Real sumLogLLF2 = 0.0;
192 for (int i=0; i<n; i++)
195 Real maxLogGamIR = -INFINITY;
196 for (int r=0; r<k; r++)
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
203 //compute det(Rho(:,:,r)) [TODO: avoid re-computations]
204 for (int j=0; j<m; j++)
206 for (int jj=0; jj<m; jj++)
207 matrixE->data[j*m+jj] = Rho[ai(j,jj,r,m,m,k)];
209 gsl_linalg_LU_decomp(matrixE, permutation, &signum);
210 Real detRhoR = gsl_linalg_LU_det(matrixE, signum);
212 //compute Y(i,:)*Rho(:,:,r)
213 for (int j=0; j<m; j++)
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)];
220 //compute X(i,:)*phi(:,:,r)
221 for (int j=0; j<m; j++)
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)];
228 //compute dotProduct < Y(:,i)*rho(:,:,r)-X(i,:)*phi(:,:,r) . Y(:,i)*rho(:,:,r)-X(i,:)*phi(:,:,r) >
229 Real dotProduct = 0.0;
230 for (int u=0; u<m; u++)
231 dotProduct += (YiRhoR[u]-XiPhiR[u]) * (YiRhoR[u]-XiPhiR[u]);
232 Real logGamIR = log(Pi[r]) + log(detRhoR) - 0.5*dotProduct;
234 //Z(i) = index of max (gam(i,:))
235 if (logGamIR > maxLogGamIR)
238 maxLogGamIR = logGamIR;
240 sumLLF1 += exp(logGamIR) / pow(2*M_PI,m/2.0);
243 sumLogLLF2 += log(sumLLF1);
246 // Assign output variable LLF
247 *LLF = -invN * sumLogLLF2;
249 //newDeltaPhi = max(max((abs(phi-Phi))./(1+abs(phi))));
250 Real newDeltaPhi = 0.0;
251 for (int j=0; j<p; j++)
253 for (int jj=0; jj<m; jj++)
255 for (int r=0; r<k; r++)
257 Real 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;
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;
271 sumDeltaPhi -= deltaPhi[0];
272 for (int u=0; u<deltaPhiBufferSize-1; u++)
273 deltaPhi[u] = deltaPhi[u+1];
274 deltaPhi[deltaPhiBufferSize-1] = newDeltaPhi;
276 sumDeltaPhi += newDeltaPhi;
278 // update other local variables
279 for (int j=0; j<m; j++)
281 for (int jj=0; jj<p; jj++)
283 for (int r=0; r<k; r++)
284 Phi[ai(jj,j,r,p,m,k)] = phi[ai(jj,j,r,p,m,k)];
294 gsl_matrix_free(matrixE);
295 gsl_matrix_free(matrixM);
296 gsl_permutation_free(permutation);
297 gsl_vector_free(work);