svdcp                 package:concor                 R Documentation

_S_V_D _f_o_r _a _C_o_l_u_m_n _P_a_r_t_i_t_i_o_n_e_d _m_a_t_r_i_x _x

_D_e_s_c_r_i_p_t_i_o_n:

     SVD for a Column Partitioned matrix x. r global successive
     solutions

_U_s_a_g_e:

     svdcp(x,H,r)

_A_r_g_u_m_e_n_t_s:

       x: is a p x q matrix

       H: is a row vector which contains the numbers qi, i=1,...,kx, of
          the partition of x with kx column blocks xi : sum q_i = q.

       r: is the wanted number of successive solutions.

_D_e_t_a_i_l_s:

     The first solution calculates 1+kx normed vectors: the vector
     u[,1] of R^p associated to the kx vectors vi[,1]'s of R^{q_i}. by
     maximizing sum_i (u[,1]'*x_i*v_i[,1])^2, with 1+kx norm
     constraints.  A value (u[,1]'*x_i*v_i[,1])^2 measures the relative
     link between R^p and R^{q_i} associated to xi. It corresponds to a
     partial squared singular value notion, since sum_i
     (u[,1]'*x_i*v_i[,1])^2=s^2, where s is the usual first singular
     value of x.

     The second solution is obtained from the same criterion, but after
     replacing each xi by xi-xi*vi[,1]*vi[,1]'.  And so on for the
     successive solutions 1,2,...,r .  The biggest number of solutions
     may be r=inf(p,qi), when the xi's are supposed with full rank;
     then rmax=min([min(H),p]).

_V_a_l_u_e:

     list with following components 

       u: is a p x r matrix; u'*u = Identity

       v: is a q x r matrix of kx row blocks vi (qi x r); vi'*vi =
          Identity

      s2: is a kx x r matrix; each column k contains kx values
          (u[,k]'*x_i*v_i[,k])^2, the partial (squared) singular values
          relative to xi

_R_e_f_e_r_e_n_c_e_s:

     Lafosse R. & Hanafi M.(1997) Concordance d'un tableau avec K
     tableaux: Definition de K+1 uples synthetiques. Revue de
     Statistique Appliquee vol.45,n.4.

_E_x_a_m_p_l_e_s:

     x<-matrix(runif(200),10,20)
     s<-svdcp(x,c(5,5,10),1)
     ss<-svd(x);ss$d[1]^2
     sum(s$s2)

