concor                package:concor                R Documentation

_R_e_l_a_t_i_v_e _l_i_n_k_s _o_f _s_e_v_e_r_a_l _s_u_b_s_e_t_s _o_f _v_a_r_i_a_b_l_e_s

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

     Relative links of several subsets of variables Yj with another set
     X. SUCCESSIVE SOLUTIONS

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

     concor(x,y,py,r)

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

     x,y: are n x p and n x q matrices of p and q centered columns

      py: is a row vector which contains the numbers qi, i=1,...,ky, of
          the ky subsets yi of y : sum(qi)=sum(py)=q. py is the
          partition vector of y

       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 Rp associated to the ky vectors vi[:,1]'s of Rqi, by
     maximizing sum_i mbox{cov}(x*u[,k],y_i*v_i[,k])^2, with 1+ky norm
     constraints on the axes. A component x*u[,k] is associated to ky
     partial components yi*vi[,k] and to a global component y*V[,k]. 
     mbox{cov}(x*u[,k],y*V[,k])^2 = sum
     mbox{cov}(x*u[,k],y_i*v_i[,k])^2. y*V[,k] is a global component of
     the components yi*vi[,k].

     The second solution is obtained from the same criterion, but after
     replacing each yi by y_i-y_i*v_i[,1]*v_i[,1]'.  And so on for the
     successive solutions 1,2,...,r.  The biggest number of solutions
     may be r=inf(n,p,qi), when the x'*yi's are supposed with full
     rank; then rmax=min(c(min(py),n,p)).  For a set of r solutions,
     the matrix u'X'YV is diagonal and the matrices u'X'Yjvj are
     triangular (good partition of the link by the solutions). 
     concor.m is the svdcp.m function applied to the matrix x'y.

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

     list with following components 

       u: is a p x r matrix of axes in Rp relative to x; u'*u =
          Identity

       v: is a q x r matrix of ky row blocks vi (qi x r) of axes in Rqi
          relative to yi; vi'*vi = Identity

       V: is a q x r matrix of axes in Rq relative to y; V'*V =
          Identity

    cov2: is a ky x r matrix; each column k contains ky squared
          covariances mbox{cov}(x*u[,k],y_i*v_i[,k])^2, the partial
          measures of link

_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:

     # To make some "GPA" : so, by posing the compromise X = Y,
     # "procrustes" rotations to the "compromise X" then are :
     # Yj*(vj*u').

     x<-matrix(runif(50),10,5);y<-matrix(runif(90),10,9)
     x<-scale(x);y<-scale(y)
     co<-concor(x,y,c(3,2,4),2)
     ((t(x%*%co$u[,1])%*%y[,1:3]%*%co$v[1:3,1])/10)^2;co$cov2[1,1] 
     t(x%*%co$u)%*%y%*%co$V

