Hi All
Since few days I am trying to figure out what in my derivation of the loadings matrix
does
not work. The question is most likely how the matrix of eigenvectors needs to be
scaled
(and how to scale them) in order that the loadings matrix comes out right. In the code
below, the eigenvalues seem correct since SAS's proc cancorr also gives me these
values
that I get from calculating them manually with SAS/IML.
read all var {PENET PCYCLE PRICE PVTSH PURHH} into X; /*the
IV*/
read all var {FEAT DISP PCUT SCOUP MCOUP} into Y; /*the
dependents*/
n = nrow (X);
o = nrow (Y); /*same as n in this
dataset*/
p = ncol (X);
q = ncol (Y); /*same as p in this
dataset*/
J = shape (1, n, n);
K = shape (1, o, o);
Xd = X - (J / n)` * X;
Sx = (1 / (n - 1)) * Xd` * Xd;
Xs = Xd * (sqrt (diag (Sx))` ** (-1))`; /*n x p standardized IV
matrix*/
Yd = Y - (J / o)` * Y;
Sy = (1 / (o - 1)) * Yd` * Yd;
Ys = Yd * (sqrt (diag (Sy))` ** (-1))`; /*o x q standardized DV
matrix*/
reset print;
Rxx = (1 / (n - 1)) * Xs` * Xs;
Rxy = (1 / (n - 1)) * Xs` * Ys;
Ryy = (1 / (n - 1)) * Ys` * Ys;
Ryx = (1 / (n - 1)) * Ys` * Xs; /*p x p correlation
matrices*/
A = eigvec (Rxx ** (-1) * Rxy * Ryy ** (-1) * Ryx);
B = eigvec (Ryy ** (-1) * Ryx * Rxx ** (-1) * Rxy); /*p x p matrix of
eigenvectors*/
aa = eigval (Rxx ** (-1) * Rxy * Ryy ** (-1) * Ryx);
Da = sqrt (diag (aa[1:p, 1]));
bb = eigval (Ryy ** (-1) * Ryx * Rxx ** (-1) * Rxy);
Db = sqrt (diag (bb[1:p, 1])); /*p x p diag matrix of canonical correlations same as
Da*/
reset noprint;
U = Xs * B;
T = Ys * A; /*n x p matrix of canonical
variates*/
reset print;
F = Rxx * B;
G = Ryy * A; /*p x p canonical loadings where A and B probably need to be
rescaled*/
/*F = (1 / (n - 1)) * Xs` * U;*/
/*G = (1 / (n - 1)) * Ys` * T;*/ /*where U and T probably need to be
rescaled*/
/*F = (1 / (n - 1)) * Xs` * Xs * B;*/
/*G = (1 / (n - 1)) * Ys` * Ys * A;*/ /*F and G can also be computed this
way*/
I really would appreciate your help. Thanks in advance.
Bye Toby
John describes canonical correlation also, but does not mention how the eigenvectors
are
scaled to yield the canonical weights.
http://www.biostat.wustl.edu/archives/html/s-news/1999-01/msg00104.html
.
.
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