toby989 <[EMAIL PROTECTED]> wrote in message
news:<[EMAIL PROTECTED]>...
>
> [...] 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. [...]
Someone has borrowed my Mulaik, so I don't know how much of this
duplicates his treatment, but here is a description of canonical
correlation that resembles Gottfried's but is a little shorter.
A -- Verbal hand-waving description
Do a complete full-rank orthogonal component analysis on each set
of variables separately. Principal components are ok but are not
necessary -- any full set will do, as long as they're orthogonal.
Rotate the components to their canonical position, in which each
component in one set correlates with only the corresponding component
in the other set. These correlations are the "canonical correlations".
The canonical loadings are the regression weights of the variables
in each set on that set's canonically positioned components.
B -- Details, in terms of data matrices
Let Y1:[n x p1] and Y2:[n x p2] be matrices of observations.
The columns (variables) may be centered or not, as desired,
and may be standardized or not, also as desired.
To simplify the presentation, assume that
either all variables are centered, or none are centered.
For future reference, let m = n-1 or n,
according as the variables are centered or not.
To simplify the presentation,
assume that Y1 and Y2 are of full column rank.
Let X1:[n x p1] and X2:[n x p2] be orthonormal bases for the
column-spaces of Y1 and Y2, and let F1 = Y1'*X1 and F2 = Y2'*X2,
so that Y1 = X1*F1' and Y2 = X2*F2',
where ' denotes matrix transposition.
Each X,F pair can be obtained from its Y by QR decomposition,
singular value decomposition, Gram-Schmidt decomposition, whatever.
Each X contains orthogonal component scores (not scaled to unit
variance), and each F contains the corresponding component loadings.
Let V1*D*V2' be the singular value decomposition of X1'*X2.
V1:[p1 x p] and V2:[p2 x p], where p = min(p1,p2).
The singular values, in the diagonal matrix D,
are the canonical correlations.
The matrices of canonical variate scores (scaled to unit variance)
are Z1 = sqrt(m)*X1*V1 and Z2 = sqrt(m)*X2*V2.
The singular vector matrices V1 and V2 are transformations that
rotate the arbitrarily positioned components X1 and X2 into canonical
position.
The loadings of the observed variables on the canonical variates
are A1 = Y1'*Z1/m and A2 = Y2'*Z2/m.
C -- Details, in terms of the moment/covariance/correlation matrices
C11 = Y1'*Y1/m, C12 = Y1'*Y2/m, and C22 = Y2'*Y2/m
Let F1 be any factoring of C11, and F2 any factoring of C22,
so that F1*F1' = C11 and F2*F2' = C22.
Let V1*D*V2' be the s.v.d. of F1^*C12*F2^', where ^ denotes matrix
inversion. The diagonal matrix D contains the canonical correlations.
The loadings of the observed variables on the canonical variates
are A1 = F1*V1 and A2 = F2*V2.
.
.
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