canonical correlation
Let X be the (T,n) matrix corresponding to the n signals and Y be a (T,p) matrix corresponding to one set of p signals. Time indexes each row of the matrix (T time samples). Let Σ11 and Σ22 be the sample covariance matrices of X and Y, respectively, and let Σ12=Σ′21 be the sample covariance matrix between X and Y. For simplicity, we suppose that all signals have zero mean.
Canonical correlation analysis (CCA) finds the linear combinations of the column of X and Y that has the largest correlation
; i.e., it finds the weight vectors (loadings) a and b that maximize:
ρ=a′Σ12b√a′Σ11a√b′Σ22b. | (1) |
We follow the derivations of Johnson and we do a change of basis: c=Σ1/211a and d=Σ1/222b.
ρ=c′Σ-1/211Σ12Σ-1/222d√c′c√d′d | (2) |
By the Cauchy-Schwartz inequality:
ρ≤√c′Σ-1/211Σ12Σ-1/222Σ-1/222Σ21Σ-1/211c√d′d√c′c√d′d=√c′Σ-1/211Σ12Σ-122Σ21Σ-1/211cc′c. | (3) |
The inequality above is an equality when Σ-1/222Σ21Σ-1/211c and d are collinear. The right hand side of the expression above is a Rayleigh quotient and it is maximum when c is the eigenvector corresponding to the largest eingenvalue of Σ-1/211Σ12Σ-122Σ21Σ-1/211 (we obtain the other rows by using the other eigenvalues
in decreasing magnitude). This results if the basis of the CCA. We can compute the two canonical variables: U1=Xa and V1=Yb.
We can continue this way to find the subsequent vectors
Title | canonical correlation |
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Canonical name | CanonicalCorrelation |
Date of creation | 2013-03-22 19:16:11 |
Last modified on | 2013-03-22 19:16:11 |
Owner | tony_bruguier (26297) |
Last modified by | tony_bruguier (26297) |
Numerical id | 4 |
Author | tony_bruguier (26297) |
Entry type | Definition |
Classification | msc 62H20 |