Let be the matrix corresponding to the signals and be a matrix corresponding to one set of signals. Time indexes each row of the matrix ( time samples). Let and be the sample covariance matrices of and , respectively, and let be the sample covariance matrix between and . For simplicity, we suppose that all signals have zero mean.
We follow the derivations of Johnson and we do a change of basis: and .
By the Cauchy-Schwartz inequality:
The inequality above is an equality when and are collinear. The right hand side of the expression above is a Rayleigh quotient and it is maximum when is the eigenvector corresponding to the largest eingenvalue of (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: and .
We can continue this way to find the subsequent vectors
|Date of creation||2013-03-22 19:16:11|
|Last modified on||2013-03-22 19:16:11|
|Last modified by||tony_bruguier (26297)|