a=l where E and F represent the residuals matrices. Linear combinations of x vectors are calculated from the latent variable ta = wjx and those for the y vectors from ua = qjy so that they maximize the covariance between X and Y explained at each dimension. wn and q0 are loading vectors. The number of latent variables can be determined by cross-validation [659].

For the first latent variable, PLS decomposition is started by selecting one column of Y,yj, as the starting estimate for ui. (Usually, the column of

Y with greatest variance is chosen.) Starting in the X data block (for the first latent variable):

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