Multivariate Covariates Regression

The principal covariates regression proposed by de Jong and Kiers [119] can also be used to develop predictive models [554]. In this method, the relationship between the predictor block X and the dependent variable block Y is expressed by finding components scores T where the A column vectors t,, i = 1 ,...,A. span the low-dimensional subspace of X that accounts for the maximum amount of variation in both X and Y.

where W is a p x A matrix of component weights, Ex and Ey contain the unique factors of X and Y, respectively, and the loading matrices Px (A x p) and Py (^4 x m) contain the regression parameters relating the variables in X and Y, respectively [119]. The model is fitted to the data in the least-squares sense by maximizing the weighted average of R2XT: the percentage of variance in X accounted for by T and i?yT, the percentage of

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