Regression with multiple regressor arrays

Abstract

Extension of standard regression to the case of multiple regressor arrays is given via the Kronecker product. The method is illustrated using ordinary least squares regression (OLS) as well as the latent variable (LV) methods principal component regression (PCR) and partial least squares regression (PLS). Denoting the method applied to PLS as mrPLS, the latter was shown to explain as much or more variance for the first LV relative to the comparable L-partial least squares regression (L-PLS) model. The same relationship holds when mrPLS is compared to PLS or n-way partial least squares (N-PLS) and the response array is 2-way or 3-way, respectively, where the regressor array corresponding to the first mode of the response array is 2-way and the second mode regressor array is an identity matrix. In a comparison with N-PLS using fragrance data, mrPLS proved superior in a validation sense when model selection was used. Though the focus is on 2-way regressor arrays, the method can be applied to n-way regressors via N-PLS.