Comments on Multilinear PLS

Recently, Bro published a paper on multilinear PLS (J. Chemometrics, 10, 47–61 (1996)) in which he proposed a generalization of PLS to multiway situations, called multilinear PLS, which is a mixture of a trilinear model (PARAFAC) and PLS. However, Bro does not give the equations for the prediction step. In this paper these prediction equations are given in both their full and closed forms. The least squares properties of the proposed multilinear PLS are established and a more comprehensive notation is given. Using this notation, it is clear that some other multiway analysis methods such as PARAFAC and Tucker1 models can be combined with PLS. Multiway methods such as Tucker2 and Tucker3 need a different approach. A framework is given for general two‐block multiway models.