Including trilinear and restricted Tucker3 models as a constraint in Multivariate Curve Resolution Alternating Least Squares

Abstract

Constrained bilinear models implemented in Multivariate Curve Resolution Alternating Least Squares (MCR-ALS) method have been revealed extremely useful in solving rotational and intensity ambiguities associated to two-way factor analysis problems in different chemistry fields and in spectroscopy. Constraints applied to resolved profiles have included non-negativity, unimodality, closure, selectivity, local rank and physical and chemical laws. These models are easily extended to three-way and multiway data sets arranged in appropriate matrix augmentation (concatenation) schemes (matricization). The bilinear model assumed in MCR-ALS may include additional constraints concerning three-way structures like those implied in PARAFAC and Tucker models. These models may be implemented in the ALS algorithm in a very flexible way covering a wide range of possible situations, including mixed modeling situations where multilinear constraints are only partly fulfilled by the physical nature of the investigated data. Detailed explanation of different implemented algorithms and of possible application will be described and discussed.