Multivariate curve resolution of multiway data using the multilinearity constraint

Abstract

The extension of Multivariate Curve Resolution‐Alternating Least Squares (MCR‐ALS) to the analysis of multiway data using the multilinearity constraint is described in detail as one step forward of previous implementations of the trilinearity and quadrilinearity constraints for the analysis of three‐ and four‐way data sets, respectively. As in previous cases, the implementation of the multilinear model for multiway data sets is done algorithmically, within the frame of the alternating least squares (ALS) optimization in the MCR‐ALS method. This implementation is tested using multiway data sets of different complexity, and the obtained results have confirmed the adequacy of the proposed approach. Special advantages of the proposed methodology are that it allows for the implementation of the constraint separately for the different components in their different modes and that it also allows for the introduction of different levels of complexity of the multilinear model, including mixed multilinear models. These two features are especially relevant because they are not present in most of the most used multiway data analysis methods at present.

Flexible Implementation of the Trilinearity Constraint in Multivariate Curve Resolution Alternating Least Squares (MCR-ALS) of Chromatographic and Other Type of Data

“…C aug (I × K, N), is the augmented factor matrix having the K individual factor (concentration) matrices also vertically concatenated, C k (I, N), for the N components, and S T (N, J), is the common factor matrix with the spectra of these N components. The ALS factorization of D aug is performed in the same way as for the individual data matrices, but with the possibility of adding additional constraints related to the multiset data structure and related to the common components (chemical speciation) in the different individual data matrices [ 1 , 13 , 14 , 25 , 37 , 40 , 47 , 48 , 49 , 50 ]…”

“…This also implies that mixed bilinear-trilinear models are feasible, i.e., the trilinear condition is applied for some of the components but not for other. This implementation of the trilinear model has already been generalized to quadrilinear, to pentalinear or in general to multilinear models as shown in [ 14 , 52 ]. When ALS finally converges and the final MCR solutions are obtained, C is made up of the concentration (column) profiles of the different components (N), which are the same for all data K matrices simultaneously analyzed, whereas Z T is made up of the profiles giving the relative amounts of these components in the different K matrices (sample profiles).…”

“…With the development of hyphenated analytical instruments, multiway datasets can be acquired very fast and the corresponding multilinear type models can be applied accordingly. The trilinear models built for multiway data have the advantage of eliminating the rotation ambiguities associated to bilinear type of modes and therefore provide unique resolution solutions [ 13 , 14 , 15 ]. Recent works about the rotation ambiguity problem associated with quantitative estimations by MCR methods can be found in the literature [ 16 , 17 ].…”

“…Recent works about the rotation ambiguity problem associated with quantitative estimations by MCR methods can be found in the literature [ 16 , 17 ]. The profiles obtained for each component using a trilinear model are unique and provide an efficient and simple way to explain the observed variance in the raw data [ 14 ].…”

“…Moreover, results obtained by MCR-ALS with trilinearity constraints have been shown to be equivalent to those obtained by other trilinear model resolution methods like the alternating trilinear decomposition (ATLD) [ 18 ], parallel factor analysis (PARAFAC), [ 19 ] and other extension methods [ 20 , 21 , 22 ]. Other multilinear decomposition methods have been proposed in the literature for the extension of the trilinear model to the quadrilinear and even higher multilinear models in the analysis of data from multi-hyphenated instruments [ 14 ]. However, in the application of all these methods for resolution purposes, the data should fulfill the underlying multilinear model for all the components obtained in the factor decomposition.…”