Matrix augmentation for breaking rank-deficiency: A case study

To attempt chemically meaningful solutions from the spectroscopic description of complex chemical systems, multivariate curve resolution based on alternating least-squares under constraints (MCR/ALS) is a powerful method based on factor analysis. It strongly relies on first insights into data sets to provide initial solutions of either the concentration profiles or the spectra. As long as the analysed data matrix is full rank, the application of singular values decomposition-based methods can be a powerful way to estimate the number of significant components in evolving systems. Nevertheless, spectroscopic data often provide rank-deficient matrices. In the case studied here, we detail a powerful column-wise matrix-augmentation strategy for achieving the condition of full rank in UV–visible spectroscopy. In particular, the potential of a variant of evolving factor analysis (EFA), adapted to analyse full rank augmented data matrices, is tested. Besides, such approach enables to build gradually the matrix of correspondence among the species, which summarises information about local rank and selectivity, defining the number and identity of components in each single matrix along the augmented data set, before attempting MCR-ALS.