A convergence criterion in alternating least squares (ALS) by global phase angle

A practical index to estimate physically or chemically meaningful convergence in self-modeling curve resolution (SMCR) is proposed. In SMCR, solutions for concentration profiles and pure component spectra are given by a sequential rotation of matrices, called alternating least square (ALS). A convergence guided by an error criterion based on the sum of squared residuals between original and reconstructed spectra brings a mathematically valid solution. However, a serious problem sometimes arises because such an error criterion can bring up an undesirable result which only satisfies purely mathematical constraint without realistic chemical or physical considerations. This situation can be seen as a kind of overfitting of a model to the original spectra. As a possible solution, an alternative error criterion based on global phase angle, which enables one to obtain chemically or physically meaningful convergence in ALS, is used. In the present study, a practical use of this error criterion is demonstrated with time-dependent IR spectra of mixture solution of oleic acid and ethanol. The result clearly reveals the optimal number of the iteration in ALS that provides a realistic model. This method may be a powerful tool of SMCR to balance both mathematically and physically meaningful information.