A new parameter for measuring the prediction uncertainty produced by rotational ambiguity in second-order calibration with multivariate curve resolution

Abstract

Multivariate curve resolution-alternating least-squares (MCR-ALS) is the model of choice when dealing with matrix data that cannot be arranged into a trilinear three-way array, that is, mostly from chromatographic origin with spectral detection. A range of feasible solutions may be found in MCR studies, due to the phenomenon of rotational ambiguity associated with bilinear decompositions of matrices. The application of chemically driven constraints is vital to achieving an adequate solution and minimizing the degree of rotational ambiguity present in the system. However, when studying complex samples, it may not be possible to recover unique solutions, even under the application of proper constraints. In such cases, it is important to be able to assess the propagation of rotation uncertainty to the estimated analyte concentrations, which stems from the existence of a finite range of feasible solutions. In this work, we present a new analytical parameter to estimate the potential uncertainty in analyte prediction brought about by rotational ambiguity, in the form of an associated root-mean-square error, named RMSERA. The proposed parameter comes in the form of a range of values, whose limits are δRA/(12)1/2 and δRA/(3)1/2, with δRA being defined as the difference between the maximum and minimum values of the analyte concentration that would be predicted by the MCR model from its concentration profiles lying in the range of feasible solutions, and corresponding to maximum and minimum area, respectively. We support our proposal on extensive simulations for systems of varying composition, and demonstrate its application on experimental data aimed at the determination of four pollutants in environmental water samples.