Quantifying the Prediction Error in Analytical Multivariate Curve Resolution Studies of Multicomponent Systems

Abstract

In multivariate curve resolution (MCR) analysis, a range of feasible solutions is often encountered, because of the rotational ambiguities associated with the bilinear decomposition of data matrices. For quantitative purposes, the analysis is usually applied to a carefully designed set of calibration and test samples having uncalibrated interferents. Under the usual minimal constraints (non-negativity, unimodality, species correspondence, etc.), concentration and spectral profiles of the analyte in the test samples are not univocally recovered, unlike those in the calibration samples, especially when profile overlapping with the interferents is significant and selective regions do not exist for the analyte. In this report, a quantitative measure of the prediction errors due to rotational ambiguities is discussed, based on the calculation of the differences between the maximum and minimum area under the analyte concentration profiles calculated by the MCR-BANDS procedure. This methodology can be applied in different analytical scenarios with any number of analytes and interferents. Both absolute and relative quantitative errors due to rotation ambiguities are estimated and discussed in both simulated and experimental examples derived from liquid chromatography with diode array detection. The proposed procedure can be generalized to most of the analytical situations where every instrumentally measured sample produces a data table or data matrix.

Second‐order multivariate calibration with the extended bilinear model: Effect of initialization, constraints, and composition of the calibration set on the extent of rotational ambiguity

“…Significant RA occurs for analyte 1, whereas only a small amount is apparent for analyte 2. The degree of RA uncertainty in the predicted analyte concentration can be estimated as previously discussed: …”

“…Moreover, Figure B also helps in interpreting the specific points in the AFS leading to maximum and minimum area. The latter ones can be estimated by the MCR‐BANDS approach recently described …”

“…The data are similar to the example data sets employed to illustrate a MATLAB graphical user interface for second‐order multivariate calibration . MCR‐BANDS was used to estimate the prediction uncertainty produced by RA …”

“…When this is not possible, the overall effect of the RA in second‐order multivariate calibration is to introduce an additional uncertainty in the estimated analyte concentrations, beyond the one stemming from signal noise or calibration concentration uncertainties. The extent of the RA uncertainty has recently been quantitated for multi‐component systems, based on the estimation of the analyte concentration profiles corresponding to their maximum and minimum areas, guided by a nonlinear optimization under nonlinear constraints . It has been proposed that the relative difference between maximum and minimum areas is a good measure of the degree of RA uncertainty that would be transmitted to the predicted analyte concentration .…”

“…The extent of the RA uncertainty has recently been quantitated for multi‐component systems, based on the estimation of the analyte concentration profiles corresponding to their maximum and minimum areas, guided by a nonlinear optimization under nonlinear constraints . It has been proposed that the relative difference between maximum and minimum areas is a good measure of the degree of RA uncertainty that would be transmitted to the predicted analyte concentration . A useful algorithm to estimate the extreme profiles and areas, MCR‐BANDS, has been described …”