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

Abstract

Extended bilinear modeling is popular in second‐order multivariate calibration, particularly when the matrix data for each sample are of chromatographic origin. Since elution time profiles vary across samples, in both shape and peak position, it is not possible to process these data in a three‐way trilinear format. In these cases, the most successful model for quantitating analytes in the presence of interferents is multivariate curve resolution‐alternating least squares (MCR‐ALS) in its extended version, ie, processing an augmented data matrix built with the matrices for a test sample and the calibration samples, appended in the direction of the elution time mode. MCR‐ALS starts with certain initial profiles and applies a set of natural constraints during the ALS phase, whose purpose is to reduce the range of feasible solutions or to lead to unique bilinear solutions if possible. In this report, a simulated second‐order three‐component system (two calibrated analytes and one uncalibrated interferent in test samples) is studied regarding the presence of rotational ambiguity in the bilinear solutions, using (a) a grid search methodology to compute the feasible solutions and (b) MCR‐ALS on a large set of test samples to estimate the average prediction errors. Various initialization schemes and constraints are probed, and the results are compared in terms of the extent of rotational ambiguity and global uncertainty in predicted concentrations.

Chromatographic Applications in the Multi-Way Calibration Field

“…It is important to notice that, even when a significant RMSE RA value can be computed by the latter equation, the MCR-ALS solutions may be driven to the correct bilinear solution by an adequate initialization procedure, as recently discussed [109]. In these cases, the value of RMSE RA may be large, although this would not be reflected in the average prediction error for a set of test samples, which may ultimately be reasonably low.…”

A New Parameter for Measuring the Prediction Uncertainty Produced by Rotational Ambiguity in Second-Order Calibration with Multivariate Curve Resolution

“…This strategy was already employed to drive the initial states at many different solutions within the range of feasible bilinear decompositions. 21 Only two constraints were imposed during the ALS fit: non-negativity in both spectral and temporal modes and correspondence between constituents and samples. It should be noticed that in the presence of uncalibrated interferents, even under non-negativity and correspondence constraints, a certain degree of rotational ambiguity remains for the analyte concentration subprofile in the test samples.…”

“…Several previous studies have focused on the analytical consequences of the extent of rotational ambiguity derived from the range of feasible solutions, which can be defined in different manners. − One alternative proposes to define this extent through the size of the area of feasible solutions (AFS), defined on an abstract space spanned by the principal component scores of the raw data. This implies rather qualitative considerations, and becomes impractical for systems with more than four components, due to the difficulties in the calculations involved for truly multicomponent systems.…”

N‐BANDS: A new algorithm for estimating the extension of feasible bands in multivariate curve resolution of multicomponent systems in the presence of noise and rotational ambiguity

“…However, even under the application of the above‐mentioned constraints and of additional ones such as local rank, selectivity, components correspondence, concentration correlation, and soft trilinearity, a certain degree of rotational ambiguity may remain in the bilinear MCR solutions, particularly in the context of analyte quantitation in test samples containing uncalibrated interferents 9–11 . This is the reason why considerable emphasis has been directed toward the estimation of the set of feasible bilinear solutions.…”

“…In the case of multivariate second‐order calibration using MCR‐ALS, analytical results on component concentrations will be also affected by the phenomenon of rotational ambiguity in the form of a prediction uncertainty contribution 22–24 . Although this contribution can be significantly decreased by the application of constraints, 9–11 it is important to be able to quantify the degree of uncertainty, which would remain after MCR‐ALS decomposition of a data matrix. In this regard, a modification of the MCR‐BANDS method has been proposed, with a different objective function to be minimized or maximized, based on the relative area of the concentration profile of each component with respect to the sum of the integrated profiles for all sample components present in the system: where || || 1 indicates the 1‐norm, defined as the sum of the absolute values of all (vector or matrix) elements 24 …”