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Bilinear decomposition of an augmented data matrix is usually complicated by the phenomenon of rotational ambiguity. If the latter is significant, quantitative and qualitative information of the recovered profiles may be less useful. Although constraints can reduce the extent of feasible regions and the degree of rotational ambiguity, the estimation of initial parameters to start the decomposition is an important phase in multivariate curve resolution-alternating least-squares (MCR-ALS) studies. Dealing with a bilinear decomposition of an augmented data matrix where rotational ambiguity persists, the question remains whether it is possible to still develop a successful calibration protocol. Indeed, literature reports indicate that various analytical systems have been experimentally developed, in which substantial rotational ambiguity exists, yet the experimental results confirmed that accurate analyte quantitation was possible. In this research, we further investigate on the effect of the initialization step for a two-component second-order multivariate calibration with the extended bilinear model. It is shown that the selection strategy based on the so-called purest variables can be helpful in achieving a correct profile resolution, depending on which data direction it is applied. Finally, some data-driven guidelines for analytical chemists are suggested, to identify the potential degree of rotational ambiguity and the correct choice of the initialization strategy. Image 1 • Initialization is an important phase in multivariate curve resolution. • Accurate results are possible even with substantial rotational ambiguity. • Initialization with pure variables implicitly benefits from selectivity. • Pure variables are helpful depending on the data mode for their estimation. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution has been applied to both simulated and experimental data sets where high or even complete overlapping occurs between component profiles in one data mode. It is shown that rotational ambiguity exists in the bilinear decomposition of the augmented data matrices built with second-order data for pure analyte standards and test samples containing uncalibrated interferents. However, even in the presence of rotational ambiguity, initialization based on the so-called purest variables in one of the data modes may allow one to develop analytical protocols with reasonable statistical indicators for the prediction of the analyte of interest. In one of the explored experimental systems, the analyte ciprofloxacin was determined in the presence of the interferent salicylate, measuring time decay-luminescence matrix data. The average prediction error was 0.02 mg L−1 in the test set, corresponding to a relative error of ca. 8%. In the second system, capillary electrophoresis with UV detection was employed to determine ciprofloxacin in aqueous samples in the presence of other fluoroquinolones, achieving analyte recoveries in the range 101–113%. Although further theoretical work may still be needed, the present analysis of the feasible component profiles after bilinear decomposition provides some clues to interpret the phenomenon. Image 1 • Multivariate curve resolution is applied to second-order data. • Strong or complete overlapping in one data mode is considered. • Rotational ambiguity is assessed. • Simulated and experimental data are studied. • Prediction results are analyzed depending on initialization and constraints. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution (MCR) methods are powerful chemometric approaches that have been significantly involved to study the complex chemical systems. The accuracy of MCR results is directly related to the applied constraints which determine the properties of the resolved profiles. Constraints have been, and still are, an active field of research in MCR studies. Different constraints have different impacts on the range of feasible solutions, so it is important to study the effect of each constraint, to examine its compliance with physico-chemical principles, and to find sufficient conditions to ensure unique solutions. In this study, we focus on known-value constraint and the requirements for reducing the range of feasible solutions under this constraint. In addition, several theoretical rules are presented to determine the minimum number of required known-values in order to get a unique solution. The theory of known-values was accessed using several simulated and experimental datasets. As shown previously, known-value information can be applied in quantitative analysis of first-order data. The prediction performance of MCR-ALS method under known-value constraint was compared to the results of the well stablished partial least squares (PLS) method in the presence of minimum number of required calibration samples. The comparison shows the advantage of using MCR-ALS method under known-value constraint over the PLS results in such experiments. [ABSTRACT FROM AUTHOR]

In the present report, an upgrade of a MATLAB graphical user interface (GUI) toolbox for implementing second-order multivariate calibration models is presented, named Multivariate Calibration 2 (MVC2). Due to the advances in the chemometric field, this software has undergone numerous changes since its last release in 2009. The new freely available MVC2 incorporates novel features and models that make it a very practical tool for data processing. Overall, the current version presents different types of algorithms, designed to provide specific degrees of model flexibility for different multi-way data scenarios: in addition to trilinear decomposition (TLD) and residual bilinearization (RBL) models, which were previously included in the 2009 version, parallel factor analysis 2 (PARAFAC2) and multivariate curve resolution (MCR) methods are now available. All these chemometric methodologies are integrated in the software for quantitative purposes, and if needed, they permit to exploit the so-called analytical second-order advantage. The program offers practical tools for data loading and visualization, model development and validation, and prediction in unknown samples. The software output also includes a comprehensive report on the analytical figures of merit (AFOMs) applicable to second-order calibration models. Specifically, a tool was incorporated for the diagnosis of the remaining rotational ambiguity in the MCR solutions. In the present report, the new features of MVC2 are illustrated through simulated examples and real calibration datasets. • Updated version of Multivariate Calibration 2 (MVC2) graphical user interface. • Trilinear decomposition (TLD) and residual bilinearization (RBL) models included. • Parallel factor analysis 2 (PARAFAC2) and Multivariate curve resolution (MCR-ALS) included. • Rotational ambiguity (RA) effect in MCR-ALS calibration models included. • Analytical figures of merit (AFOMs) for second-order calibration models included. [ABSTRACT FROM AUTHOR]

[Display omitted] • Directions to evaluate the ambiguity in MCR-ALS solutions are given. • The impact of ambiguity on prediction uncertainty is assessed. • The effect of different constraints is shown on the ambiguity extent. • Recommendations are given to report uncertainties in MCR-ALS protocols. The aims of the present tutorial are the following: (1) to show the importance of applying all the chemically sensible constraints which are necessary to retrieve interpretable component profiles when processing second-order analytical data with multivariate curve resolution – alternating least-squares (MCR-ALS), (2) to demonstrate that the constraints allow one to reduce or even remove the impact of the rotational ambiguity in the MCR-ALS solutions, and (3) to provide details on how to estimate the uncertainty in the predicted analyte concentrations which is derived from any remaining ambiguity with a recently developed N-BANDS method. Both simulated and experimental data sets are examined for this purpose. Recommendations are given for including the ambiguity uncertainty in addition to the statistical reports and analytical figures of merit in every MCR-ALS calibration protocol, to assess the reliability of the analytical results. [ABSTRACT FROM AUTHOR]

The possibility of building an interference-free calibration with first-order instrumental data with multivariate curve resolution-alternating least-squares (MCR-ALS) has been a recent topic of interest. When the protocols were successful, MCR-ALS proved to be suitable for the extraction of chemically meaningful information from first-order calibration datasets, even in the presence of unexpected species, i.e., constituents present in the test samples but absent in the calibration set. This may represent an interesting advantage over classical first-order models, e.g. partial least-squares regression (PLS). However, the predictive capacity of MCR-ALS models can be severely affected by rotational ambiguity (RA), which is usually present in first-order datasets when interferents occur, and has not been always characterized in the published analytical protocols. The aim of this report is to discuss important issues regarding MCR-ALS modelling of first-order data for a calibration scenario with a single analyte and one interferent through simulated and experimental data. Specifically, the question of when and why MCR-ALS allows one to build interference-free calibration models with first-order data is studied in terms of signal overlapping, extent of RA, and especially the role of ALS initialization procedures in prediction performance. The aim is to alert analytical chemists that interference-free MCR-ALS with first-order data may not always be successful. [Display omitted] • The second-order advantage with first-order data is investigated. • Multivariate curve resolution may allow for the second-order advantage in this case. • Selectivity and initialization are crucial issues. • The role of rotational ambiguity is studied. [ABSTRACT FROM AUTHOR]

Rotational ambiguity is a phenomenon with the potential of generating an uncertainty in the estimation of analyte concentrations in protocols based on matrix instrumental data processed by multivariate curve resolution - alternating least-squares (MCR-ALS). This is particularly relevant when the second-order advantage is to be achieved, i.e., when selected analytes are determined in unknown samples having unexpected constituents, not considered in the calibration set of samples. It is therefore imperative that analytical chemists developing second-order multivariate calibration methods using MCR-ALS acknowledge the relevance of this issue, and more importantly, have access to the required tools to size the relative impact of this potential source of uncertainty on the estimated analyte concentrations. The purpose of this tutorial is to provide a down-to-earth view of rotational ambiguity, by studying in detail a synthetic example mimicking a typical chromatographic-spectral experiment, where a set of calibration samples is joined with an unknown sample having an uncalibrated interference. After explaining the background information needed to understand the origin of the phenomenon, the available tools for the estimation of the feasible MCR-ALS solutions and the derived uncertainty on analyte predictions will be discussed. A multi-component experimental system will also be discussed, stressing the fact that rotational ambiguity uncertainties, however small, should always be estimated and reported. [Display omitted] • Rotational ambiguity is ubiquitous in multivariate curve resolution. • The phenomenon is explained in simple terms. • Simulated and experimental data are described. • The potential impact on prediction uncertainty is underlined. [ABSTRACT FROM AUTHOR]