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Evaluation of uncertainties due to rotational ambiguities and noise propagation is essential to ascertain the reliability of Multivariate Curve Resolution estimations. When ambiguity is present, every resolved profile can be represented by a band of feasible solutions instead of by a unique profile. In the presence of experimental noise the estimation of this band of feasible solutions is more difficult and uncertain. The aim of this work is to show how experimental noise and profiles overlapping affect the reliability of feasible solutions. For this purpose, feasible solutions of several two and three components data systems with different profiles overlapping and noise levels have been systematically investigated. Results obtained using simulated and experimental data showed that increasing noise levels and profiles overlapping in one mode (e.g. concentration profiles) make more uncertain the estimation of feasible solutions in the other mode (e.g. spectral profiles), and this uncertainty is propagated in a non-uniform and complex way. [ABSTRACT FROM AUTHOR]

A major problem that limit the use of the model-free analysis methods, like positive matrix factorization (PMF), for the resolution of multivariate data is that usually, there is rotational ambiguity in the results. Implementing adequate constraints may reduce such ambiguities. In some special cases, non-negativity constraints may serve as a sufficient condition to uniquely resolve profiles. In this study, effect of adding new variables on the extent of rotational ambiguity is investigated, and profiles with lower rotation have been obtained on the basis of data set structure. The displacement (DISP) method of error estimation (EE) for the factor elements incorporated into the US Environmental Protection Agency's version of positive matrix factorization (PMF) has been applied to capture the uncertainty of PMF analyses due to rotational ambiguity. To explore these considerations, results are presented for synthetic including submicron particle size distributions (12–470 nm) measured between January 2008 and December 2010 at the New York State Department of Environmental Conservation (NYS DEC) site in Rochester, NY. Generally, added informative variables can reduce the ambiguity in the profiles. The results of the PMF analysis showed the effect of adding new variables to feasible solutions led to significantly decreased (up to 44%) DISP intervals. More robust results with lower rotational uncertainty became possible and the results were more easily interpreted. This approach can be applied to any other field that utilize quantitative factor analysis methods. [ABSTRACT FROM AUTHOR]

Soft modelling or multivariate curve resolution (MCR) are well-known methodologies for the analysis of multivariate data in many different application fields. Results obtained by soft modelling methods are very likely impaired by rotational and scaling ambiguities, i.e. a full range of feasible solutions can describe the data equally well while fulfilling the constraints of the system. These issues are severely limiting the applicability of these methods and therefore, they can be considered as the most challenging ones. The purpose of the current review is to describe and critically compare the available methods that attempt at determining the range of ambiguity for the case of 3-component systems. Theoretical and practical aspects are discussed, based on a collection of simulated examples containing noise-free and noisy data sets as well as an experimental example. [ABSTRACT FROM AUTHOR]

In this work it is shown that soft-modeling methods can be generalized to multivariate calibration of first-order data sets. Second-order data sets are natural type of data to be analyzed with soft-modeling methods. Recently these methods, especially MCR-ALS are applied for analyzing first-order data. In first-order multivariate calibration procedures such as PLS, interferences in the unknown set are also contained in the standards used to calibrate the system. In this work the reliability of the analyte quantitations in the current multivariate calibration practices with soft-modeling method is investigated. It is shown that applying soft-modeling methods in analyzing first-order data results in unique concentration values for the analyte of interest under two main conditions: first, all the present components in the unknown set should be concluded in the calibration set. Second, the known concentration of the analyte in the calibration set should be incorporated as a constraint during the iterations. A simulated data containing three components and an experimental system with synthetic mixtures containing three artificial dyes – amaranth, tartrazine and sunset yellow – was analyzed. Grid search minimization approach was used to analyze rotational ambiguity associated to the resolution of the components present in the simulated and experimental data sets. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution methods, frequently used in analyzing bilinear data sets, result in ambiguous decomposition in general. Implementing the adequate constraints may lead to reduce the so-called rotational ambiguity drastically, and in the most favorable cases to the unique solution. However, in some special cases, non-negativity constraint as minimal information of the system is a sufficient condition to resolve profiles uniquely. Although, several studies on exploring the uniqueness of the bilinear non-negatively constrained multivariate curve resolution methods have been made in the literature, it has still remained a mysterious question. In 1995, Manne published his profile-based theorems giving the necessary and sufficient conditions of the unique resolution. In this study, a new term, i.e., data-based uniqueness is defined and investigated in details, and a general procedure is suggested for detection of uniquely recovered profile(s) on the basis of data set structure in the abstract space. Close inspection of Borgen plots of these data sets leads to realize the comprehensive information of local rank, and these argumentations furnish a basis for data-based uniqueness theorem. The reported phenomenon and its exploration is a new stage (it can be said fundament) in understanding and describing the bilinear (matrix-type) chemical data in general. Our proposed detection tool is restricted to three-component systems because of the visual limitations of the Borgen plot, but the theorem is universal for systems with more than three components. A recently published experimental four-component system is used for illustrating this theorem in the case of systems with more than three components. [ABSTRACT FROM AUTHOR]

Abstract: Differential pulse voltammetry technique assisted by chemometric methods such as multivariate curve resolution-alternating least squares (MCR-ALS) has been proposed as a valuable approach for (+)-catechin determination in the presence of gallic acid at the surface of multiwalled carbon nanotube paste electrode. Central composite design and response surface methodology were used to optimize the influencing parameters. To determine (+)-catechin in the presence of unexpected electroactive interference with a very high degree of overlapping, second-order electrochemical data were generated by changing the pulse height as an instrumental parameter. After potential shift correction, MCR-ALS results show that second-order calibration could be applied with great success for electroanalytical determination of highly overlapped electroactive species. The linear least-squares calibration curve based on the area under concentration profile was provided over the range of 0.10–2.69μM for (+)-catechin, whereas detection limit was found to be 0.017μM. Also in this study, the effect of rotational ambiguity associated with a particular MCR solution under a set of constraints was investigated. [Copyright &y& Elsevier]

Highlights: [•] A new MCR method based on particle swarm optimization (PSO) is developed. [•] Multi-component simulated GC–MS and HPLC–DAD data are successfully resolved. [•] Performance of MCR-PSO algorithm is compared with MCR-ALS and MCR-FMIN. [•] MCR-PSO is used for high-throughput analysis of real chromatographic data. [Copyright &y& Elsevier]

Abstract: Rotational ambiguity is a major problem in the application of soft-modeling analysis to a variety of multivariate mixture resolution problems and particularly important in the analysis of kinetic data. Soft-modeling analyses rely on constraints that restrict the concentration profiles and/or the spectral responses of all components. The main goal of this work is to demonstrate how a hard-modeling constraint on concentration profiles drastically decreases the extent of the rotational ambiguity. Therefore, in the present paper the discussion is focused on systems in which hard-modeling information is available. The results of simulated examples reveal that the utilized hard constraint decreases the rotational ambiguity in estimated concentration profile even components that do not take part in the explicit model. In addition, the rate constant of known reaction is determined in this method. [Copyright &y& Elsevier]

Abstract: In this article, we examine Independent Component Analysis (ICA) and the concept of Mutual information (MI) as a quantitative measure of independence from the point of view of analytical chemistry. We compare results obtained by different ICA methods with results obtained by Multivariate Curve Resolution Alternating Least Squares (MCR-ALS). These results have shown that, when non-negativity constraints are applied, values of MI increase considerably and the resolved components cannot anymore be considered to be independent (i.e. they can only be considered to be the “least dependent” components). MI values of profiles resolved by MCR-ALS and ICA did not differ significantly when non-negativity constraints were applied. In addition, since data-fitting values were also practically the same in both cases, the solutions provided by them should be considered equivalent from a mathematical point of view and within the region of feasible solutions for the particular set of applied constraints. We therefore conclude that the solutions based only on the independence concept are not necessarily better from the point of view of analytical chemistry than those obtained by other proposed MCR methods. Results obtained in this work also show that ICA can be considered an alternative tool for resolving mixed signals only in a limited number of cases. [Copyright &y& Elsevier]

Abstract: Nowadays self-modeling/multivariate curve resolution algorithms have become very popular in chemometrics, i.e. for evaluating analytical chemical measurements. The developments split into two directions: (1) finding band solution and (2) finding unique solution. For band solutions the task is to find the band boundaries of the feasible regions. The size of the range calculated in this way can be considered as the measure of the rotational ambiguity. In this paper the developed methods are compared and some theoretical and practical considerations are given according to the improper and proper calculations. [Copyright &y& Elsevier]