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The results by soft modeling multivariate curve resolution methods often are not unique and are questionable because of the rotational ambiguity. It means a range of feasible solutions equally fits experimental data and fulfills the constraints. Regarding to chemometric literature, the reduction of the rotational ambiguity in multivariate curve resolution problems is a major challenge in order to construct effective chemometric methods. It is worth to study the effects of applying constraints on the reduction of the rotational ambiguity, since it can help us to choose the useful constraints for multivariate curve resolution methods for analyzing data sets. The aim of this work is to demonstrate the impact of monotonicity and unimodality constraints on the full set of all feasible, nonnegative solutions. We compared the results of two constraints in different two- and three-component systems. To reach this goal, two simulated kinetic and equilibrium data sets are used. Moreover, an experimental data set related to a mixture of two uniprotic acid solutions at different pH values as a model for equilibrium systems is used to extend the discussions to real cases. It is shown in this work that monotonicity is a meaningful chemistry-based constraint which in some cases is more effective than unimodality. • Reduction of both AFS-sets by the monotonicity and unimodality. constraints in SMCR. • Computation of monotone-unimodal areas in outer polygon for U-Space. • The results of two constraints, monotonicity and unimodality, in different two- and three-component systems were compared. [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]

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]

The boundaries of the set of feasible bilinear solutions for three-component systems have been estimated in the presence of noise, as a sequel of a previous report on two-component data sets. Simulations involving a significant degree of overlap among component profiles in both matrix modes have been analyzed using FACPACK and the recently described sensor-wise N-BANDS model. The latter computes the upper and lower envelopes of the set of profiles showing maximum and minimum intensity at each sensor. For three and more components, the graphical representation of the extreme profiles corresponding to maximum and minimum signal contribution function provided by methods such as MCR-BANDS and N-BANDS do not enclose, in general, the full set of feasible profiles. The present results were obtained under non-negativity as the only constraint on the profiles, and were successfully compared with those obtained using a noise addition procedure. A three-component experimental system was also analyzed regarding the combined effect of rotational ambiguity and instrumental noise. • The combined effect of rotational ambiguity and noise is studied. • Sensor wise N-BANDS and FACPACK algorithms are applied. • Results for a simulated three-component system are compared. • An experimental system is analyzed using both models. [ABSTRACT FROM AUTHOR]

Three models are compared regarding the estimation of the boundaries of the range of feasible bilinear solutions brought about by rotational ambiguity in the presence of noise. A two-component system having significant overlapping between the component profiles in both data modes has been studied using FACPACK, based on the so-called polygon inflation algorithm, and two procedures which estimate the extreme profiles corresponding to maximum and minimum signal contribution function: MCR-BANDS and N-BANDS. These latter two models operate by applying different constraints during the optimization phase. Non-negativity was applied as the only constraint on the retrieved profiles. The results were compared with a simulation based on the addition of identically and independently distributed noise at two levels. An experimental system was also analyzed regarding the combined effect of rotational ambiguity and instrumental noise. • The combined effect of rotational ambiguity and noise is studied • MCR-BANDS, N-BANDS and FACPACK algorithms are applied • Results for a simulated two-component system are compared • Two experimental systems are analyzed using N-BANDS [ABSTRACT FROM AUTHOR]

Abstract: In this study the effect of the rotational ambiguity on the accuracy of quantitative results obtained with the help of MCR methods is investigated in detail by simulating different examples chosen from real systems analyzed in previous reported researches. In order to analyze the rotational ambiguity, the systematic grid search minimization algorithm in two and three component systems is employed. The main objective here is to demonstrate that applying not enough constraint to restrict the feasible region makes the estimated concentration for the analyte of interest deviate considerably from the true solution. [Copyright &y& Elsevier]

Rotational ambiguity is a major problem in the application of factor analysis methods to the mixture resolution problem. However, self-modeling curve resolution of spectrochemical data is often able to achieve good separations given fixed spectral characteristics including true zero absorbance values and typically smaller measurement uncertainties. For air pollution mixture problems such as the identification and quantification of sources of ambient particulate matter (PM), there are problems of the lack of true zero values in either the source profiles or their contributions to the ambient concentrations, variability in the source profiles, and typically much larger measurement uncertainties. Recent work has provided several approaches to reducing rotational ambiguity and providing improved resolution including preprocessing the data to remove the meteorologically induced covariance, applying constraints based on a priori knowledge, and analysis of multiple site data. Examples of these applications to airborne PM have shown improved results. • Rotational ambiguity produce significant uncertainty in factor analysis results. • For source apportionment of ambient particle data, there are multiple tools available. • Constraints can be applied based on external, a priori information. • Extraneous meteorological covariance can be reduced by data preprocessing. • Addition of other variables such as pollutant gases can reduce the ambiguity. [ABSTRACT FROM AUTHOR]