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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]

The objective of self modeling curve resolution (SMCR) methods is to decompose a second-order bilinear data matrix into a range of chemically meaningful matrices without any knowledge about the chemical or physical model describing the considered system. In addition, SMCR methods are efficient approaches to deeply investigate data structures by finding not only one of the solutions but all possible ones. Multi-set data analysis can be a powerful tool to decrease the range of feasible solutions in the absence of appropriate conditions for unique resolution. Using SMCR methods, we have investigated the impact of multi-set data analysis on the accuracy of soft modeling results. Interestingly, the feasible regions of individual and simultaneous analysis are compared in a common abstract space. It is demonstrated how such global analysis can result in the reduction of rotational ambiguity in soft modeling analysis. Moreover, as a systematic study, different factors are considered in order to discover the advantages and limitations of multi-set data analysis and lead to a proper design for more accurate results. [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]

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]

Abstract: Although, in many cases parallel factor analysis (PARAFAC) resolves the trilinear data arrays to the true physical factors that form the data, i.e., unique solution can be found, the algorithm does not always converge to chemically meaningful solutions. Kiers and Smilde [J. Chemom. 1995; 9: 179–195] rigorously proved that unique decomposition does not hold in cases with ‘rank overlap’. They showed when PARAFAC is applied on a three-way data array which has rank overlap in one of its loading modes; the solution obtained is not unique and at best cannot be easily compared with the underlying physical factors owing to a rotational ambiguity. An aspect which is significantly less documented in the previous publications is the reliable detection of rotational ambiguities in multi-way methods. A few reported methods are based on bilinear models for calculating the feasible bands of three-way data. In this paper we propose a method to calculate feasible bands of resolved profiles of components in three-way methods and visualize the rotational ambiguity in three-way data in the results of the three-way methods. Most of discussion is in the PARAFAC algorithm. The principle behind the algorithm is described in detail and tested for simulated data set. Completely general and exhaustive results are presented for the two-component cases. In particular, the effect of the noise is investigated and a comparison is made between feasible solutions obtained from PARAFAC and matrix-augmented with trilinearity. It is shown that the results obtained from both methods are identical. [ABSTRACT FROM AUTHOR]

The boundaries of the range of feasible profiles in the bilinear decomposition of a data matrix are estimated by suitably modifying the already published N-BANDS algorithm. Instead of maximizing and minimizing the profile norms, the new sensor-wise N-BANDS approach calculates the profiles showing extreme values of the elements at each sensor in both data modes. This produces two sets of profiles for each sensor, whose envelops define the boundaries of the range of feasible solutions: the upper envelop for maximum element values represents the upper boundary, and the lower envelop the lower boundary. A three-component data set was studied in this manner under non-negativity constraints in both data modes. The sensor-wise N-BANDS results agree with those provided by the FACPACK program, which computes the area of feasible solutions and the associated set of feasible profiles. Computer times required for truly multi-component systems appear to be feasible using sensor-wise N-BANDS (up to eight sample components were checked). As an example, a six-component system was analyzed with the presently described approach, with results which are consistent with the expectations based on the consideration of the degree of profile overlapping among the participating species. • The boundaries of the range of bilinear solutions are estimated. • A modification of the N-BANDS algorithm is applied. • Results for a three-component system are compared with other methods. • Multi-component systems can be studied in a reasonable time. [ABSTRACT FROM AUTHOR]

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]

Since mass spectra are sparse in nature, the "sparsity" has been recently proposed as a constraint in multivariate curve resolution-alternating least squares (MCR-ALS) methods for analyzing LC/GC-MS data. There are different ways for implementation of sparsity constraint, and the majority of such methods rely on imposing a penalty term on the norms of recovered mass spectra. However, the main question is which penalty method is more appropriate for implementation of sparsity constraint in MCR methods. In order to address this question, two- and three-component LC/GC-MS data were simulated and used for the case study, in the work. The areas of feasible solutions (AFS) were calculated using the grid search and Nelder-Mead simplex algorithms. Moreover, different measures of sparsity such as L 0 -norm, L 1 -norm, and L 2 /L 1 for all mass spectra in the AFSs were calculated and visualized as contour plots. The results revealed that all mentioned measures of sparsity find the sparsest solution in AFS. However, from the optimization point of view, L 1 -norm and L 2 /L 1 are easier to implement than L 0 -norm. Approximation methods for solving L 0 -norm problem can take advantage of both unique solution and fast optimization. The results of L 0 -norm, L 1 -norm, and L 2 /L 1 criteria were compared with other sparsity measures such as Shannon entropy, Hoyer, Kurtosis, and Gini indices. The results revealed that L 1 -norm sparsity measure coincides with L 0 -norm, Hoyer, Kurtosis, and Gini indices from the accuracy point of view. Finally, the feasibility of least absolute shrinkage and selection operator (Lasso) was assessed for implementation of L 1 -norm penalty and finding the sparsest solution in MCR. It was found that for small values of λ parameter, Lasso is able to find the sparsest solution with the minimum sum of squares of errors. Thorough optimization of λ is necessary for obtaining accurate results. The lasso-MCR-ALS algorithm was tested with the real GC-MS datasets related to the analysis of Iranian red jujube oil. • The effects of L 0 -norm, L 1 -norm and L 2 /L 1 penalty terms have been explored for implementation of sparsity constraint in MCR. • The values of Shannon entropy, Hyper, Kurtosis and Gini indices were calculated in areas of feasible solutions (AFS). • The minimum values of L 0 -norm and L 1 -norm coincided with maximum Hoyer, Kurtosis and Gini indices in AFSs. • The feasibility of Lasso algorithm with different λ values was assessed for implementation of sparsity constraint in MCR. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution methods suffer from the non-uniqueness of the solutions of the nonnegative matrix factorization problem. The solution ambiguity can be considerably reduced by equality constraints in the form of known spectra or concentration profiles. Two measures are suggested that indicate the impact of the equality constraints. The representation of these measures in the area of feasible solutions show strong variations in the restrictiveness of equality constraints. The measures are tested for a three-component model problem and experimental data sets from the hydroformylation process and a catalyst cluster formation. • The strongly varying restrictiveness of the equality constraint in MCR computations is analyzed by two estimators. • The estimator values are visualized in the area of feasible solutions (AFS) by means of an additional color coordinate. • The additional AFS color coordinate improves the information content and can be generalized to other indicator variables. [ABSTRACT FROM AUTHOR]