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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]

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]

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]

the chemometric processing of second-order chromatographic-spectral data is usually carried out with the aid of multivariate curve resolution-alternating least-squares (MCR-ALS). When baseline contributions occur in the data, the background profile retrieved with MCR-ALS may show abnormal lumps or negative dips at the position of the remaining component peaks. The phenomenon is shown to be due to remaining rotational ambiguity in the obtained profiles, as confirmed by the estimation of the boundaries of the range of feasible bilinear profiles. To avoid the abnormal features in the retrieved profile, a new background interpolation constraint is proposed and described in detail. Both simulated and experimental data are employed to support the need of the new MCR-ALS constraint. In the latter case, the estimated analyte concentrations agreed with those previously reported. The developed procedure helps to reduce the extent of rotational ambiguity in the solution and to better interpret the results on physicochemical grounds. [Display omitted] • A new background interpolation constraint is discussed for MCR-ALS. • The constraint allows one to correctly retrieved baseline profiles. • Physical interpretation and quantitative analysis may be greatly improved. • The new constraint can be easily added to the already available in MCR-ALS. [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]

The objective of the present work is development of joint approximate diagonalization of eigenmatrices (JADE) as a member of independent component analysis (ICA) family, for the analysis of gas chromatography-mass spectrometry (GC–MS) and comprehensive two-dimensional gas chromatography-mass spectrometry (GC × GC–MS) data to address incomplete separation problem occurred during the analysis of complex sample matrices. In this regard, simulated GC–MS and GC × GC–MS data sets with different number of components, different degree of overlap and noise were evaluated. In the case of simultaneous analysis of multiple samples, column-wise augmentation for GC–MS and column-wise super-augmentation for GC × GC–MS was used before JADE analysis. The performance of JADE was evaluated in terms of statistical parameters of lack of fit (LOF), mutual information (MI) and Amari index as well as analytical figures of merit (AFOMs) obtained from calibration curves. In addition, the area of feasible solutions (AFSs) was calculated by two different approaches of MCR-BANDs and polygon inflation algorithm (FACPACK). Furthermore, JADE performance was compared with multivariate curve resolution-alternating least squares (MCR-ALS) and other ICA algorithms of mean-field ICA (MFICA) and mutual information least dependent component analysis (MILCA). In all cases, JADE could successfully resolve the elution and spectral profiles in GC–MS and GC × GC–MS data with acceptable statistical and calibration parameters and their solutions were in AFSs. To check the applicability of JADE in real cases, JADE was used for resolution and quantification of phenanthrene and anthracene in aromatic fraction of heavy fuel oil (HFO) analyzed by GC × GC–MS. Surprisingly, pure elution and spectral profiles of target compounds were properly resolved in the presence of baseline and interferences using JADE. Once more, the performance of JADE was compared with MCR-ALS in real case. On this matter, the mutual information (MI) values were 1.01 and 1.13 for resolved profiles by JADE and MCR-ALS, respectively. In addition, LOD values (μg/mL) were respectively 1.36 and 1.24 for phenanthrene and 1.26 and 1.09 for anthracene using MCR-ALS and JADE which showed outperformance of JADE over MCR-ALS. [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]

In the present contribution, a new approach based on mutual information (MI) is proposed for exploring the independence of feasible solutions in two component systems. Investigating how independent are different feasible solutions can be a way to bridge the gap between independent component analysis (ICA) and multivariate curve resolution (MCR) approaches and, to the best of our knowledge, has not been investigated before. For this purpose, different chromatographic and hyperspectral imaging (HSI) datasets were simulated, considering different noise levels and different degrees of overlap for two-component systems. Feasible solutions were then calculated by both grid search (GS) and Lawton-Sylvester (LS) plots. MI map which is the plot of MI vs. rotation matrix elements was used to estimate the degree of independence between different solutions. Inspection of the results showed that the different solutions in the feasible bands correspond to different MI values and that those values are lower for spectral profiles (more independent) than for concentration profiles (more dependent) as expected from the duality concept and the opposite is true. In addition, component profiles are found near more dependent solutions for concentration profiles and near less dependent solutions for spectral profiles which is due to the fact that "independence" constraint was applied to the spectral profiles in ICA algorithms. The performance of three well-known ICA algorithms (mean-field independent component analysis (MF-ICA), mutual information-based least dependent component analysis (MILCA) and joint approximate diagonalization of eigenmatrices (JADE)) as well as MCR-alternating least squares (MCR-ALS) were investigated. MI maps showed that the solutions of MF-ICA and MCR-ALS are in the feasible bands but the MILCA and JADE solutions which are just based on the independence maximization are outside the MI maps. [Display omitted] • MI map is proposed as a new way for exploring the independence of feasible solutions. • Performance of MI map was compared with error map to find meaningful solutions. • MI map was proposed for comparing the performance of different ICA algorithms. • Different simulated and real chromatographic and hyperspectral imaging data was tested. • MF-ICA and MCR-ALS solutions were inside the GS and LS feasible bands. [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]

[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 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]

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]

Analysis of fragrance composition is very important for both the fragrance producers and consumers. Unraveling of fragrance formulation is necessary for quality control, competitor and trace analysis. Gas chromatography–mass spectrometry (GC–MS) has been introduced as the most appropriate analytical technique for this type of analysis, which is based on Kovats index and MS database. The most straightforward method to analyze a GC–MS dataset is to integrate those peaks that can be recognized by their mass profiles. But, because of common problems of chromatographic data such as spectral background, baseline offset and specially overlapped peaks, accurate quantitative and qualitative analysis could be failed. Some chemometric modeling techniques such as bilinear multivariate curve resolution (MCR) methods have been introduced to overcome these problems and obtained well resolved chromatographic profiles. The main drawback of these methods is rotational ambiguity or nonunique solution that is represented as area of feasible solutions (AFS). Polygonal inflation algorithm (PIA) is an automatic and simple to use algorithm for numerical computation of AFS. In this study, the extent of rotational ambiguity in curve resolution methods is calculated by MCR-BAND toolbox and the PIA. The ability of the PIA in resolving GC–MS data sets is evaluated by simulated GC–MS data in comparison with other popular curve resolution methods such as multivariate curve resolution alternative least square (MCR-ALS), multivariate curve resolution objective function minimization (MCR-FMIN) by different initial estimation methods and independent component analysis (ICA). In addition, two typical challenging area of total ion chromatogram (TIC) of commercial fragrances with overlapped peaks were analyzed by the PIA to investigate the possibility of peak deconvolution analysis. [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]

Abstract: In recent years source apportionment of observed PM2.5 has been improved by incorporating meteorological information as additional factors in receptor modelling studies using ME-2. In this study we replace one of these meteorological factors, namely, parameterisation by wind speed, with a parameterisation based on hourly observations of the naturally occurring terrestrial gas Radon-222 (radon), and compare results of the two parameterisations over five years at an inland site in the Greater Sydney Region. The efficacy of the wind speed and radon parameterisation techniques is assessed by comparing regressions between the daily contributions from identified elemental fingerprints estimated from the wind speed and radon multi-linear models against those obtained from the corresponding bi-linear model (while the two models are solved simultaneously). The radon parameterisation yielded improved regressions for all source fingerprints, most notably Smoke and Autos (r 2 = 0.67 and 0.65, respectively, compared to 0.57 and 0.47 when the wind speed parameterisation was used). Both parameterisation schemes were equally effective in attributing PM2.5 to wind sectors known to contain sources characterised by the observed fingerprints. Our findings demonstrate that incorporating radon as a parameter in ME-2 can lead to an improved PM2.5 source apportionment than that obtained using meteorological parameters alone, particularly for inland sites with distributed sources. [Copyright &y& Elsevier]

Highlights: [•] Novel comprehensive determination of feasible solutions for 4-component systems. [•] The method works in the presence of noise. [•] Visualization of data structure in both concentration and spectral subspaces. [ABSTRACT FROM AUTHOR]

Highlights: [•] Studying the effect of the equality constraint on the rotational ambiguity. [•] Extending grid search method using SPSO to find solutions are corresponding. [•] The equality constraint effect on complementary space shown by linear algebra. [Copyright &y& Elsevier]

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]

Highlights: [•] Calculation of rotational ambiguity in PARAFAC extends to three components systems. [•] Selective window in profiles causes uniqueness in PARAFAC on rank deficient systems. [•] An independent factor in two modes with selective window recovered uniquely by PARAFAC. [•] Kruskal condition in sufficient, not necessary for R =2 and R =3. [•] Effect of selective windows on uniqueness in PARAFAC is investigated systematically. [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: 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: 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: 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]

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]

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]

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]

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]

Rotational ambiguity in factor analyses prevents users from obtaining accurate source apportionment results. The rotational space in positive matrix factorization (PMF) can be reduced by constraining the solution with a priori information such as source profiles. However, the only prior report on constraints using information on the source contributions was their use to ensure compatibility in the simultaneous analyses of PM 2.5 and PM 10 data. By combining data from three monitoring sites affected by a gear casting plant in Xi'an as an example, a methodology for improving the accuracy of PMF results by constraining source contributions using wind information was explored. Seven common factors derived from individual PMF analyses for each of the three sites (INDUS, URBAN, and RURAL) with different location characteristics, were then combined in a multisite PMF analysis. The factors were interpreted as nitrate with all site average contributions of 28.7%, sulfate (22.5%), coal combustion (19.3%), road traffic (12.8%), biomass burning (6.4%), soil (5.4%), and metallurgical industry (4.9%). Except for the INDUS site, contributions of metallurgical industry to the URBAN and RURAL sites were pulled down maximally to reduce the rotational space. The constrained solution substantially improved the results over the base run. The local and regional nature of the sources were identified by coefficient of divergence combined with Pearson correlation analysis, and further quantitatively estimated using Lenschow approach. On average, local sources contributed for 52.4% and 47.7% of the PM 2.5 mass concentrations at the INDUS and URBAN site respectively. The metallurgical industry showed the highest local contributions while sulfate was primarily regional. For the multisite analysis where there are considerable point source emissions, this methodology highlights the role of local wind directions to inform constraints on the results and obtaining more reliable solutions. Unlabelled Image • PM 2.5 composition measured at three sites were investigated using multisite PMF. • G-value constraints were developed with information of winds and point source location. • The constrained solution substantially improved the results over the base run. • The local and regional nature of the sources were estimated using Lenschow approach. [ABSTRACT FROM AUTHOR]

• Multiset data analysis can be a proper way to increase the accuracy of multivariate curve resolution (MCR) results, which is dropped due to rotational ambiguity. • Unique resolution of the data can only be obtained under uniqueness conditions, i.e. resolution theorems and databased-uniqueness. However, the conditions for obtaining the unique solution by resolution theorems are entirely dependent on the estimation of selectivity and local rank information. • With multiset analysis, there is a possibility to design the calibration data sets that provide conditions for resolution theorems through the application of correspondence constraint. • Simultaneous analysis of multiple data sets, even without applying correspondence, can improve the accuracy of results and in some cases obtain the unique solution based on databased-uniqueness. Multiset data analysis can be a proper way to increase the accuracy of multivariate curve resolution (MCR) results, which is dropped due to rotational ambiguity. Resolution theorems and databased-uniqueness introduce the conditions for obtaining the unique solutions. However, the conditions for resolution theorems are dependent on the estimation of local rank information. With multiset analysis, there is a possibility to design the calibration data sets that provide conditions for resolution theorems. Therefore, correspondence constraint, which express the presence or absence of the components in each data set, can be applied in simultaneous analysis to implement the local rank information. Furthermore, simultaneous analysis even without applying correspondence can improve the accuracy of results and in some cases obtain the unique solution based on databased-uniqueness. The results can be explained by the duality concept, which explains the relationship between row and column vector spaces. Two simulated examples and one real excitation-emission fluorescence data were analyzed to explore the unique solutions. The results illustrate how a proper design of multiset data can lead to a unique solution in MCR methods while providing further information about the system. Therefore, not only multiset analysis can improve the results in well-designed data sets, but also can provide the situation to apply correspondence and lead to a unique resolution. [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]