Parallel vector analysis based soft resolution of spectral data from pH‐metric titration using symmetry constraint

In this study, a soft method is proposed to calculate concentration and spectral profiles for the two‐way spectral data from dissociation equilibria of polyprotic acids (H_nA). This method has four main distinct steps: (i) a fixed size moving window evolving factor analysis (FSMWEFA) was used to identify the local rank map, (ii) WFA was applied to calculate the concentration profiles of H_nA and Aⁿ⁻ (selection of the window for application of WFA was performed using EFA), (iii) PVA was used to calculate H_{n − 1}A to HA spectral profiles, and (iv) a symmetry constraint, in addition to the non‐negativity constraint, was utilized to obtain the unique concentration and spectral profiles from different acceptable sets of profiles. In the absence of any selective region in the spectral data, the proposed soft method resulted in unique solution without rotational ambiguity. This study is the first application of symmetry constraint on concentration profiles. The rotational ambiguity drastically decreased on considering the constraint of symmetry of the H_{n − 1}A and HA concentration profiles, in addition to non‐negativity of profiles. Simulated examples were used to confirm these approaches. Effect of closeness of dissociation constants on the estimated values of constants was investigated. The results showed that when the difference between pK_a values is more than 1.2, the obtained errors in the estimation of pK_a values are less than about 6.5%. The considered real data were from pH‐metric titration of fluorescein. The obtained spectral and concentration profiles and the estimated pK_a values for fluorescein were in good agreement with the previously reported data. Copyright © 2009 John Wiley & Sons, Ltd.     

Reduction of the rotational ambiguity of curve resolution techniques under partial knowledge of the factors. Complementarity and coupling theorems

Multivariate curve resolution techniques allow to uncover from a series of spectra (of a chemical reaction system) the underlying spectra of the pure components and the associated concentration profiles along the time axis. Usually, a range of feasible solutions exists because of the so‐called rotational ambiguity. Any additional information on the system should be utilized to reduce this ambiguity. Sometimes the pure component spectra of certain reactants or products are known, or the concentration profiles of the same or other species are available. This valuable information should be used in the computational procedure of a multivariate curve resolution technique. The aim of this paper is to show how such supplemental information on the components can be exploited. The knowledge of spectra leads to linear restrictions on the concentration profiles of the complementary species and vice versa. Further, affine–linear restrictions can be applied to pairs of a concentration profile and the associated spectrum of a species. These (affine) linear constraints can also be combined with the usual non‐negativity restrictions. These arguments can reduce the rotational ambiguity considerably. In special cases, it is possible to determine the unknown concentration profile or the spectrum of a species only from these constraints. Copyright © 2012 John Wiley & Sons, Ltd.     

On uniqueness and selectivity in three-component parallel factor analysis

Unambiguous recovery of profiles is a distinguishable advantage of Parallel Factor Analysis (PARAFAC) as a trilinear model and has made it a promising exploratory tool for data analysis. Linear dependency in profiles destroys trilinearity and will increase ambiguity in the curve resolution of three-way data sets. PARAFAC uniqueness deteriorates totally or partially in data sets with linearly dependent loadings. Exploiting a reliable method for determination and direct visualization of feasible bands in the PARAFAC model can be helpful not only in full characterization of uniqueness conditions but also in the investigation of the effects of constraints on the PARAFAC feasible solutions. The purpose of this paper is twofold. First, the calculation of rotational ambiguity in the PARAFAC model extends to three components system. The principle behind the algorithm is described in detail and tested for simulated and real data sets. Completely general and thoroughly investigated results are presented for the three component cases. Secondly, the effects of selective regions in the profiles on the resolution of systems that suffered from the rank deficiency problem, due to rank overlap, are emphasized. In the case of two-way data sets the effect of selectivity constraint on the unique recovery of profiles was investigated and applied. However, to our knowledge, in this report, for the first time, the effect of the presence of selective windows in the profiles, on the unique resolution of three-way data sets has been systematically investigated.     

Determination and visualization of rotational ambiguity in four-component systems

One of the main problems that limit the use of model-free analysis methods for the resolution of multivariate data is that usually there is rotational ambiguity in the result. While methods for the complete definition of rotational ambiguity for two- and three-component systems have been published recently, the comprehensive and general resolution of rotational ambiguity for four-component systems has eluded chemists for several decades. We have developed an extension of self-modelling curve resolution for a mixture of four-components. The performance of the method was verified by applying it to resolve simulated and real data sets.     

On the area of feasible solutions and its reduction by the complementarity theorem

Multivariate curve resolution techniques in chemometrics allow to uncover the pure component information of mixed spectroscopic data. However, the so-called rotational ambiguity is a difficult hurdle in solving this factorization problem. The aim of this paper is to combine two powerful methodological approaches in order to solve the factorization problem successfully. The first approach is the simultaneous representation of all feasible nonnegative solutions in the area of feasible solutions (AFS) and the second approach is the complementarity theorem. This theorem allows to formulate serious restrictions on the factors under partial knowledge of certain pure component spectra or pure component concentration profiles.     

Model‐free multivariate curve resolution combined with model‐based kinetics: algorithm and applications

Multivariate curve resolution techniques are powerful tools to extract from sequences of spectra of a chemical reaction system the number of independent chemical components, their associated spectra, and the concentration profiles in time. Usually, these solutions are not unique because of the so‐called rotational ambiguity. In the present work, we reduce the non‐uniqueness by enforcing the consistency of the computed concentration profiles with a given kinetic model. Traditionally, the kinetic modeling is realized in a separate step, which follows the multivariate curve resolution procedure. In contrast to this, we consider a hybrid approach that combines the model‐free curve resolution technique with the model‐based kinetic modeling in an overall optimization. For a two‐component model problem, the range of possible solutions is analyzed, and its reduction to a single, unique solution by means of the hybrid kinetic modeling is shown. The algorithm reduces the rotational ambiguity and improves the quality of the kinetic fitting. Numerical results are also presented for a multi‐component catalytic reaction system that obeys the Michaelis–Menten kinetics. Copyright © 2012 John Wiley & Sons, Ltd.     

Investigation of the equality constraint effect on the reduction of the rotational ambiguity in three-component system using a novel grid search method

The obtained results by soft modeling multivariate curve resolution methods often are not unique and are questionable because of rotational ambiguity. It means a range of feasible solutions equally fit experimental data and fulfill the constraints. Regarding to chemometric literature, a survey of useful constraints for the reduction of the rotational ambiguity is a big challenge for chemometrician. It is worth to study the effects of applying constraints on the reduction of rotational ambiguity, since it can help us to choose the useful constraints in order to impose in multivariate curve resolution methods for analyzing data sets. In this work, we have investigated the effect of equality constraint on decreasing of the rotational ambiguity. For calculation of all feasible solutions corresponding with known spectrum, a novel systematic grid search method based on Species-based Particle Swarm Optimization is proposed in a three-component system.     

Chemometrics study of the kinetics of the Griess reaction

The kinetics of the Griess reaction in which 3‐nitroaniline acts as a nitrosation agent and 1‐naphtylamine as a coupling reagent was studied by chemometrics methods. The kinetic reaction was investigated under pH 1.0 and 25°C by UV‐vis spectrophotometry. The concentrations of nitrite, 3‐nitroaniline and 1‐naphtylamine were such that a second‐order kinetic reaction took place. Data explorations based on principal component analysis and multivariate curve resolution–alternating least squares were performed to obtain information about the reaction. Calculation of band boundaries of the multivariate curve resolution–alternating least squares solutions showed that the rotational ambiguities associated with the calculation of spectra and concentration profiles have been completely removed. The decrease in the ambiguity of the recovered solutions was closely related to the application of the equality constraint. The results of the exploratory data analysis showed that the kinetic reaction proceeds through a two‐step mechanism. Moreover, the two‐steps are second order. Data analysis approaches based on hard modeling and global hard modeling were used to resolve profiles of the reactants, intermediates and products and to evaluate the rate constants. Copyright © 2013 John Wiley & Sons, Ltd.     

Hard–soft modeling parallel factor analysis to solve equilibrium processes

PARAFAC model is the most famous model for analyzing three‐way data. However, this method does not converge to chemically meaningful solutions when applied to three‐way problems involving rank overlap profiles at least in one mode. Rank overlap can be simply found where components have similar spectral profiles or analytes appearing in identical proportions throughout an experiment. However, an appropriate selection of the initial parameters and constraints such as non‐negativity and unimodality can still make PARAFAC model useful in this regard. Although such constraints reduce rotational freedom in PARAFAC solution, they are generally insufficient to wholly eliminate the rotational problem. The goal of the present paper is to incorporate hard modeling constraint in the soft‐modeled PARAFAC algorithm to overcome non‐uniqueness problem in the equilibrium processes involving linearly dependent factors at least in one mode. The hard constraint is introduced to force some or all of the concentration profiles to fulfill an equilibrium model that is refined at each iteration cycle of the optimization process of PARAFAC. The proposed approach is called hard–soft PARAFAC (HSPARAFAC). When the rank overlap species obeys equilibrium model in HSPARAFAC, the unique results are obtained even in the presence of non‐modeled interferences. The new modification in the treatment of equilibrium data sets yields more satisfactory results than the exclusive PARAFAC algorithm. Simulated and real examples with rank overlap problem are used to confirm this statement. Copyright © 2011 John Wiley & Sons, Ltd.     

Characterization and determination of stability constants of copper(II)-l-histidine complexation system by using multivariate curve resolution method of visible spectra and two hard modeling methods in aqueous solutions

Multivariate curve resolution, alternating least-squares is applied to spectra data obtained in the study of Cu(II) complexation by l-histidine. The combination of several chemometric techniques based on factor analysis (FA), singular value decomposition (SVD), evolving factor analysis (EFA), and multivariate curve resolution with constrained alternating least-squares (ALS) is used to determine the number of species and their distribution diagram. This multivariate analysis data treatment simultaneously reveals the species Cu, CuL, CuLH, CuL₂, CuL₂H, and CuLOH, through the calculated concentration profiles and allows the assignment of numerically obtained pure individual spectra. Formation constants of these species were calculated by hard-modeling methods applied potentiometric and spectrophotometric measurements.     

A review of recent methods for the determination of ranges of feasible solutions resulting from soft modelling analyses of multivariate data

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.     

Application of soft- and hard-modelling approaches to resolution of kinetics of electron donor-acceptor complex formation of 2,3-dichloro-5,6-dicyano-1,4-benzoquinone with imipramine in different solutions

Kinetics of electron donor-acceptor (EDA) complex formation of imipramine and 2,3-dichloro-5,6-dicyano-1,4-benzoquinone (DDQ) was investigated spectrophotometrically in acetonitrile, 1,2-dichloroethane, and chloroform solutions using soft- and hard-modelling approaches. From the results of exploratory analysis of kinetic data and the spectral changes by soft-modelling approaches, evolving factor analysis (EFA) and orthogonal projection approach (OPA), a consecutive two-steps reaction with two intermediates was proposed for the process in acetonitrile and 1,2-dichloroethane media and one with a single intermediate in chloroform solution. Secondly, by applying, multivariate nonlinear least squares hard-modelling approach on the collected experimental kinetic data matrix, the nonlinear parameters (rate constants) as well as the linear parameters (spectral profiles) were obtained by fitting the collected experimental kinetic data matrix to the proposed model. Small values of standard deviation in the resulting parameters and sum of squares of the residuals (ssq) obtained showed the proper selection of the model. Furthermore, the values of lack of fit and percent of explained variance confirmed the correct identified models. Identification of the model with the aid of soft-modelling approaches followed by application of the hard-modelling approaches decreases significantly the rotational ambiguity associated with the obtained concentration and spectral profiles. Variations in the kinetic constants were in complete agreement with the model proposed and the solvent polarities.     

On the ambiguity of the reaction rate constants in multivariate curve resolution for reversible first-order reaction systems

If for a chemical reaction with a known reaction mechanism the concentration profiles are accessible only for certain species, e.g. only for the main product, then often the reaction rate constants cannot uniquely be determined from the concentration data. This is a well-known fact which includes the so-called slow-fast ambiguity.     

A fast polygon inflation algorithm to compute the area of feasible solutions for three‐component systems. I: concepts and applications

The multicomponent factorization of multivariate data often results in nonunique solutions. The so‐called rotational ambiguity paraphrases the existence of multiple solutions that can be represented by the area of feasible solutions (AFS). The AFS is a bounded set that may consist of isolated subsets. The numerical computation of the AFS is well understood for two‐component systems and is an expensive numerical process for three‐component systems. In this paper, a new fast and accurate algorithm is suggested that is based on the inflation of polygons. Starting with an initial triangle located in a topologically connected subset of the AFS, an automatic extrusion algorithm is used to form a sequence of growing polygons that approximate the AFS from the interior. The polygon inflation algorithm can be generalized to systems with more than three components. The efficiency of this algorithm is demonstrated for a model problem including noise and a multicomponent chemical reaction system. Further, the method is compared with the recent triangle‐boundary‐enclosing scheme of Golshan, Abdollahi, and Maeder (Anal. Chem. 2011, 83, 836–841). Copyright © 2013 John Wiley & Sons, Ltd.     

Definition and detection of data-based uniqueness in evaluating bilinear (two-way) chemical measurements

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.     

Estimation of rotation ambiguity in multivariate curve resolution with charged particle swarm optimization (cPSO‐MCR)

Rotation ambiguity (RA) in multivariate curve resolution (MCR) is an undesirable case, when the physicochemical constraints are not sufficiently strong to provide a unique resolution of the data matrix of the mixtures into spectra and concentration profiles of individual chemical components. RA is often met in MCR of overlapped chromatographic peaks, kinetic and equilibrium data, and fluorescence two‐dimensional spectra. In case of RA, a single candidate solution has little practical value. So, the whole set of feasible solutions should be characterized somehow. It is a quite intricate task in a general case. In the present paper, a method was proposed to estimate RA with charged particle swarm optimization (cPSO), a population‐based algorithm. The criteria for updating the particles were modified, so that the swarm converged to the steady state, which spanned the set of feasible solutions. The performance of cPSO‐MCR was demonstrated on test functions, simulated datasets, and real‐world data. Good accordance of the cPSO‐MCR results with the analytical solutions (Borgen plots) was observed. cPSO‐MCR was also shown to be capable of estimating the strength of the constraints and of revealing RA in noisy data. As compared with analytical methods, cPSO‐MCR is simpler to implement, expands to more than three chemical compounds, is immune to noise, and can be easily adapted to virtually all types of constraints and objective functions (constraint based or residue based). cPSO‐MCR also provides natural visual information about the level of RA in spectra and concentration profiles, similar to the methods of two extreme solutions (e.g., MCR‐BANDS). Copyright © 2014 John Wiley & Sons, Ltd.     

Investigating the effect of flexible constraints on the accuracy of self‐modeling curve resolution methods in the presence of perturbations

Self‐modeling curve resolution methods have continuously been improved during recent years. Many efforts have been made on curve resolution methods to reduce the rotational ambiguity by means of different types of constraints. Choosing proper constraints and cost functions is critically important for the reduction of the rotational ambiguity because the constraints have a direct influence on the accuracy of the area of feasible solution (AFS). In this work, we introduce a new improved cost function, which serves to apply nonnegativity, unimodality, equality, and closure constraints. We also investigate the reduction of the AFS under hard and soft constraints. Another point of this work is to evaluate the accuracy and precision of the reduced AFS in the presence of noise and perturbations, under hard and soft implementation of nonnegativity, unimodality, equality, and closure constraints. A comparison is given between the reduced AFS with soft constraints (small deviations from constraints are accepted) and the reduced AFS under hard constraints (restrictedly forced constraints). A graphical visualization of this comparison is presented for various model problems. The results show that an AFS computation with soft constraints provides more reliable results, especially in the presence of noise. The test problems substantiate significant advantages of soft constraints over hard constraints because the obtained profiles are closer to the potentially true noisy profiles, which contain small deviations from ideal responses. Using tunable parameters ϵ,γ,ω,δ is one of the advantages of soft constrained cost function that allows the small deviations from ideal responses. Ultimately, soft constraints can help to reduce the lack‐of‐fit, and they are a proper instrument to handle the effect of noise on the AFS. Copyright © 2016 John Wiley & Sons, Ltd.     

Free energy profile along a discretized reaction path via the hyperplane constraint force and torque

By employing mechanical work analogies, we derive a convenient computational approach for evaluation of the free energy profile (FEP) along some discretized path defined as a sequence of hyperplanes. A hyperplane is fully specified by any of its point and a tangent vector. The FEP is obtained as an integral of two components. The translational component of the free energy is computed by integrating the hyperplane constraint force. The rotational component is evaluated via the hyperplane torque. Both ingredients--the constraint force and the hyperplane torque-are evaluated on each hyperplane independently. The integration procedure utilizes a set of reference points defining a point of rotation on each hyperplane, and these points can be chosen before or after the sampling takes place. A shift in the reference points redistributes the FEP contributions between the translational and rotational components. For systems where the FEP is dominated by the potential energy differences, reference points residing on the minimum energy path present a natural choice. We demonstrate the validity of our approach on two examples, a simple two-dimensional (2D) potential, and a seven-atom Lennard-Jones cluster. In each case, we compare the numerical FEP with the harmonic approximation estimates. Our results for the 2D potential are also verified by the data available in the literature. In both cases, the rotational component of the FEP represents a sizable contribution to the total FEP, so ignoring it would yield clearly incorrect results.     

Fluorescence,fluorescence-excitation and absorption spectra of trans-1-(2-anthryl)-2-phenylethene conformers

Fluorescence spectra of trans-1-(2-anthryl)-2-phenylethene (APE) obtained under varying conditions of excitation wavelength and oxygen concentration in toluene are resolved into two distinct components by application of principal component analysis with self-modeling (PCA-SM). Self-modeling is guided by the constraint that Stem-Volmer quenching plots for the individual conformers be independent of excitation wavelength. The same process applied on a matrix of fluorescence-excitation spectra leads to resolved conformer-specific fluorescence-excitation spectra. Consistency between the fluorescence and the fluorescence-excitation spectrum of each conformer is established. The fluorescence-excitation spectr and literature fluorescence quantum yields are used to resolve the UV absorption spectrum of APE. The pure conformer spectra obtained in this work are compared with those from earlier PCA-SM treatments in which self-modeling procedures were based on the Lawton and Sylvestre nonnegativity constraint and on a maximal spectral dissimilarity constraint.     

Explicit equations for the height and position of the first component shock for binary mixtures with competitive Langmuir isotherms under ideal conditions

Explicit equations for the height c(1)(S) and retention time t(R,1) of the pure first component shock in the case of a narrow rectangular injection pulse of a binary mixture with competitive Langmuir isotherms were derived within the frame of the equilibrium theory. The height of the first shock is obtained as an only positive root of a quartic equation. Hence, it was shown that, for binary Langmuir systems, the individual concentration profiles at the column outlet can be expressed entirely in closed-form. In addition, a novel, simple parametric representation that gives the trajectory of the first shock in the distance-time diagram as a function of c(1)(S) was derived. The practical relevance of the new equations was demonstrated by utilizing them for optimization of batch chromatography. It was shown that c(1)(S) increases and t(R,1) decreases with increasing duration of injection for given feed concentrations when the pure first component plateau is eroded during elution. The derivative of the cycle time with respect to the duration of injection is always more than unity. For this reason, the maximum productivity of more retained component is obtained when the duration of injection is selected so that the purity constraint can be fulfilled by having 100% yield. For the less retained component, an implicit expression for the maximum productivity was derived. When the injected loadings are constant, t(R,1) decreases with increasing feed concentrations while c(1)(S) and the cycle time are independent of them. In addition, the productivities of both components always increase with increasing feed concentrations.