Comments on a recently published paper ‘some surprising properties of multivariate curve resolution‐alternating least squares (MCR‐ALS) algorithms’

Some of the results given in a recently published paper in this journal concerning some surprising properties of the multivariate curve resolution‐alternating least squares (MCR‐ALS) method are discussed. My results showed that the surprising properties of MCR‐ALS refer only to the slow linear convergence properties of ALS algorithms and to rounding error computer calculations. Results obtained by MCR‐ALS for the first data example were correct and no significant differences were observed in the resolved profiles. In the second more complex data example, large rotation ambiguities were present for the spectrum profile of the very minor second component which was not correctly estimated by MCR‐ALS. However, even in this case, the subspaces spanned by the MCR‐ALS solutions were also very close to the correct ones apart from slow convergence properties of the MCR‐ALS algorithm in this case. Copyright © 2009 John Wiley & Sons, Ltd.     

Multivariate curve resolution-particle swarm optimization: A high-throughput approach to exploit pure information from multi-component hyphenated chromatographic signals

Multivariate curve resolution-particle swarm optimization (MCR-PSO) algorithm is proposed to exploit pure chromatographic and spectroscopic information from multi-component hyphenated chromatographic signals. This new MCR method is based on rotation of mathematically unique PCA solutions into the chemically meaningful MCR solutions. To obtain a proper rotation matrix, an objective function based on non-fulfillment of constraints is defined and is optimized using particle swarm optimization (PSO) algorithm. Initial values of rotation matrix are calculated using local rank analysis and heuristic evolving latent projection (HELP) method. The ability of MCR-PSO in resolving the chromatographic data is evaluated using simulated gas chromatography–mass spectrometry (GC–MS) and high-performance liquid chromatography–diode array detection (HPLC–DAD) data. To present a comprehensive study, different number of components and various levels of noise under proper constraints of non-negativity, unimodality and spectral normalization are considered. Calculation of the extent of rotational ambiguity in MCR solutions for different chromatographic systems using MCR-BANDS method showed that MCR-PSO solutions are always in the range of feasible solutions like true solutions. In addition, the performance of MCR-PSO is compared with other popular MCR methods of multivariate curve resolution-objective function minimization (MCR-FMIN) and multivariate curve resolution-alternating least squares (MCR-ALS). The results showed that MCR-PSO solutions are rather similar or better (in some cases) than other MCR methods in terms of statistical parameters. Finally MCR-PSO is successfully applied in the resolution of real GC–MS data. It should be pointed out that in addition to multivariate resolution of hyphenated chromatographic signals, MCR-PSO algorithm can be straightforwardly applied to other types of separation, spectroscopic and electrochemical data.     

The application to wastewaters of chemometric approaches to handling problems of highly complex matrices

This overview covers current chemometric methodologies using second-order advantage to solve problems of analyzing highly complex matrices. Among the existing algorithms, it focuses on those most frequently used (e.g., the standard for second-order approaches to data analysis, PARAFAC (parallel factor analysis), and MCR-ALS (multivariate curve resolution alternating least squares), as well as the most recently implemented BLLS (bilinear least-squares), and U-PLS/RBL (unfolded partial least squares/residual bilinearization)). All of these are based on linear models. The overview also covers ANN/RBL (artificial neural networks followed by residual bilinearization), which achieves the second-order advantage in systems involving non-linear behavior. In addition, the overview deals with the drawbacks of these approaches, as well as other drawbacks that are inherent in the analytical techniques to question.     

Application of modern chemometric methods to the study of equilibria in solutions

The possibility of the spectrometric-chemometric study of equilibria in solutions is demonstrated for substances with strongly overlapping spectra, in particular, using the independent component analysis (MILCA and SIMPLISMA algorithms) and the alternating least squares algorithm (MCR-ALS). Using the chemometric approach allows one to resolve spectral curves, identify species present in the solution, and calculate the characteristics of equilibria. The proposed approach is illustrated on a series of examples (study of a tautomeric equilibrium and complexation reactions).     

Molecular conformations from distance matrices

Two algorithms are introduced that show exceptional promise in finding molecular conformations using distance geometry on nuclear magnetic resonance data. The first algorithm is a gradient version of the majorization algorithm from multidimensional scaling. The main contribution is a large decrease in CPU time. The second algorithm is an iterative algorithm between possible conformations obtained from the first algorithm and permissible data points near the configuration. These ideas are similar to alternating least squares or alternating projections on convex sets. The iterations significantly improve the conformation from the first algorithm when applied to the small peptide E. coli STh enterotoxin. © 1993 John Wiley & Sons, Inc.     

Classical and modern algorithms for the evaluation of data from sensor-arrays

Sensors have found wide application in process control, environmental analysis, and other analytical problems in recent years. Optical sensor arrays can be used to monitor organic solvent vapour mixtures by use of reflectometric interference spectroscopy. Lack in selectivity of the sensitive polymer films requires multivariate algorithms for evaluation. Two major aspects are of interest: the random error of calibration and the interpretation of the influence of a single sensor in an array with redundant information. Due to the partial selectivity of the different sensitive layers, non-linearities, cross-sensitivities, and differences in sensitivity, the selection of the most suitable sensitive polymer layers is not trivial. Model based algorithms allow the interpretation of variables whereas the model free algorithms provide better results concerning the random error of calibration. We choose the pruning algorithm to optimize a neural network topology in order to obtain the qualitative information on the sensor elements from the remaining links between the input layer and the hidden layer. We compare these results to the ones obtained for linear and non-linear PLS1 by partial least squares (PLS1) and calculate the errors for the calibration.     

Comparison of robust regression methods based on least-median and adaptive kalman filtering approaches applied to linear calibration data

Five algorithms for data analysis are evaluated for their abilities to discriminate against outliers in small data sets (4–10 points). These methods included least-squares regression, the least absolute -deviation method, the least median of squares method, and two techniques based on an adaptive Kalman filter. For data sets consisting of 4–9 points with one outlier, the average errors in the estimation of the slope were found to be 18.9 % by least-squares, 17.7% by the least absolute deviation method, 0.5% by the least median of squares algorithm, 9.1% by an adaptive Kalman filter algorithm, and 0.9% by a zero-lag adaptive Kalman filter algorithm. Based on these results, the conclusion is that the zero-lag adaptive Kalman filter and the least median of squares approaches are best suited for the detection of outliers in small calibration data sets.     

Classical and modern algorithms for the evaluation of data from sensor-arrays

Sensors have found wide application in process control, environmental analysis, and other analytical problems in recent years. Optical sensor arrays can be used to monitor organic solvent vapour mixtures by use of reflectometric interference spectroscopy. Lack in selectivity of the sensitive polymer films requires multivariate algorithms for evaluation. Two major aspects are of interest: the random error of calibration and the interpretation of the influence of a single sensor in an array with redundant information. Due to the partial selectivity of the different sensitive layers, non-linearities, cross-sensitivities, and differences in sensitivity, the selection of the most suitable sensitive polymer layers is not trivial. Model based algorithms allow the interpretation of variables whereas the model free algorithms provide better results concerning the random error of calibration. We choose the pruning algorithm to optimize a neural network topology in order to obtain the qualitative information on the sensor elements from the remaining links between the input layer and the hidden layer. We compare these results to the ones obtained for linear and non-linear PLS1 by partial least squares (PLS1) and calculate the errors for the calibration. Received: 3 December 1996 / Revised: 27 February 1997 / Accepted: 4 March 1997     

A hybrid genetic algorithm for estimating the equilibrium potential of an ion-selective electrode

Non-linear equations can be used to model the measured potential of ion-selective electrodes (ISEs) as a function of time. This can be done by using non-linear least squares regression to fit parameters of non-linear equations to an ISE response curve. In iterative non-linear least squares regression (which can be considered as local optimisers), the determination of starting parameter estimates that yield convergence to the global optimum can be difficult. Starting values away from the global optimum can lead to either abortive divergence or convergence to a local optimum. To address this issue, a global optimisation technique was used to find initial parameter estimates near the global optimum for subsequent further refinement to the absolute optimum. A genetic algorithm has been applied to two non-linear equations relating the measured potential from selected ISEs to time. The parameter estimates found from the genetic algorithm were used as starting values for non-linear least squares regression, and subsequent refinement to the absolute optimum. This approach was successfully used for both expressions with measured data from three different ISEs; namely, calcium, chloride and lead ISEs.     

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.     

Rejoinder to ‘Comments on a recently published paper “Some surprising properties of multivariate curve resolution‐alternating least squares (MCR‐ALS) algorithms”’

This rejoinder addresses comments recently published in this journal on the paper titled ‘Some surprising properties of multivariate curve resolution‐alternating least squares (MCR‐ALS) algorithms’. It is explained again that the revealed discrepancy of MCR‐ALS algorithms, i.e. the sub‐ and even the final solutions can be outside the range of the data matrix, does exist and this theoretical fact could not be refuted by Tauler. Copyright © 2009 John Wiley & Sons, Ltd.     

Quantification of analytes in overlapping peaks from capillary electrophoresis using multivariate curve resolution-alternating least squares methods

Application of multivariate curve resolution with alternating least squares (ALS) methods to second-order data from capillary electrophoresis diode array detector (CE-DAD) is shown. Second-order data are easily obtained by setting individual data matrix of CE run one in top of the other. Initial qualitative solutions obtained by evolving factor analysis can be further optimized by simultaneous analysis of multiple electrophoresis run data with ALS regression. Quantification is achieved by the comparison of the analyte peak areas with that of pure standards. During the ALS regression procedure, the following constraints were applied: (i) both concentrations and unit pure spectra of the resolved components must be positive; (ii) elution profiles have an unimodal shape; (iii) correspondence exists between common species in the different data matrices; (iv) the pure spectrum of each species is the same in all runs where it is present. The above methods were applied for the determination of dinitrotoluene (DNT) isomeric compounds in overlapping peaks from CE.     

Distance algorithm based procedure for non‐negative least squares

In chemistry and many other scientific disciplines, non‐negativity‐constrained estimation of models is of practical importance. The time required for estimating true least squares non‐negativity‐constrained models is typically many times longer than that for estimating unconstrained models. That is why it is necessary to find faster and faster non‐negative least squares (NNLS) algorithms. Very recently, the distance algorithm has been developed, and this algorithm can be adapted to solve NNLS regression task faster (in some cases) than the conventional algorithms. Based on some simulated investigation, DA_NNLS was the fastest for small‐sized and medium‐sized linear regression tasks. The visualization (geometry) of the NNLS task being solved by our new algorithm is discussed as well. Besides linear algebra, convex geometrical concepts and tools are suggested to investigate, to use, and to develop in chemometrics for exploiting the geometry of chemometry. Copyright © 2014 John Wiley & Sons, Ltd.     

ICE: a new method for the multivariate curve resolution of hyperspectral images

The iterated constrained endmembers (ICE) algorithm is a new method of unmixing hyperspectral images that combines aspects of multivariate curve resolution (MCR) methods in chemometrics and unmixing algorithms in remote sensing. Like many MCR methods, ICE also estimates pure components, or endmembers, via alternating least squares; however, it is explicitly based on a convex geometry model and estimation is carried out in a subspace of reduced dimensionality defined by the minimum noise fraction (MNF) transform. In this paper, we describe the ICE algorithm and its properties. We also illustrate its use on a hyperspectral image of cervical tissue. The unmixing of hyperspectral images presents some unique challenges, and we also outline where further development is required. © 2008 John Wiley & Sons, Ltd.     

A flexible trilinear decomposition algorithm for three-way calibration based on the trilinear component model and a theoretical extension of the algorithm to the multilinear component model

There is a great deal of interest in decompositions of multilinear component models in the field of multi-way calibration, especially the three-way case. A flexible novel trilinear decomposition algorithm of the trilinear component model as a modification of an alternating least squares algorithm for three-way calibration is proposed. The proposed algorithm (constrained alternating trilinear decomposition, CATLD) is based on an alternating approximate least-squares scheme, in which two extra terms are added to each loss function, making it more efficient and flexible. The analysis of simulated three-way data arrays shows that it converges fast, is insensitive to initialization, and is insensitive to the overestimated number of components used in the decomposition. The analysis of real excitation–emission matrix (EEM) fluorescence and real high performance liquid chromatography–photodiode array detection (HPLC–DAD) data arrays confirms the results of the simulation studies, and shows that the proposed algorithm is favorable not only for EEMs but also for HPLC–DAD data. The three-way calibration method based on the CATLD algorithm is very efficient and flexible for direct quantitative analysis of multiple analytes of interest in complex systems, even in the presence of uncalibrated interferents and varying background interferents. Additionally, a theoretical extension of the proposed algorithm to the multilinear component model (constrained alternating multilinear decomposition, CAMLD) is developed.     

Application of maximum likelihood multivariate curve resolution to noisy data sets

In this work, two different maximum likelihood approaches for multivariate curve resolution based on maximum likelihood principal component analysis (MLPCA) and on weighted alternating least squares (WALS) are compared with the standard multivariate curve resolution alternating least squares (MCR‐ALS) method. To illustrate this comparison, three different experimental data sets are used: the first one is an environmental aerosol source apportionment; the second is a time‐course DNA microarray, and the third one is an ultrafast absorption spectroscopy. Error structures of the first two data sets were heteroscedastic and uncorrelated, and the difference between them was in the existence of missing values in the second case. In the third data set about ultrafast spectroscopy, error correlation between the values at different wavelengths is present. The obtained results confirmed that the resolved component profiles obtained by MLPCA‐MCR‐ALS are practically identical to those obtained by MCR‐WALS and that they can differ from those resolved by ordinary MCR‐ALS, especially in the case of high noise. It is shown that methods that incorporate uncertainty estimations (such as MLPCA‐ALS and MCR‐WALS) can provide more reliable results and better estimated parameters than unweighted approaches (such as MCR‐ALS) in the case of the presence of high amounts of noise. The possible advantage of using MLPCA‐MCR‐ALS over MCR‐WALS is then that the former does not require changing the traditional MCR‐ALS algorithm because MLPCA is only used as a preliminary data pretreatment before MCR analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

Partial least squares and random sample consensus in outlier detection

A novel outlier detection method in partial least squares based on random sample consensus is proposed. The proposed algorithm repeatedly generates partial least squares solutions estimated from random samples and then tests each solution for the support from the complete dataset for consistency. A comparative study of the proposed method and leave-one-out cross validation in outlier detection on simulated data and near-infrared data of pharmaceutical tablets is presented. In addition, a comparison between the proposed method and PLS, RSIMPLS, PRM is provided. The obtained results demonstrate that the proposed method is highly efficient.     

Gaussian expansions of orbital products for the evaluation of two-electron integrals

A method is described for evaluating multicenter integrals over contracted gaussian-type orbitals by the use of gaussian expansion of orbital products. The expansions are determined by the method of non-linear least squares with constraints. There is no restriction upon the symmetry of the orbital product and the method is applicable to all products arising from s, p and d-type orbitals. Results are given to indicate the accuracy of the method.     

Characterization of the Acid-Base Properties of Humic Substances by Chemometric Analysis of Synchronous Fluorescence and pH Potentiometric Data

Abstract

A procedure for the analysis of the acid-base characteristics of humic substances based on a self-modeling analysis of synchronous fluorescence spectra, collected at varying pH, and on a non-linear least squares adjustment of potentiometic pH data, is described. The data analysis methodology consists of two steps: first, the number of acid-base systems and the corresponding spectra and distribution diagrams are calculated by evolving factor analysis (EFA) with concentration constraints of the spectroscopic data; second, the potentiometric data is analyzed by a standard non-linear least square procedure using as fixed parameters the number of components and their pKas, determined in the first step of the analysis. As an example, for a sample of marine fulvic acids studied at pH between 2 and 11, four acid-base systems were found with average pKas: 3.1, 4.8, 8.0 and 10.0. The concentrations of the corresponding systems were: 2.55(5), 1.95(7), 0.14(4) and 1.8(3) meq/g.