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.     

Simultaneous Anodic Stripping Voltammetric Determination of Pb and Cd,Using a Vibrating Gold Microwire Electrode,Assisted by Chemometric Techniques

Simultaneous anodic stripping voltammetric determination of Pb and Cd is restricted on gold electrodes as a result of the overlapping of these two peaks. This work describes the quantitative determination of a binary mixture system of Pb and Cd, at low concentration levels (up to 15.0 and 10.0 µg L^?1 for Pb and Cd, respectively) by differential pulse anodic stripping voltammetry (DPASV; deposition time of 30 s), using a green electrode (vibrating gold microwire electrode) without purging in a chloride medium (0.5 M NaCl) under moderate acidic conditions (HCl 1.0 mM), assisted by chemometric tools. The application of multivariate curve resolution alternating least squares (MCR‐ALS) for the resolution and quantification of both metals is shown. The optimized MCR‐ALS models showed good prediction ability with concentration prediction errors of 12.4 and 11.4 % for Pb and Cd, respectively. The quantitative results obtained by MCR‐ALS were compared to those obtained with partial least squares (PLS) and classical least squares (CLS) regression methods. For both metals, PLS and MCR‐ALS results are comparable and superior to CLS. For Cd, as a result of the peak shift problem, the application of CLS was unsuitable. MCR‐ALS provides additional advantage compared to PLS since it estimates the pure response of the analytes signal. Finally, the built up multivariate calibration models, based either in MCR‐ALS or PLS regression, allowed to quantify concentrations of Pb and Cd in surface river water samples, with satisfactory results.     

Phase Sensitive Alternating Current Polarography: A Chemometric Approach for the Selection of Phase Angles

A method based on the combined use of multivariate curve resolution by alternating least squares (MCR‐ALS) with phase sensitive alternating current polarography (ACP) is proposed to evaluate the phase angle where capacitive current is minimized in a much more accurate way than the visual inspection of ACP signals. The method allows, through the analysis of series of AC polarograms measured at different phase angles out the potential, to distinguish between faradaic and capacitive contributions. Then the angle at which the capacitive current is negligible can be shown and, in some cases, the influence of adsorption on measured currents minimized.     

Independent component analysis (ICA) algorithms for improved spectral deconvolution of overlapped signals in 1H NMR analysis: application to foods and related products

The major challenge facing NMR spectroscopic mixture analysis is the overlapping of signals and the arising impossibility to easily recover the structures for identification of the individual components and to integrate separated signals for quantification. In this paper, various independent component analysis (ICA) algorithms [mutual information least dependent component analysis (MILCA); stochastic non‐negative ICA (SNICA); joint approximate diagonalization of eigenmatrices (JADE); and robust, accurate, direct ICA algorithm (RADICAL)] as well as deconvolution methods [simple‐to‐use‐interactive self‐modeling mixture analysis (SIMPLISMA) and multivariate curve resolution‐alternating least squares (MCR‐ALS)] are applied for simultaneous ¹H NMR spectroscopic determination of organic substances in complex mixtures. Among others, we studied constituents of the following matrices: honey, soft drinks, and liquids used in electronic cigarettes. Good quality spectral resolution of up to eight‐component mixtures was achieved (correlation coefficients between resolved and experimental spectra were not less than 0.90). In general, the relative errors in the recovered concentrations were below 12%. SIMPLISMA and MILCA algorithms were found to be preferable for NMR spectra deconvolution and showed similar performance. The proposed method was used for analysis of authentic samples. The resolved ICA concentrations match well with the results of reference gas chromatography–mass spectrometry as well as the MCR‐ALS algorithm used for comparison. ICA deconvolution considerably improves the application range of direct NMR spectroscopy for analysis of complex mixtures. Copyright © 2014 John Wiley & Sons, Ltd.     

On the implementation of spatial constraints in multivariate curve resolution alternating least squares for hyperspectral image analysis

Hyperspectral imaging (HSI) is a method for exploring spatial and spectral information associated with the distribution of the different compounds in a chemical or biological sample. Amongst the multivariate image analysis tools utilized to decompose the raw data into a bilinear model, multivariate curve resolution alternating least squares (MCR‐ALS) can be applied to obtain the distribution maps and pure spectra of the components of the sample image. However, a requirement is to have the data in a two‐way matrix. Thus, a preliminary step consists of unfolding the raw HSI data into a single‐pixel direction. Consequently, through this data manipulation, the information regarding pixel neighboring is lost, and spatial information cannot directly be constrained on the component profiles in the current MCR‐ALS algorithm. In this short communication, we propose an adaptation of the MCR‐ALS framework, enabling the potential implementation of any variation of spatial constraint. This can be achieved by adding, at each least‐squares step, refolding/unfolding of the distribution maps for the components. The implementation of segmentation, shape smoothness, and image modeling as spatial constraints is proposed as a proof of concept. Copyright © 2015 John Wiley & Sons, Ltd.     

Multivariate curve resolution of mixed bacterial DNA sequence spectra: identification and quantification of bacteria in undefined mixture samples

A comprehensive understanding of factors that influence microbial competition and cooperation, their diversity and processes will be greatly beneficial in many research areas. Current tools for microflora determinations are far from suitable for high‐throughput monitoring of development in complex microbial communities. Here, we describe the application of a calibration free method, multivariate curve resolution with alternating least squares (MCR‐ALS), for identification and quantification of different microbes in mixture samples. The idea is to utilize MCR‐ALS to enable close monitoring of ecology in a variety of microbial communities. The data from two designed experiments consisting of DNA sequence spectra measured on mixtures were analysed with MCR‐ALS using no prior information on the data except for appropriate constraints, such as non‐negativity and closure. The results were compared both to the known true concentrations as well as to the results obtained from the well‐established multivariate calibration method partial least squares (PLS) regression. MCR‐ALS performed as well as PLS regression, successfully extracting all pure bacterial spectra and quantitative information on these, with 97.81% and 97.91% explained variance for the first and the second data set, respectively. Copyright © 2008 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.     

Polynomial multivariate least‐squares regression for modeling nonlinear data applied to in‐depth characterization of chromatographic resolution

Well‐established, linear multivariate calibration methods such as multivariate least‐squares regression (MLR), principal component regression (PCR), or partial least squares (PLS) have two limitations: (i) measured data must be linearly related to the response variables and (ii) predictor variables x_{n = 1, …, N} cannot be coupled to each other. For evaluation of nonlinear data, however, these restrictions need to be overcome and thus polynomial multivariate least‐squares regression (PMLR or “response surfaces”) has been introduced here. PMLR is based on multivariate least squares but incorporates all combinations of predictor variables up to a user‐selected polynomial order (e.g., including u or v = 0). Because of the inclusion of such coupled terms and their powers, PMLR models are better adapted to model nonlinear data and can help to enhance the prediction step's accuracy and precision. PMLR has been based on MLR because it facilitates—unlike PCR or PLS—a physical and chemical interpretation of the predictors. Hence, the origins and the relevance of nonlinear and/or coupled predictors can be investigated. The details of the PMLR algorithm and its implementation are presented along with a method for model optimization utilizing gradients of response surfaces. Newly developed PMLR models up to quintic order have been applied to predict a chromatograph's peak resolution as a function of six‐instrument parameters. It has been demonstrated that PMLR is better capable than MLR and PCR to describe these nonlinear and coupled instrument parameters. In addition, the novel software tool has been utilized for model optimization to determine instrument parameters, which result in the best chromatographic resolution. Copyright © 2011 John Wiley & Sons, Ltd.     

Two‐way and three‐way approaches to ultra high performance liquid chromatography–photodiode array dataset for the quantitative resolution of a two‐component mixture containing ciprofloxacin and ornidazole

Two‐way and three‐way calibration models were applied to ultra high performance liquid chromatography with photodiode array data with coeluted peaks in the same wavelength and time regions for the simultaneous quantitation of ciprofloxacin and ornidazole in tablets. The chromatographic data cube (tensor) was obtained by recording chromatographic spectra of the standard and sample solutions containing ciprofloxacin and ornidazole with sulfadiazine as an internal standard as a function of time and wavelength. Parallel factor analysis and trilinear partial least squares were used as three‐way calibrations for the decomposition of the tensor, whereas three‐way unfolded partial least squares was applied as a two‐way calibration to the unfolded dataset obtained from the data array of ultra high performance liquid chromatography with photodiode array detection. The validity and ability of two‐way and three‐way analysis methods were tested by analyzing validation samples: synthetic mixture, interday and intraday samples, and standard addition samples. Results obtained from two‐way and three‐way calibrations were compared to those provided by traditional ultra high performance liquid chromatography. The proposed methods, parallel factor analysis, trilinear partial least squares, unfolded partial least squares, and traditional ultra high performance liquid chromatography were successfully applied to the quantitative estimation of the solid dosage form containing ciprofloxacin and ornidazole.     

Linear calibrations in chromatography: The incorrect use of ordinary least squares for determinations at low levels,and the need to redefine the limit of quantification with this regression model

Ordinary least squares is widely applied as the standard regression method for analytical calibrations, and it is usually accepted that this regression method can be used for quantification starting at the limit of quantification. However, it requires calibration being homoscedastic and this is not common. Different calibrations have been evaluated to assess whether ordinary least squares is adequate to quantify estimates at low levels. All calibrations evaluated were linear and heteroscedastic. Despite acceptable values for precision at limit of quantification levels were obtained, ordinary least squares fitting resulted in significant and unacceptable bias at low levels. When weighted least squares regression was applied, bias at low levels was solved and accurate estimates were obtained. With heteroscedastic calibrations, limit values determined by conventional methods are only appropriate if weighted least squares are used. A “practical limit of quantification” can be determined with ordinary least squares in heteroscedastic calibrations, which should be fixed at a minimum of 20 times the value calculated with conventional methods. Biases obtained above this “practical limit” were acceptable applying ordinary least squares and no significant differences were obtained between the estimates measured using weighted and ordinary least squares when analyzing real‐world samples.     

A regularized nonnegative canonical polyadic decomposition algorithm with preprocessing for 3D fluorescence spectroscopy

We consider blind source separation in chemical analysis focussing on the 3D fluorescence spectroscopy framework. We present an alternative method to process the Fluorescence Excitation‐Emission Matrices (FEEM): first, a preprocessing is applied to eliminate the Raman and Rayleigh scattering peaks that clutter the FEEM. To improve its robustness versus possible improper settings, we suggest to associate the classical Zepp's method with a morphological image filtering technique. Then, in the second stage, the Canonical Polyadic (CP or Candecomp/Parafac) decomposition of a nonnegative three‐way array has to be computed. In the fluorescence spectroscopy context, the constituent vectors of the loading matrices should be nonnegative (since standing for spectra and concentrations). Thus, we suggest a new nonnegative third order CP decomposition algorithm (NNCP) based on a nonlinear conjugate gradient optimization algorithm with regularization terms and periodic restarts. Computer simulations performed on real experimental data are provided to enlighten the effectiveness and robustness of the whole processing chain and to validate the approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Comparative Study of Hard‐ and Soft‐Modeling Algorithms for Kinetic Data Processing

This article critically compares the efficacy of three algorithms, namely Alternating Least‐squares Multi Curve Resolution (ALS‐MCR), Hard Modeling Alternating Least‐squares (HM‐ALS), and classical Hard Modeling Multi Curve Resolution (HM‐MCR) in finding the true values of rate constants associated with a kinetic model. Simulated experiments on the simple system () indicate that soft‐modeling ALS‐MRC methodology, which is subject only to linear constraints, does not ensure that experimental responses are correctly deconvolved, thus preventing further calculations to determine the true rate constants. Inclusion of the kinetic model in the ALS scheme, which gives rise to the HM‐ALS methodology, was found to yield a correct assessment of the rate coefficients but had a large computational cost. Numerical experiments employing a more complex model () were also carried out, mainly to evaluate strategies for performing efficient searches on multidimensional multimodal least‐squares surfaces using HM‐ALS and HM‐MCR. This study again revealed the efficiency and reliability of classical HM‐MCR methods. Results from simulations were corroborated by analysis of data from an experimental study of chromate reduction by hydrogen peroxide; the mechanism of which is similar in complexity to those considered in simulations. The present work suggests that HM‐MCR algorithms implementing a multiminimum search strategy are the method of choice for analyzing two‐dimensional kinetic data.     

Scale‐adaptive tensor algebra for local many‐body methods of electronic structure theory

While the formalism of multiresolution analysis, based on wavelets and adaptive integral representations of operators, is actively progressing in electronic structure theory (mostly on the independent‐particle level and, recently, second‐order perturbation theory), the concepts of multiresolution and adaptivity can also be utilized within the traditional formulation of correlated (many‐particle) theory based on second quantization and the corresponding (generally nonorthogonal) tensor algebra. In this article, we present a formalism called scale‐adaptive tensor algebra, which introduces an adaptive representation of tensors of many‐body operators via the local adjustment of the basis set quality. Given a series of locally supported fragment bases of a progressively lower quality, we formulate the explicit rules for tensor algebra operations dealing with adaptively resolved tensor operands. The formalism suggested is expected to enhance the applicability of certain local correlated many‐body methods of electronic structure theory, for example, those directly based on atomic orbitals (or any other localized basis functions in general). © 2014 Wiley Periodicals, Inc.     

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.     

Second‐Order Data Obtained from Differential Pulse Voltammetry: Determination of Lead in River Water Using Multivariate Curve Resolution‐Alternating Least‐Squares (MCR‐ALS)

Multivariate curve resolution by alternating least squares (MCR‐ALS) with the aim of achieving the electrochemical second order advantage has been applied to potential‐time second‐order data. In this work a simple way is reported as a first approach towards generation of the instrumental electrochemical second‐order data by differential pulse voltammetry (DPV). A linear dependency exists in the pulse duration profiles of the electroactive species in the mixture samples. Rank deficiency of the mixture data matrix is broken by matrix augmentation. Due to existence of potential shift in the obtained data, MCR‐ALS could not be achieved the convergence on the augmented data. So this shift was corrected with potential shift correction algorithm. Results of MCR‐ALS after shift correction show that the proposed method could be efficiently used for determination of Pb²⁺ in the presence of unexpected interferents in the river water sample.     

Area correlation constraint for the MCR−ALS quantification of cholesterol using EEM fluorescence data: A new approach

This work demonstrates the use of a new additional constraint for the Multivariate Curve Resolution−Alternating Least Squares (MCR−ALS) algorithm called “area correlation constraint”, introduced to build calibration models for Excitation Emission Matrix (EEM) data. We propose the application of area correlation constraint MCR−ALS for the quantification of cholesterol using a simulated data set and an experimental data system (cholesterol in a ternary mixture). This new constraint includes pseudo-univariate local regressions using the area of resolved profiles against reference values during the alternating least squares optimization, to provide directly accurate quantifications of a specific analyte in concentration units. In the two datasets investigated in this work, the new constraint retrieved correctly the analyte and interference spectral profiles and performed accurate estimations of cholesterol concentrations in test samples. This the first study using the proposed area constraint using EEM measurements. This new constraint approach emerges as a new possibility to be tested in general cases of second-order multivariate calibration data in the presence of unknown interferents or in more involved higher order calibration cases.     

Simultaneous determination of potassium guaiacolsulfonate, guaifenesin, diphenhydramine HCl and carbetapentane citrate in syrups by using HPLC-DAD coupled with partial least squares multivariate calibration

A simple and rapid analytical procedure was proposed for the determination of chromatographic peaks by means of partial least squares multivariate calibration (PLS) of high-performance liquid chromatography with diode array detection (HPLC-DAD). The method is exemplified with analysis of quaternary mixtures of potassium guaiacolsulfonate (PG), guaifenesin (GU), diphenhydramine HCI (DP) and carbetapentane citrate (CP) in syrup preparations. In this method, the area does not need to be directly measured and predictions are more accurate. Though the chromatographic and spectral peaks of the analytes were heavily overlapped and interferents coeluted with the compounds studied, good recoveries of analytes could be obtained with HPLC-DAD coupled with PLS calibration. This method was tested by analyzing the synthetic mixture of PG, GU, DP and CP. As a comparison method, a classsical HPLC method was used. The proposed methods were applied to syrups samples containing four drugs and the obtained results were statistically compared with each other. Finally, the main advantage of HPLC-PLS method over the classical HPLC method tried to emphasized as the using of simple mobile phase, shorter analysis time and no use of internal standard and gradient elution.     

A simple,fast, and accurate thermodynamic‐based approach for transfer and prediction of gas chromatography retention times between columns and instruments Part I: Estimation of reference column geometry and thermodynamic parameters

The transfer of retention times based on thermodynamic models between columns can aid in separation optimization and compound identification in gas chromatography. Although earlier investigations have been reported, this problem remains unsuccessfully addressed. One barrier is poor predictive accuracy when moving from a reference column or system to a new target column or system. This is attributed to challenges associated with the accurate determination of the effective geometric parameters of the columns. To overcome this, we designed least squares‐based models that account for geometric parameters of the columns and thermodynamic parameters of compounds as they partition between mobile and stationary phases. Quasi‐Newton‐based algorithms were then used to perform the numerical optimization. In this first of three parts, the model used to determine the geometric parameters of the reference column and the thermodynamic parameters of compounds subjected to separation is introduced. As will be shown, the overall approach significantly improves the predictive accuracy and transferability of thermodynamic data (and retention times) between columns of the same stationary phase chemistry. The data required for the determination of the thermodynamic parameters and retention time prediction are obtained from fast and simple experiments. The proposed model and optimization algorithms were tested and validated using simulated and experimental data.     

Principal component analysis with boundary constraints

Observed data often belong to some specific intervals of values (for instance in case of percentages or proportions) or are higher (lower) than pre‐specified values (for instance, chemical concentrations are higher than zero). The use of classical principal component analysis (PCA) may lead to extract components such that the reconstructed data take unfeasible values. In order to cope with this problem, a constrained generalization of PCA is proposed. The new technique, called bounded principal component analysis (B‐PCA), detects components such that the reconstructed data are constrained to belong to some pre‐specified bounds. This is done by implementing a row‐wise alternating least squares (ALS) algorithm, which exploits the potentialities of the least squares with inequality (LSI) algorithm. The results of a simulation study and two applications to bounded data are discussed for evaluating how the method and the algorithm for solving it work in practice. Copyright © 2007 John Wiley & Sons, Ltd.     

Quinone‐based switches for candidate building blocks of molecular junctions with QTAIM and the stress tensor

The current work investigates candidate building blocks based on molecular junctions from hydrogen transfer tautomerization in the benzoquinone‐like core of an azophenine molecule with QTAIM and the recently introduced stress tensor trajectory analysis. We find that in particular the stress tensor trajectories are well suited to describe the mechanism of the switching process. The effects of an Fe‐dopant atom coordinated to the quinone ring, as well as F and Cl substitution of different ring‐hydrogens, are investigated and the new QTAIM and stress tensor analysis is used to draw conclusions on the effectiveness of such molecules as molecular switches in nanosized electronic circuits. We find that the coordinated Fe‐dopant greatly improves the switching properties, both in terms of the tautomerization barrier that has to be crossed in the switching process and the expected conductance behavior, while the effects of hydrogen substitution are more subtle. The absence of the Fe‐dopant atom led to impaired functioning of the switch “OFF” mechanism as well as coinciding with the formation of closed‐shell H—H bond critical points that indicated a strained or electron deficient environment. Our analysis demonstrates promise for future use in design of molecular electronic devices.