A fully robust PARAFAC method for analyzing fluorescence data

Parallel factor analysis (PARAFAC) is a widespread method for modeling fluorescence data by means of an alternating least squares procedure. Consequently, the PARAFAC estimates are highly influenced by outlying excitation–emission landscapes (EEM) and element‐wise outliers, like for example Raman and Rayleigh scatter. Recently, a robust PARAFAC method that circumvents the harmful effects of outlying samples has been developed. For removing the scatter effects on the final PARAFAC model, different techniques exist. Newly, an automated scatter identification tool has been constructed. However, there still exists no robust method for handling fluorescence data encountering both outlying EEM landscapes and scatter. In this paper, we present an iterative algorithm where the robust PARAFAC method and the scatter identification tool are alternately performed. A fully automated robust PARAFAC method is obtained in that way. The method is assessed by means of simulations and a laboratory‐made data set. Copyright © 2009 John Wiley & Sons, Ltd.     

Predicting the drug concentration in starch acetate matrix tablets from ATR-FTIR spectra using multi-way methods

The amounts of drug and excipient were predicted from ATR-FTIR spectra using two multi-way modelling techniques, parallel factor analysis (PARAFAC) and multi-linear partial least squares (N-PLS). Data matrices consisted of dissolved and undissolved parallel samples having different drug content and spectra, which were collected at axially cut surface of the flat-faced matrix tablets. Spectra were recorded comprehensively at different points on the axially cut surface of the tablet. The sample drug concentrations varied between 2 and 16% v/v. The multi-way methods together with ATR-FTIR spectra seemed to represent an applicable method for the determination of drug and excipient distribution in a tablet during the release process. The N-PLS calibration method was more robust for accurate quantification of the amount of components in the sample whereas the PARAFAC model provided approximate relative amounts of components.     

Vibrational overtone combination spectroscopy (VOCSY)—a new way of using IR and NIR data

This work explores a novel method for rearranging 1st order (one-way) infra-red (IR) and/or near infra-red (NIR) ordinary spectra into a representation suitable for multi-way modelling and analysis. The method is based on the fact that the fundamental IR absorption and the first, second, and consecutive overtones of NIR absorptions represent identical chemical information. It is therefore possible to rearrange these overtone regions of the vectors comprising an IR and NIR spectrum into a matrix where the fundamental, 1st, 2nd, and consecutive overtones of the spectrum are arranged as either rows or columns in a matrix, resulting in a true three-way tensor of data for several samples. This tensorization facilitates explorative analysis and modelling with multi-way methods, for example parallel factor analysis (PARAFAC), N-way partial least squares (N-PLS), and Tucker models. The vibrational overtone combination spectroscopy (VOCSY) arrangement is shown to benefit from the “order advantage”, producing more robust, stable, and interpretable models than, for example, the traditional PLS modelling method. The proposed method also opens the field of NIR for true peak decomposition—a feature unique to the method because the latent factors acquired using PARAFAC can represent pure spectral components whereas latent factors in principal component analysis (PCA) and PLS usually do not.     

Chemometrics assisted spectrophotometric determination of pyridine in water and wastewater

The paper reports a direct method for the determination of pyridine in water and wastewater samples based on ultraviolet spectrophotometric measurements using multi-way modeling techniques. Parallel factor analysis (PARAFAC) and multi-way partial least squares (N-PLS) regression methods were employed for the decomposition of spectra and quantification of pyridine. The study was carried out in the pH range of 1.0-12.0 and concentration range of 0.67-51.7 μg mL⁻¹ of pyridine. Both the three-way PARAFAC and tri-PLS1 models successfully predicted the concentration of pyridine in synthetic (spiked) river water and field wastewater samples. The mean recovery obtained from PARAFAC regression model were 97.39% for the spiked and 99.84% for the field wastewater samples, respectively. The sensitivity and precision of the method for pyridine determination were 0.58% and 5.95%, respectively. The N-PLS regression model yielded mean recoveries of 99.29% and 100.18% for the spiked and field wastewater samples, respectively. The prediction accuracy of the methods was evaluated through the root mean square error of prediction (RMSEP). For PARAFAC, it was 0.65 and 0.82 μg mL⁻¹ for spiked river water and field wastewater samples, respectively, while for N-PLS, it was 0.25 and 0.37 μg mL⁻¹, respectively. Both the PARAFAC and N-PLS methods, thus, yielded satisfactory results for the prediction of pyridine concentration in water and wastewater samples.     

Classification of weathered petroleum oils by multi-way analysis of gas chromatography-mass spectrometry data using PARAFAC2 parallel factor analysis

The application of multi-way parallel factor analysis (PARAFAC2) is described for the classification of different kinds of petroleum oils using GC-MS. Oils were subjected to controlled weathering for 2, 7 and 15 days and PARAFAC2 was applied to the three-way GC-MS data set (MSxGCxsample). The classification patterns visualized in scores plots and it was shown that fitting multi-way PARAFAC2 model to the natural three-way structure of GC-MS data can lead to the successful classification of weathered oils. The shift of chromatographic peaks was tackled using the specific structure of the PARAFAC2 model. A new preprocessing of spectra followed by a novel use of analysis of variance (ANOVA)-least significant difference (LSD) variable selection method were proposed as a supervised pattern recognition tool to improve classification among the highly similar diesel oils. This lead to the identification of diagnostic compounds in the studied diesel oil samples.     

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.     

PARAFAC and PARAFAC2 calibration models for antihypertensor Nifedipine quantification

Two-way data structures were obtained by acquiring UV-vis spectra as function of the time of the alkaline hydrolysis reaction of the antihypertensor Nifedipine in dimethylsulfoxide (DMSO). Sets of three-way data structures were obtained from the analysis of solutions with different concentrations of Nifedipine generated by standard additions to DMSO, Nifedipine standard and a pharmaceutical formulation. PARAFAC and PARAFAC2 methods were used in the analysis of these multi-way data structures and calibration models were developed for Nifedipine quantification in pharmaceutical formulations. For all the three-way data structures a better model fit was found with the PARAFAC2 suggesting that the experimental data sets have deviations from trilinearity. The best concentrations estimations were found with the PARAFAC2 model in the analysis of a [concentration × time (s) × wavelength (nm)] three-way data structure which allows the quantification of Nifedipine in pharmaceutical formulations.     

Modeling nonbilinear total synchronous fluorescence data matrices with a novel adapted partial least squares method

A new residual modeling algorithm for nonbilinear data is presented, namely unfolded partial least squares with interference modeling of non bilinear data by multivariate curve resolution by alternating least squares (U-PLS/IMNB/MCR-ALS). Nonbilinearity represents a challenging data structure problem to achieve analyte quantitation from second-order data in the presence of uncalibrated components. Total synchronous fluorescence spectroscopy (TSFS) generates matrices which constitute a typical example of this kind of data. Although the nonbilinear profile of the interferent can be achieved by modeling TSFS data with unfolded partial least squares with residual bilinearization (U-PLS/RBL), an extremely large number of RBL factors has to be considered. Simulated data show that the new model can conveniently handle the studied analytical problem with better performance than PARAFAC, U-PLS/RBL and MCR-ALS, the latter modeling the unfolded data. Besides, one example involving TSFS real matrices illustrates the ability of the new method to handle experimental data, which consists in the determination of ciprofloxacin in the presence of norfloxacin as interferent in water samples.     

Robustified least squares support vector classification

Support vector machine (SVM) algorithms are a popular class of techniques to perform classification. However, outliers in the data can result in bad global misclassification percentages. In this paper, we propose a method to identify such outliers in the SVM framework. A specific robust classification algorithm is proposed adjusting the least squares SVM (LS‐SVM). This yields better classification performance for heavily tailed data and data containing outliers. Copyright © 2009 John Wiley & Sons, Ltd.     

Classification of data with missing elements and outliers

Missing elements and outliers can often occur in experimental data. The presence of outliers makes the evaluation of any least squares model parameters difficult, while the missing values influence the adequate identification of outliers. Therefore, approaches that can handle incomplete data containing outliers are highly valued. In this paper, we present the expectation-maximization robust soft independent modeling of class analogy approach (EM-S-SIMCA) based on the recently introduced spherical SIMCA method. Several important issues like the possibility of choosing the complexity of the model with the leverage correction procedure, the selection of training and test sets using methods of uniform design for incomplete data and prediction of new samples containing missing elements are discussed. The results of a comparison study showed that EM-S-SIMCA outperforms the classic expectation-maximization SIMCA method. The performance of the method was illustrated on simulated and real data sets and led to satisfactory results.     

Multivariate curve resolution based chromatographic peak alignment combined with parallel factor analysis to exploit second-order advantage in complex chromatographic measurements

In the present contribution, a new combination of multivariate curve resolution-correlation optimized warping (MCR-COW) with trilinear parallel factor analysis (PARAFAC) is developed to exploit second-order advantage in complex chromatographic measurements. In MCR-COW, the complexity of the chromatographic data is reduced by arranging the data in a column-wise augmented matrix, analyzing using MCR bilinear model and aligning the resolved elution profiles using COW in a component-wise manner. The aligned chromatographic data is then decomposed using trilinear model of PARAFAC in order to exploit pure chromatographic and spectroscopic information. The performance of this strategy is evaluated using simulated and real high-performance liquid chromatography-diode array detection (HPLC-DAD) datasets. The obtained results showed that the MCR-COW can efficiently correct elution time shifts of target compounds that are completely overlapped by coeluted interferences in complex chromatographic data. In addition, the PARAFAC analysis of aligned chromatographic data has the advantage of unique decomposition of overlapped chromatographic peaks to identify and quantify the target compounds in the presence of interferences. Finally, to confirm the reliability of the proposed strategy, the performance of the MCR-COW-PARAFAC is compared with the frequently used methods of PARAFAC, COW-PARAFAC, multivariate curve resolution-alternating least squares (MCR-ALS), and MCR-COW-MCR. In general, in most of the cases the MCR-COW-PARAFAC showed an improvement in terms of lack of fit (LOF), relative error (RE) and spectral correlation coefficients in comparison to the PARAFAC, COW-PARAFAC, MCR-ALS and MCR-COW-MCR results.     

Enzyme activity measurement via spectral evolution profiling and PARAFAC

The recent advances in multi-way analysis provide new solutions to traditional enzyme activity assessment. In the present study enzyme activity has been determined by monitoring spectral changes of substrates and products in real time. The method relies on measurement of distinct spectral fingerprints of the reaction mixture at specific time points during the course of the whole enzyme catalyzed reaction and employs multi-way analysis to detect the spectral changes. The methodology is demonstrated by spectral evolution profiling of Fourier Transform Infrared (FTIR) spectral fingerprints using parallel factor analysis (PARAFAC) for pectin lyase, glucose oxidase, and a cellulase preparation.     

A strategy for enhancing the reliability of near-infrared spectral analysis

Near-infrared (NIR) spectroscopy will present a more promising tool for quantitative measurement if the reliability of the calibration model is further improved. To achieve this purpose, a new partial least squares (PLSs) technique based on Monte Carlo (MC) resampling is proposed, which is named as MCPLS. In this method, the outliers are firstly removed based on probability statistics. Then, the models without outliers are averaged and combined into a single prediction model as done in a consensus modeling, which can greatly enhance the reliability of PLS calibration. To validate the effectiveness and universality of the proposed method, it was applied to two different sets of NIR spectra. It was found that MCPLS could effectively avoid the swamping and masking effects caused by multiple outliers. The results show that the method is of value to enhance the reliability of PLS model involving complex NIR matrices with a small number of outliers.     

Self-modeling curve resolution method applied for the evaluation of dissolution testing data: A case study of meloxicam-mannitol binary systems

This paper introduces some chemometric methods, i.e., self-modeling curve resolution (SMCR), multivariate curve resolution-alternating least squares (MCR-ALS) and parallel factor analysis (PARAFAC and PARAFAC2), which are used to evaluate in vitro dissolution testing data detected by a UV-vis spectrophotometer on meloxicam-mannitol binary systems. These systems were chosen because of their relative simplicity to apply as part of the validation process illustrating the effectiveness of the developed and applied chemometric method. The paper illustrates the failure of PARAFAC methods used before for pharmaceutical data evaluations as well, and we suggest application of the feasible band form given by SMCR as a more general procedure.Steps to improve the dissolution behavior of drugs have become among the most interesting aspects of pharmaceutical technology, and our results show that a larger particle size of meloxicam is advantageous for dissolution. Instead of the use of only one characteristic wavelength, appropriate chemometric methods can furnish more information from dissolution testing data, i.e., the individual dissolution rate profiles and the individual spectra for all the components can be obtained without resorting to any separation techniques such as HPLC.     

A review on advanced oxidation processes: From classical to new perspectives coupled to two- and multi-way calibration strategies to monitor degradation of contaminants in environmental samples

This paper offers a critical review from classical to new perspectives of advanced oxidation processes (AOPs) coupled to two- and multi-way calibration strategies based on multivariate curve resolution – alternating least-squares (MCR-ALS) and parallel factory analysis (PARAFAC) with various analytical techniques to monitor the degradation of contaminants in environmental samples. It focuses on the generation of highly reactive hydroxyl (HO•) radicals (classical AOPs with emphasis on Fenton, photo-Fenton and ozonation processes) and emerging reactive sulphate (SO₄^•−) radicals (new perspectives of AOPs) for effective degradation of recalcitrant compounds. Other new perspectives of AOPs were also addressed, namely semiconductor photocatalysis (TiO₂/UV), combination of processes involving at least one AOP (hybrid or single-step processes and sequential or two-step processes), novel advanced electrochemical oxidation technologies (electro-Fenton and electro-photo-Fenton) and nanocatalytic heterogeneous Fenton technology with high specific surface area. Literature reports since 2008 for real applications in the environmental remediation based on AOPs (from classical to new perspectives) coupled to PARAFAC and MCR-ALS with first-, second- and third-order data were reviewed and the improvements obtained were briefly discussed. The two- and multi-way calibration strategies allow one the successful decomposition of first-, second- and third-order data collected from different analytical techniques. Therefore, the respective profiles obtained allowed qualitative (spectral profiles) and quantitative (concentration profiles) analysis of complex samples during the degradation of contaminants through the second-order advantage. Finally, trends of future research directions for AOPs coupled to various analytical techniques and advanced chemometric models were provided.     

Analysis of lipoproteins using 2D diffusion-edited NMR spectroscopy and multi-way chemometrics

This study represents the first application of multi-way calibration by N-PLS and multi-way curve resolution by PARAFAC to 2D diffusion-edited ¹H NMR spectra. The aim of the analysis was to evaluate the potential for quantification of lipoprotein main- and subfractions in human plasma samples. Multi-way N-PLS calibrations relating the methyl and methylene peaks of lipoprotein lipids to concentrations of the four main lipoprotein fractions as well as 11 subfractions were developed with high correlations (R = 0.75-0.98). Furthermore, a PARAFAC model with four chemically meaningful components was calculated from the 2D diffusion-edited spectra of the methylene peak of lipids. Although the four extracted PARAFAC components represent molecules of sizes that correspond to the four main fractions of lipoproteins, the corresponding concentrations of the four PARAFAC components proved not to be correlated to the reference concentrations of these four fractions in the plasma samples as determined by ultracentrifugation. These results indicate that NMR provides complementary information on the classification of lipoprotein fractions compared to ultracentrifugation.     

Least median of squares: a robust method for outlier and model error detection in regression and calibration

The least median of squares method is a robust regression method, which means that it is not sensitive to outliers or other violations of the assumption of the usual normal model. This contrasts with the conventional regression method, which minimizes the sum of squares. It is demonstrated that the proposed method can be used to detect or correct for outliers or model errors in calibration applications and in comparing two procedures.     

Robust partial least squares model for prediction of green tea antioxidant capacity from chromatograms

In this paper a robust version of the partial least squares model (partial robust M-regression, PRM) was built to predict the total antioxidant capacity of green tea extracts. In order to construct a calibration model, chromatograms obtained by a fast high-performance liquid chromatographic method on a monolithic silica column were related with the total antioxidant capacity of green tea extracts as determined by the Trolox antioxidant capacity method. Since natural samples are the subject of the study, some outlying samples are present in the data, as shown in an earlier work. Therefore, to construct reliable calibration models, they were detected and removed prior to modeling. With the applied robust partial least squares approach, where a weighting scheme is embedded to down-weight the negative influence of outliers upon the model it is possible to construct a robust calibration model, without prior identification of outlying objects. It was shown that a robust model, allowing satisfactory prediction for test samples, can be used in controlling green tea antioxidant capacity based on their chromatograms. The constructed robust partial least squares model was shown to have virtually the same fit and predictive power as the classical partial least squares model when outlying samples were removed from the data.     

Robust curve fitting method for optical spectra by least median squares (LMedS) estimator with particle swarm optimization (PSO).

A curve fitting technique for optical spectra based on a robust estimator, least median squares (LMedS), is introduced in this study. For the effective calculation of LMedS, particle swarm optimization (PSO) is also introduced. Unlike a standard curve fitting method using least squares (LS) estimator, the method based on LMedS estimator is less influenced by outliers in experimental data. Two kinds of data sets, simulated data with outliers and temperature-dependent near-infrared (NIR) spectra of oleic acid (OA) are applied for the demonstration of the proposed method. The results clearly reveal that, compared with the LS estimator, the proposed method can effectively reduce undesirable effects of low SN ratio and can yield more accurate fitting results.     

Multivariate methods on the excitation emission matrix fluorescence spectroscopic data of diesel-kerosene mixtures: a comparative study

Quantitative determination of kerosene fraction present in diesel has been carried out based on excitation emission matrix fluorescence (EEMF) along with parallel factor analysis (PARAFAC) and N-way partial least squares regression (N-PLS). EEMF is a simple, sensitive and nondestructive method suitable for the analysis of multifluorophoric mixtures. Calibration models consisting of varying compositions of diesel and kerosene were constructed and their validation was carried out using leave-one-out cross validation method. The accuracy of the model was evaluated through the root mean square error of prediction (RMSEP) for the PARAFAC, N-PLS and unfold PLS methods. N-PLS was found to be a better method compared to PARAFAC and unfold PLS method because of its low RMSEP values.