Algorithm combination strategy to obtain the second‐order advantage: simultaneous determination of target analytes in plasma using three‐dimensional fluorescence spectroscopy

Although a number of algorithms have established to obtain the well‐known second‐order advantage that quantifies analytes of interest in the presence of interferents, each has associated problems. In this work, for the first time, the optimization procedure of trilinear decomposition has been divided into three subparts, and a novel strategy is developed for assembling the advantages of the alternating trilinear decomposition (ATLD) algorithm, the self‐weighted alternating trilinear decomposition (SWATLD) algorithm, and the parallel factor analysis (PARAFAC) algorithm. The performance of the proposed strategy was evaluated using a simulated data set, a published fluorescence data set together with a new fluorescence data set that simultaneously quantifies procaine and tetracaine in plasma. Results show that the novel method can accurately and effectively estimate the qualitative and quantitative information of analytes of interest. Besides, the resolved profiles are very stable with respect to the number of components as long as the employed number is chosen to be equal or larger than the underlying one. Additionally, the study confirms that better prediction can be obtained by the new strategy when compared with ATLD, SWATLD, and PARAFAC as well as the strategy that employs direct trilinear decomposition method as initial values for PARAFAC. Moreover, the strategy can be directly extended to third‐order or higher‐order data analysis. Copyright © 2012 John Wiley & Sons, Ltd.     

A study on the differential strategy of some iterative trilinear decomposition algorithms: PARAFAC‐ALS,ATLD, SWATLD,and APTLD

This study presents an in‐depth discussion of the differential properties of various iterative trilinear decomposition algorithms, including Parallel Factor Analysis‐Alternating Least Squares (PARAFAC‐ALS), Alternating Trilinear Decomposition (ATLD), Self‐Weighted Alternating Trilinear Decomposition (SWATLD), and Alternating Penalty Trilinear Decomposition (APTLD). The shape of each algorithm's objective function (“convex” or “strictly convex”) is related to the algorithm's sensitivity to the estimated component number of the trilinear system. Different situations near the objective solution are analyzed both theoretically and numerically. The wall of perturbation generated by deviations in the iterative steps prevents the PARAFAC algorithm from achieving the objective solution when the component number is overestimated. This may explain, from a calculational perspective, why the PARAFAC algorithm could not obtain the objective solution or any equivalent thereto (although equivalents might still be chemically meaningful optimal solutions). The different effects of deviation and residual on the algorithms are demonstrated by numerical analysis in this paper. The convergence rate can be improved by the use of high‐performance computing strategy of the specific algorithm. The concept of solution set discussed in this paper complements the theory of the uniqueness of trilinear decomposition. Copyright © 2014 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.     

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.     

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.     

Vertex vector sequential projection for the resolution of three-way data

Usually, the PARAFAC2 method is utilized for handling retention time shifts in resolving chromatographic three-way data. It requires all profiles shift the same amount, which, unfortunately, seems unlikely the case in the practice of chromatographic analysis of multi-component samples. The present authors deal with the problem by unfolding the three-way data array along a certain direction into one matrix and setting up a multi-bilinear model. Then, a new method called vertex vector sequential projection (VVSP) is proposed to select pure variables and then the alternating least squares (ALS) procedure is used to iteratively improve the fit of the data to the multi-bilinear model. With a good estimate that is as close as possible to the pure variables, a fast convergence can be expected. Moreover, no prerequisite on the shifting is required and the multi-bilinear model provides a plausible manner to make use of the multi-sample information. An additional advantage is that the present fitting procedure is easier to adjust when constraints such as non-negativity, unimodality, etc., are to be imposed on the loading matrix. The proposed method is evaluated with simulated and real chemical data sets. Satisfactory resolution results are obtained, which demonstrates the performance of the proposed method.     

A scalable optimization approach for fitting canonical tensor decompositions

Tensor decompositions are higher‐order analogues of matrix decompositions and have proven to be powerful tools for data analysis. In particular, we are interested in the canonical tensor decomposition, otherwise known as CANDECOMP/PARAFAC (CP), which expresses a tensor as the sum of component rank‐one tensors and is used in a multitude of applications such as chemometrics, signal processing, neuroscience and web analysis. The task of computing CP, however, can be difficult. The typical approach is based on alternating least‐squares (ALS) optimization, but it is not accurate in the case of overfactoring. High accuracy can be obtained by using nonlinear least‐squares (NLS) methods; the disadvantage is that NLS methods are much slower than ALS. In this paper, we propose the use of gradient‐based optimization methods. We discuss the mathematical calculation of the derivatives and show that they can be computed efficiently, at the same cost as one iteration of ALS. Computational experiments demonstrate that the gradient‐based optimization methods are more accurate than ALS and faster than NLS in terms of total computation time. Copyright © 2011 John Wiley & Sons, Ltd.     

Non‐rotational Tucker3 core simplification

A simple and efficient approach is reported to estimate the sparsest Tucker3 model for a considered linear dependent multiway data array using PARAFAC profiles. Employing the least possible number of non‐zero core elements equal to the pseudo array rank of data, a better and easier interpretation of the data array is possible. The approach does not require any prior information. The type of rank deficiency, that is rank overlap or closure in different modes, and the Tucker3 core size can be determined from a congruency factor while running the algorithm. The replacement method (RM) of optimization is applied to determine the pattern (positions and values) of non‐zero elements in the sparsest core of the Tucker3 model. Full rank and rank deficient simulated data sets in different conditions as well as an experimental 3D fluorescence data set from gold nanoparticle (AuNP) interaction with HIV genome are successfully used for evaluating the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

Determination of Riboflavin in Human Plasma by Excitation‐Emission Matrix Fluorescence and Multi‐Way Analysis

Direct determination of riboflavin (Fig. 1), a vitamin, in human plasma was accomplished based on excitation‐emission matrix (EEM) fluorescence measurements and multi‐way chemometrics method based on parallel factor analysis (PARAFAC). The PARAFAC trilinear model, without restrictions and using one factor was used in the data analysis. The excitation wavelength range was from 380 to 460 nm and the emission was recorded from 480 to 600 nm. The calibration set was constructed with sixteen standard solutions in a concentration range of 0.02–0.38 μg mL^?1 for riboflavin. The capabilities of the method for the analysis were evaluated by determination of riboflavin in synthetic and real samples with satisfactory results. The accuracy of the methods, evaluated through the root mean square error of prediction (RMSEP), was 0.0059 for riboflavin by the PARAFAC model. Also, partial least squares (PLS) model was built at one excitation wavelength and used to determine a set of synthetic and real samples. The best model was obtained with PARAFAC. This result shows that molecular fluorescence spectroscopy can be used for the development of robust analytical methods for the direct determination of riboflavin in complex backgrounds such as human plasma.     

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.     

A new third‐order calibration method with application for analysis of four‐way data arrays

A novel third‐order calibration algorithm, alternating weighted residue constraint quadrilinear decomposition (AWRCQLD) based on pseudo‐fully stretched matrix forms of quadrilinear model, was developed for the quantitative analysis of four‐way data arrays. The AWRCQLD algorithm is based on the new scheme that introduces four unique constraint parts to improve the quality of four‐way PARAFAC algorithm. The tested results demonstrated that the AWRCQLD algorithm has the advantage of faster convergence rate and being insensitive to the excess component number adopted in the model compared with four‐way PARAFAC. Moreover, simulated data and real experimental data were analyzed to explore the third‐order advantage over the second‐order counterpart. The results showed that third‐order calibration methods possess third‐order advantages which allow more inherent information to be obtained from four‐way data, so it can improve the resolving and quantitative capability in contrast with second‐order calibration especially in high collinear systems. Copyright © 2011 John Wiley & Sons, Ltd.     

Multidimensional scaling and visualization of large molecular similarity tables

Multidimensional scaling (MDS) is a collection of statistical techniques that attempt to embed a set of patterns described by means of a dissimilarity matrix into a low‐dimensional display plane in a way that preserves their original pairwise interrelationships as closely as possible. Unfortunately, current MDS algorithms are notoriously slow, and their use is limited to small data sets. In this article, we present a family of algorithms that combine nonlinear mapping techniques with neural networks, and make possible the scaling of very large data sets that are intractable with conventional methodologies. The method employs a nonlinear mapping algorithm to project a small random sample, and then “learns” the underlying transform using one or more multilayer perceptrons. The distinct advantage of this approach is that it captures the nonlinear mapping relationship in an explicit function, and allows the scaling of additional patterns as they become available, without the need to reconstruct the entire map. A novel encoding scheme is described, allowing this methodology to be used with a wide variety of input data representations and similarity functions. The potential of the algorithm is illustrated in the analysis of two combinatorial libraries and an ensemble of molecular conformations. The method is particularly useful for extracting low‐dimensional Cartesian coordinate vectors from large binary spaces, such as those encountered in the analysis of large chemical data sets. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 488–500, 2001     

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.     

On the use of the observation‐wise k‐fold operation in PCA cross‐validation

Cross‐validation (CV) is a common approach for determining the optimal number of components in a principal component analysis model. To guarantee the independence between model testing and calibration, the observation‐wise k‐fold operation is commonly implemented in each cross‐validation step. This operation renders the CV algorithm computationally intensive, and it is the main limitation to apply CV on very large data sets. In this paper, we carry out an empirical and theoretical investigation of the use of this operation in the element‐wise k‐fold (ekf) algorithm, the state‐of‐the‐art CV algorithm. We show that when very large data sets need to be cross‐validated and the computational time is a matter of concern, the observation‐wise k‐fold operation can be skipped. The theoretical properties of the resulting modified algorithm, referred to as column‐wise k‐fold (ckf) algorithm, are derived. Also, its performance is evaluated with several artificial and real data sets. We suggest the ckf algorithm to be a valid alternative to the standard ekf to reduce the computational time needed to cross‐validate a data set. Copyright © 2015 John Wiley & Sons, Ltd.     

Matrix-free analysis of aflatoxins in pistachio nuts using parallel factor modeling of liquid chromatography diode-array detection data

In the present study a second-order calibration strategy for high performance liquid chromatography with diode-array detection (HPLC-DAD) has been developed using parallel factor analysis (PARAFAC) and has been applied for simultaneous determination of aflatoxins B₁, B₂, G₁ and G₂ in pistachio nuts in the presence of matrix interferences. Sample preparation was based on solvent extraction (SE) followed by solid phase extraction (SPE) on Bond Elut C18 cartridges. Since the sample preparation procedure was not selective to the analytes of interest, exploiting second-order advantage to obtain concentrations of individual analytes in the presence of uncalibrated interfering compounds seemed necessary. Appropriate pre-processing steps have been applied to correct background signals and the effect of retention time shifts. Transferred calibration data set obtained from standardization of solvent based calibration data has been used in prediction step. The results of PARAFAC on a set of spiked and naturally contaminated pistachio nuts indicated that the four aflatoxins could be successfully determined. The method was validated and multivariate analytical figures of merit were calculated. The advantages of the proposed method are using a low-cost SPE step relative to standard method of aflatoxin analysis (immune affinity column assay), a unique and simple isocratic elution program for all samples and a calibration transfer for saving both chemicals and time of analysis. This study show that coupling of SPE-HPLC-DAD with PARAFAC as a powerful second-order calibration method can be considered as an alternative method for resolution and quantification of aflatoxins in the presence of unknown interferences obtained through analysis of highly complex matrix of pistachio samples and cost per analysis can be reduced significantly.     

Evaluation of variation matrix arrays by parallel factor analysis

PARAFAC is one of the most widely used algorithms for trilinear decomposition. The uniqueness properties of the PARAFAC model are very attractive regardless of whether one is interested curve resolution or not. The fact that PARAFAC provides one unique solution simplifies interpretation of the model. But in three‐way data arrays the uniqueness condition can only be expected when k_A + k_B + k_C ≥ 2F + 2, where F is the number of components and k's are the Kruskal ranks of loadings A to C. As much as second order instruments produce data of varying complexity depending upon the nature of the analytical techniques being combined, with some three‐way data it is possible for patterns generated by the underlying sources of variation to have sufficient independent effects in two modes, yet nonetheless be proportional in a third mode. For example, in three‐way data for spectrophotometric titrations of weak acids or bases (pH‐wavelength‐sample), a rank deficiency may occur in two modes, that is closure rank deficiency in the pH mode and proportionality rank deficiency in the sample direction because each analyte will have acidic and basic forms that are linear combinations in the sample mode. The goal of the present paper is to overcome the non‐uniqueness problem in the second order calibration of monoprotic acids mixtures. The solution contains two steps: first each pH‐absorbance matrix is pretreated by subtraction of the first spectrum from each spectrum in the data matrix. This pretreated data matrix is called the variation matrix. Second, by stacking the variation matrices, a three‐way trilinear variation data array will be obtained without the proportional linear dependency problem that can be resolved uniquely by PARAFAC. It is shown, although unique results are not guaranteed by the Kruscal's condition for the original three‐way data, this condition is fulfilled for pretreated three‐way data. Hence, the variation array may be uniquely decomposed by the PARAFAC algorithm. Studies on simulated as well as real data array reveal the applicability of the proposed method to this kind of problem in the second order calibration of monoprotic acids. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of parallel factor analysis to total synchronous fluorescence spectrum of dilute multifluorophoric solutions: Addressing the issue of lack of trilinearity in total synchronous fluorescence data set

In recent years, total synchronous fluorescence (TSF) spectroscopy has become popular for the analysis of multifluorophoric systems. Application of PARAFAC, a popular deconvolution tool, requires trilinear structure in the three-way data array. The present work shows that TSF based three-way array data set of dimension sample × wavelength × Δλ does not have trilinear structure and hence it should not be subjected to PARAFAC analysis. This work also proposes that a TSF data set can be converted to an excitation–emission matrix fluorescence (EEMF) like data set which has trilinear structure, so that PARAFAC analysis can be performed on it. This also enables the retrieval of PARAFAC-separated component TSF spectra.     

Approaching the direct exponential curve resolution algorithm from a maximum likelihood perspective

The implementation of maximum likelihood parallel factor analysis (MLPARAFAC) in conjunction with the direct exponential curve resolution algorithm (DECRA) is described. DECRA takes advantage of the intrinsic exponential structure of some bilinear data sets to produce trilinear data by a simple shifting scheme, but this manipulation generates an error structure that is not optimally handled by traditional three-way chemometrics methods such as TLD and PARAFAC. In this work, the effects of these violations are studied using simulated and experimental data used in conjunction with the well-established TLD and PARAFAC. The results obtained by both methods are compared with the results obtained by MLPARAFAC, which is a method designed to optimally accomodate a variety of measurement error structures. The impact on the estimates of different parameters linked to the data sets and the DECRA method is investigated using simulated data. The results indicate that PARAFAC produces estimates of much poorer quality than TLD and MLPARAFAC. Also, it was found that the quality TLD estimates was comparable or only marginally poorer than the MLPARAFAC estimates. A number of commonly used algorithms were also compared to MLPARAFAC using two sets of published experimental data from kinetic studies. The MLPARAFAC estimates of rate constants were more precise than the other methods examined.     

A novel strategy for solving matrix effect in three-way data using parallel profiles with linear dependencies

This work presents a novel strategy for solving matrix effects using the second-order advantage and a new method called PARAllel profiles with LINear Dependencies (PARALIND). PARALIND is a generalization of parallel factor analysis (PARAFAC) and was developed to extend its use to problems with linearly dependent factors where normal PARAFAC analysis will fail to provide meaningful results. Such linearly dependent factors occur in standard addition with second-order data such as fluorescence excitation emission matrices (EEM). By successive standard addition of an analyte, the concentrations of the remaining components (interferences) remain constant and introduce linear dependency between interference concentrations in the samples. This theoretically leads to rank deficiency in the score matrix holding the relative concentrations when using PARAFAC for modeling. In practice, PARAFAC models of such data will mostly provide solutions where the score matrix is not rank deficient but a function of the noise in the data. This problem is shown to be solved by using PARALIND. In order to evaluate the applicability of the method a simulated as well as an experimental data set is tested. The results from experimental data relate to the direct determination of salicylic acid (SA), the main product of aspirin degradation, in undiluted human plasma by spectrofluorimetry.     

A PARAFAC algorithm using penalty diagonalization error (PDE) for three-way data array resolution

A modified parallel factors analysis (PARAFAC) algorithm with the penalty diagnolization error (PDE) was developed. This algorithm can overcome the slow convergence problem of the traditional PARAFAC method and is insensitive to the number of components, i.e., it is much faster than PARAFAC and insensitive to overestimation of the dimensionality of the model. The characteristic performance was demonstrated by treating simulated and real excitation-emission fluorescence data for samples of naphthalene, 1-naphthol and 2-naphthol with satisfactory results.