Direct Trilinear Decomposition Research Articles

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

ABSTRACTAlthough 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.

Direct decomposition of three-way arrays using a non-negative approximation

Non-negative matrix approximation (NNMA) has been used in diverse scientific fields, but it still has some major limitations. In the present study a novel trilinear decomposition method, termed three-way NNMA (TWNNMA), was developed. The method decomposes three-way arrays directly without unfolding and overcomes the restriction of locking zero elements in the deduced multiplicative update rules by adding a positive symmetric matrix. Direct trilinear decomposition was used as the TWNNMA initialization method and experimental results confirm that this greatly accelerated the convergence. An obvious advantage of TWNNMA is the uniqueness of the non-negative solution, which facilitates a better understanding of the underlying physical realities of complex data. TWNNMA was applied in complex systems such as chemical kinetics, second-order calibration and analysis of GC–MS data. The results demonstrate that TWNNMA, differing from previous trilinear decomposition methods, is comparable to existing second-order calibration methods and represents a promising resolution method for complex systems.

Multi-way standardization of an AC voltammetric analyzer for electrometallization baths

Two standardization techniques were implemented for transferring three multi-way calibration models for determining of the suppressor concentration in industrial copper electrometallization baths used in semiconductor manufacturing. Parallel Factor analysis (PARAFAC) for multi-way array decomposition coupled with Inverse Least Squares (ILS) (PARAFAC/ILS) regression, Direct Trilinear Decomposition (DTLD) coupled with ILS (DTLD/ILS), and Multi-linear Partial Least Squares (N-PLS) regressions were employed to develop and test calibration models based on trilinear Alternating Current (AC) voltammetric data. The calibration transfer was performed by Direct Standardization (DS) and Reverse Standardization (RS) on the voltammetric data. The pairwise performance of standardization techniques and decomposition/regression techniques was studied comparatively in order to assess reliability and robustness of analytical model transfer.

Application of Multiway Chemometric Techniques for Analysis of AC Voltammetric Data

AbstractThree multiway calibration techniques have been applied for determining of the suppressor concentration in industrial copper electrometallization baths used in semiconductor manufacturing. Parallel factor analysis (PARAFAC) for multiway array decomposition coupled with inverse least squares (ILS) regression (PARAFAC/ILS), direct trilinear decomposition (DTLD) coupled with ILS (DTLD/ILS), and multilinear partial least squares (N‐PLS) regression were employed to develop and test calibration models based on trilinear AC voltammetric data. All techniques employed comparatively produce reliable calibration model and provide quantitative information about its robustness.

Use of pseudo-sample extraction and the projection technique to estimate the chemical rank of three-way data arrays

Determining the rank of a trilinear data array is a first step in subsequent trilinear component decomposition. Different from estimating the rank of bilinear data, it is more difficult to decide the significant number of component to fit the trilinear decompositions exactly. General methods of rank estimation utilize the information contained in the singular values but ignore information from eigenvectors. In this paper, a rank estimating method specifically for trilinear data arrays is proposed. It uses the idea of direct trilinear decomposition (DTLD) to compress the cube matrix into two pseudo sample matrices which are then decomposed by singular value decomposition. Two eigenvectors combined with the projection technique are used to estimate the rank of trilinear data arrays. Simulated trilinear data arrays with homoscedastic and heteroscedastic noise, different noise levels, high collinearity, and real three-way data arrays have been used to illustrate the feasibility of the proposed method. Compared with other factor-determining methods, for example use of the factor indication function (IND), residual percentage variance (RPV), and the two-mode subspace comparison approach (TMSC), the results showed that the new method can give more reliable answers under the different conditions applied.

Comparison of three‐way resolution methods for non‐trilinear chemical data sets

AbstractResolution of three‐way chemical data sets can be tackled using two families of chemometric methods: those assuming trilinear structure in the data set, such as direct trilinear decomposition (DTD) or parallel factor analysis (PARAFAC); and those which decompose the three‐way data set according to a model lacking this structure, such as TUCKER3 or multivariate curve resolution–alternating least squares (MCR–ALS). The first group of methods provides unique solutions, whereas the second group gives solutions subject to rotational ambiguities. DTD and PARAFAC are thus the choice to deal with chemical data sets with trilinear structure. However, in the analysis of chemical data with non‐trilinear structure, which is most commonly found in practice, the more ambiguous solutions given by TUCKER3 and MCR–ALS could be balanced by the major flexibility in the modelling of profiles. To assess this possibility, three‐way resolution methods from the two mentioned families are applied to simulated and real data sets designed to show typical non‐trilinear chemical situations, caused by shifts and shape changes in profiles. Copyright © 2001 John Wiley & Sons, Ltd.

Three-way alternating least squares using three-dimensional tensors in MATLAB®

This paper describes an improved three-way alternating least-squares multivariate curve resolution algorithm that makes use of the recently introduced multi-dimensional arrays of MATLAB®. Multi-dimensional arrays allow for a convenient way to apply chemically sound constraints, such as closure, in the third dimension. The program is designed for kinetic studies on liquid chromatography with diode array detection but can be used for other three-way data analysis. The program is tested with a large number of synthetic data sets and its flexibility is demonstrated, especially when non-trilinear data sets are fit. In this case, the algorithm finds a solution with a better fit than direct trilinear decomposition (DTD). When trilinear data are used, the optimal fit is not as good as when a direct decomposition method is used. Most real data sets, however, have some degree of non-trilinearity. This makes this method a better choice to analyze non-trilinear, three-way data than direct trilinear decomposition.

Three-Way Analysis of Fluorescence Spectra of Polycyclic Aromatic Hydrocarbons with Quenching by Nitromethane

The application of trilinear decomposition (TLD) to the analysis of fluorescence excitation-emission matrices of mixtures of polycyclic aromatic hydrocarbons (PAHs) is described. The variables constituting the third-order tensor are excitation wavelength, emission wavelength, and concentration of a fluorescence quencher (nitromethane). The addition of a quencher to PAH mixtures selectively reduces the fluorescence intensity of mixture components according to the Stern-Volmer equation. TLD allows the three-way matrix to be decomposed to give unique solutions for the excitation spectrum, emission spectrum, and quenching profiles for each component. The availability of spectra and calculated Stern-Volmer constants can aid in the identification of unknown components. Preprocessing of the data to correct for Rayleigh/Raman scatter and primary absorption by the quencher is necessary. Both three-component (anthracene, pyrene, 1-methylpyrene) and four-component (fluoranthene, anthracene, pyrene, 2,3-benzofluorene) synthetic mixtures are successfully resolved by TLD using quencher concentrations up to 100 mM. Results are compared using both alternating least-squares and direct trilinear decomposition algorithms. The reproducibility of extracted Stern-Volmer constants is determined from replicate experiments. To illustrate the application of TLD to a real sample, a chromatographic cut from the analysis of a light gas oil sample was used. Analysis of the TLD extracted spectra and quenching constants suggests the presence of three classes of polycyclic aromatic hydrocarbons consistent with data from a second dimension of chromatography and mass spectrometry.

Second-Order Standard Addition Method Based on Alternating Trilinear Decomposition

In this study, a second-order standard addition method based on alternating trilinear decomposition (ATLD-SOSAM) was developed. It was applied to second-order HPLC-DAD data and compared to methods employing direct trilinear decomposition (DTLD) and PARAFAC. The results show that ATLD-SOSAM is slightly superior to both DTLD-SOSAM and PARAFAC-SOSAM.

Three-way data analysis of pollutant degradation profiles monitored using liquid chromatography-diode array detection

There are a wide variety of environmental contaminants that need to be better characterized. It is important to know the degradation pathways for these pollutants in order to better understand their effect on the environment. The hydrolysis of Glean® (chlorsulfuron), a sulfonylurea herbicide, is currently under study in this laboratory. Liquid chromatography coupled with diode array detection (LC–DAD) offers a means of studying these types of reactions. With this approach, three-way data are obtained—absorbance measurements as a function of chromatographic retention time, UV wavelength and reaction time. The direct trilinear decomposition (DTD) algorithm was applied to these data, presuming an internal trilinear structure in the data set. Upon inspection of the results, however, deviations from trilinearity were found that caused chemically meaningless results to be obtained using this approach. This difficulty was addressed by using the three-way multivariate curve resolution–alternating least squares approach (MCR–ALS). This technique allows for the optional use of the trilinearity constraint, and some additional constraints related to the features of the chromatographic and spectral profiles can be included. In the present work the application of this approach has been evaluated for resolving the overlapped responses that arise when measuring herbicide degradation reaction profiles. Copyright © 1999 John Wiley & Sons, Ltd.

Comparison between the direct trilinear decomposition and the multivariate curve resolution-alternating least squares methods for the resolution of three-way data sets

Direct trilinear decomposition (DTD) and multivariate curve resolution-alternating least squares (MCR-ALS) methods are two of the most representative three-way resolution procedures. The former, non-iterative, is based on the resolution of the generalized eigenvector/eigenvalue problem and the latter, iterative, is focused on the optimization of initial estimates by using data structure and chemical constraints. DTD and MCR-ALS have been tested on a variety of three-way simulated data sets having common sources of variation in real response profiles, such as signal shift, broadening or shape distortions caused by noise. The effect of these factors on the resolution results has been evaluated through the analysis of several parameters related to the recovery of both qualitative and quantitative information and to the quality of the overall data description. Conclusions inferred from the simulated examples help to clarify the performance of both methods on a real example and to provide some general guidelines to understand better the potential of each method.

Kinetic Detection of Overlapped Amino Acids in Thin-Layer Chromatography with a Direct Trilinear Decomposition Method

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTKinetic Detection of Overlapped Amino Acids in Thin-Layer Chromatography with a Direct Trilinear Decomposition MethodMin. Gui, Sarah C. Rutan, and Armel. AgbodjanCite this: Anal. Chem. 1995, 67, 18, 3293–3299Publication Date (Print):September 15, 1995Publication History Published online1 May 2002Published inissue 15 September 1995https://doi.org/10.1021/ac00114a028RIGHTS & PERMISSIONSArticle Views203Altmetric-Citations43LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (2 MB) Get e-Alerts Get e-Alerts

Eliminating complex eigenvectors and eigenvalues in multiway analyses using the direct trilinear decomposition method

AbstractThe direct trilinear decomposition method (DTDM) is an algorithm for performing quantitative curve resolution of three‐dimensional data that follow the so‐called trilinear model, e.g. chromatography–spectroscopy or emission‐excitation fluorescence. Under certain coditions complex eigenvalues and eigenvectors emerge when the generalized eigenproblem is solved in DTDM. Previous publications never treated those cases. In this paper we show how similarity transformations can be used to eliminate the imaginary part of the complex eigenvalues and eigenvectors, thereby increasing the usefulness of DTDM in practical applications. The similarity transformation technique was first used by our laboratory to solve the similar problem in the generalized rank annihilation method (GRAM). Because unique elution profiles and spectra can be derived by using data matrices from three or more samples simultaneously, DTDM with similarity transformations is more efficient than GRAM in the case where there are many samples to be investigated.

A new receptor model: A direct trilinear decomposition followed by a matrix reconstruction

AbstractIn many cases, monitoring data for ambient airborne particles can be organized in the form of a three‐way data table with one way for chemical species, one for sampling periods and one for sites. A direct trilinear decomposition followed by a matrix reconstruction (DTDMR) is developed to analyze such a data table as a whole. The three‐way data set is composed into three two‐way matrices by a direct trilinear decomposition (DTD). The column vectors of each of the matrices are called ‘source profiles’, ‘emission patterns’ and ‘site coefficients’ respectively. Particulate sources are identified by examining both their source profiles and emission patterns. After the sources have been identified, emission patterns and site coefficients are used to produce a three‐way matrix that gives estimates of mass contributions of sources to the samples collected at every site in every period. By simulation study, not only has the method been verified, but a good indicator has been found that shows the number of factors (i.e. sources) in the system. Unlike other receptor models, DTDMR does not require source profile data and does not involve trial‐and‐error procedures. Since DTDMR identifies sources based on variations in two dimensions, it has a higher potential to distinguish two sources that have similar chemical compositions. The DTDMR model has provided excellent results with simulated data and has been applied in a real world three‐way data set.

Tensorial resolution: A direct trilinear decomposition

AbstractModern instrumentation in chemistry routinely generates two‐dimensional (second‐order) arrays of data. Considering that most analyses need to compare several samples, the analyst ends up with a three‐dimensional (third‐order) array which is difficult to visualize or interpret with the conventional statistical tools.Some of these data arrays follow the so‐called trilinear model, These trilinear arrays of data are known to have unique factor analysis decompositions which correspond to the true physical factors that form the data, i.e. given the array ℝ, a unique solution can be found in many cases for each order X, Y and Z. This is in contrast to the well‐known second‐order bilinear data factor analysis, where the abstract solutions obtained are not unique and at best cannot be easily compared with the underlying physical factors owing to a rotational ambiguity.Trilinear decompositions have had the disadvantage, however, that a non‐linear optimization with many parameters is necessary to reach a least‐squares solution. This paper will introduce a method for reducing the problem to a rectangular generalized eigenvalue–eigenvector equation where the eigenvectors are the contravariant form (pseudo‐inverse) of the actual factors. It is shown that the method works well when the factors are linearly independent in at least two orders (e.g. Xir and Yjr are full rank matrices).Finally, it is shown how trilinear decompositions relate to multicomponent calibration, curve resolution and chemical analysis.