Estimating the chemical rank of three-way data arrays by a simple linear transform incorporating Monte Carlo simulation

Estimating an appropriate chemical rank of a three-way data array is very important to second-order calibration. In this paper, a simple linear transform incorporating Monte Carlo simulation approach (LTMC) to estimate the chemical rank of a three-way data array was suggested. The new method determines the chemical rank through performing a simple linear transform procedure on the original cube matrix to produce two subspaces by singular value decomposition. One of two subspaces is derived from the original three-way data array itself and the other is derived from a new three-way data array produced by the linear transformation of the original one. Projection technique incorporating the Monte Carlo approach acts as distinguishing criterion to choose the appropriate component number of the system. Simulated three-way trilinear data arrays with different noise types (homoscedastic and heteroscedastic), various noise level as well as high collinearity are used to illustrate the feasibility of the new method. The results have shown that the new method could yield accurate results with different conditions appended. The feasibility of the new method is also confirmed by two real arrays, HPLC-DAD data and excitation-emission fluorescent data. All the results are compared with the other three factor-determining methods: factor indicator function (IND), core consistency diagnostic (CORCONDIA) and two-mode subspace comparison (TMSC) approach. It shows that the newly proposed algorithm can objectively and quickly determine the chemical rank to fit the trilinear model.     

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.      

Analytical solution and meaning of feasible regions in two-component three-way arrays

Although many efforts have been directed to the development of approximation methods for determining the extent of feasible regions in two- and three-way data sets; analytical determination (i.e. using only finite-step direct calculation(s) instead of the less exact numerical ones) of feasible regions in three-way arrays has remained unexplored. In this contribution, an analytical solution of trilinear decomposition is introduced which can be considered as a new direct method for the resolution of three-way two-component systems. The proposed analytical calculation method is applied to the full rank three-way data array and arrays with rank overlap (a type of rank deficiency) loadings in a mode. Close inspections of the analytically calculated feasible regions of rank deficient cases help us to make clearer the information gathered from multi-way problems frequently emerged in physics, chemistry, biology, agricultural, environmental and clinical sciences, etc. These examinations can also help to answer, e.g., the following practical question: “Is two-component three-way data with proportional loading in a mode actually a three-way data array?” By the aid of the additional information resulted from the investigated feasible regions of two-component three-way data arrays with proportional profile in a mode, reasons for the inadequacy of the seemingly trilinear data treatment methods published in the literature (e.g., U-PLS/RBL-LD that was used for extraction of quantitative and qualitative information reported by Olivieri et al. (Anal. Chem. 82 (2010) 4510–4519)) could be completely understood.     

基于化学子空间对线性变换稳定的秩估计方法

提出了一个新的化学计量学方法,即线性变换下稳定的化学子空间法,用于二维数据的化学秩估计.该方法的基本思想是具有化学意义的子空间对在一个方向上的二维数据的线性变换是最稳定的.据此,两个新的定量指标投影残差法和子空间夹角法被提出用于衡量两个子空间之间的差异性.对两个近红外数据的分析结果表明,这一方法为二维数据的化学秩估计提供了一个很有价值的工具.     

旋转微扰法用于确定化学体系的组分数

本文提出了一个新的用于估计化学体系组分的化学计量学方法,方法的基础在于具有化学意义的特征子空间对微扰和数据的旋转变换具有较大的稳定性,通过观察多次实验中投影残差的标准方差来确定化学秩。对实际化学体系的分析结果表明,该法为组分数的估计提供了一个有价值的工具。     

Investigation of kinetics and thermodynamics of DNA hybridization by means of 2-D fluorescence spectroscopy and soft/hard modeling techniques

Reversible hybridization reaction plays a key role in fundamental biological processes, in many laboratory techniques, and also in DNA based sensing devices. Comprehensive investigation of this process is, therefore, essential for the development of more sophisticated applications. Kinetics and thermodynamics of the hybridization reaction, as a second order process, are systematically investigated with the aid of the soft and hard chemometric methods. Labeling two complementary 21 mer DNA single strands with FAM and Texas red fluorophores, enabled recording of the florescence excitation−emission matrices during the experiments which led to three-way data sets. The presence of fluorescence resonance energy transfer in excitation and emission modes and the closure in concentration mode, made the three-way data arrays rank deficient. To acquire primary chemical information, restricted Tucker3 as a soft method was employed. Herein a model-based method, hard restricted trilinear decomposition, is introduced for in depth analysis of rank deficient three-way data sets. By employing proposed hard method, the nonlinear model parameters as well as the correct profiles could be estimated. In addition, a simple constraint is presented to extract chemically reasonable output profiles regarding the core elements of restricted Tucker3 model.     

Development of a method for analysis of Iranian damask rose oil: combination of gas chromatography-mass spectrometry with Chemometric techniques

Gas chromatography-mass spectrometry (GC-MS) combined with Chemometric resolution techniques were proposed as a method for the analysis of volatile components of Iranian damask rose oil. The essential oil of damask rose was extracted using hydrodistillation method and analyzed with GC-MS in optimized conditions. A total of 70 components were identified using similarity searches between mass spectra and MS database. This number was extended to 95 components with concentrations higher than 0.01% accounting for 94.75% of the total relative content using Chemometric techniques. For the first time in this work, an approach based upon subspace comparison is used for determination of the chemical rank of GC-MS data. The peak clusters were resolved using heuristic evolving latent projection (HELP) and multivariate curve resolution-alternating least square (MCR-ALS) by applying proper constraints, and the combination of both methods for some cases. It is concluded that a thorough analysis of the complex mixtures such as Iranian damask rose requires sophisticated GC-MS coupled with the Chemometric techniques.     

红外光谱分析数据特征指纹的可视化表达方法

提出一类红外光谱分析数据特征指纹的可视化表达方法。该方法采用基于贝叶斯准则的多元统计方法对原始数据进行投影变换，再以地维灰度图对变换后数据进行可视化表征，形成基于计算机图像的虚拟化学指纹图谱。将其用于鉴别3种不同产地中药当归样品的结果表明，它能有效提取红外光谱分析数据的特征指纹，实现以虚拟指纹图谱对药材产地的分类鉴别，从而为辨识复杂化学物质体系提供了新的技术手段。     

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

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

Determination of rank by median absolute deviation (DRMAD): a simple method for determining the number of principal factors responsible for a data matrix

Median absolute deviation (MAD) is a well‐established statistical method for determining outliers. This simple statistic can be used to determine the number of principal factors responsible for a data matrix by direct application to the residual standard deviation (RSD) obtained from principal component analysis (PCA). Unlike many other popular methods the proposed method, called determination of rank by MAD (DRMAD), does not involve the use of pseudo degrees of freedom, pseudo F‐tests, extensive calibration tables, time‐consuming iterations, nor empirical procedures. The method does not require strict adherence to normal distributions of experimental uncertainties. The computations are direct, simple to use and extremely fast, ideally suitable for online data processing. The results obtained using various sets of chemical data previously reported in the chemical literature agree with the early work. Limitations of the method, determined from model data, are discussed. An algorithm, written in MATLAB format, is presented in the Appendix. Copyright © 2008 John Wiley & Sons, Ltd.     

Three-way Resolution of the Overlapping Ultrahigh-performance Liquid Spectrochromatograms for the Analysis of a Quaternary Mixture Using Parallel Factor Analysis

This paper presents a new application of three-way parallel factor analysis (3W-PARAFAC) model to the coeluting spectrochromatograms for the quantitative resolution of a quaternary mixture system consisting of paracetamol, propyphenazone, and caffeine with aspirin as an internal standard. Spectrochromatograms of calibration standards, validation sets, and unknown samples were recorded as a function of retention time and wavelength in the range of 0.0–2.5?min and 200–400?nm, respectively, using ultra-performance liquid chromatography with photodiode array detection (UPLC-PDA). Three-way UPLC-PDA data array X (retention time?×?wavelength?×?sample) was obtained from the data matrices of the spectrochromatograms. 3W-PARAFAC decomposition of three-way UPLC-PDA data array provided three loading matrices corresponding to chromatographic mode, spectral mode, and relative concentration mode. Quantitative estimation of paracetamol, propyphenazone, and caffeine in analyzed samples was accomplished using the relative concentration mode obtained by the deconvolution of the UPLC-PDA data set. The validity and ability of 3W-PARAFAC model were checked by analyzing independent test samples. It was observed from analyses that 3W-PARAFAC method has potential to uniquely resolve strongly overlapping peaks of analyzed compounds in a spectrochromatogram, which was obtained under experimental conditions consisting of the lower flow rate, short run time, and simple mobile phase composition. The proposed three-way chemometric approach was successfully applied to the simultaneous quantification of paracetamol, propyphenazone, and caffeine in tablets. Experiments showed that the determination results were in good agreement with label amount in commercial pharmaceutical preparation.     

三维荧光二阶校正法用于未知混合物中苯酚含量的测定

将三维荧光光谱技术与秩消失因子分析、广义秩消失因子分析和交替三线性分解3种二阶校正方法相结合,建立了测定未知混合物中苯酚含量的三维荧光二阶校正新方法。设定在激发波长240~280 nm和发射波长280~360 nm范围内测定未知混合物中苯酚的三维荧光光谱,构建三维响应数据阵,运用基于三线性分解的二阶校正算法进行解析。结果表明,当模拟样品的组分数为2时,秩消失因子分析、广义秩消失因子分析和交替三线性分解3种方法测定苯酚的预测均方根误差分别为0.33,1.18和0.15,平均回收率分别为101.6%,115.6%和101.9%;当组分数为3时,3种方法的预测均方根误差则分别为1.61,1.80和0.51,平均回收率分别为134.2%,133.9%和107.1%;将其分别应用于实际样品中苯酚的测定,结果满意,且交替三线性分解法的测定结果优于秩消失因子分析法和广义秩消失因子分析法。     

Mitigating model deficiency in three-way data analysis by the combination of background constraining and iterative correcting techniques

PARAFAC is a popular model for trilinear data analysis in analytical chemistry. The prerequisite for the successful application of PARAFAC in analytical chemistry is that the three-way data array should follow a trilinear model, which is always violated by the presence of deviations such as Rayleigh scattering in fluorescence spectroscopy. In order to mitigate the influence of model deviations, background constraining and iterative correcting techniques are advocated in this contribution. The method established on these two techniques can nearly eliminate the effect of model deviation on the chemical loading parameters estimated. Compared with other methods for mitigating model deviations, the proposed method requires no prior knowledge about the chemical loading parameters. It is also unnecessary to assign weights to data entities as the weighted PARAFAC of Anderson does. Its implementation is comparable to PARAFAC-ALS and can be programmed to be completely automatic. Its performance has been demonstrated by fluorescent and chromatographic experiments.     

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.     

Differentially expressed genes selection via Laplacian regularized low-rank representation method

With the rapid development of DNA microarray technology and next-generation technology, a large number of genomic data were generated. So how to extract more differentially expressed genes from genomic data has become a matter of urgency. Because Low-Rank Representation (LRR) has the high performance in studying low-dimensional subspace structures, it has attracted a chunk of attention in recent years. However, it does not take into consideration the intrinsic geometric structures in data.In this paper, a new method named Laplacian regularized Low-Rank Representation (LLRR) has been proposed and applied on genomic data, which introduces graph regularization into LRR. By taking full advantages of the graph regularization, LLRR method can capture the intrinsic non-linear geometric information among the data. The LLRR method can decomposes the observation matrix of genomic data into a low rank matrix and a sparse matrix through solving an optimization problem. Because the significant genes can be considered as sparse signals, the differentially expressed genes are viewed as the sparse perturbation signals. Therefore, the differentially expressed genes can be selected according to the sparse matrix. Finally, we use the GO tool to analyze the selected genes and compare the P-values with other methods.The results on the simulation data and two real genomic data illustrate that this method outperforms some other methods: in differentially expressed gene selection.     

Local resolution of two-way data from multicomponent equilibria

A self-modeling curve resolution (SMCR) method is proposed to calculate concentration and spectral profiles for the two-way spectral data from an equilibrium containing several chemical components. The proposed method has three main distinctive steps: (i) fixed size moving window evolving factor analysis (FSMWEFA) is used to identify the selective and zero concentration regions for a desired component, (ii) orthogonal projection resolution (OPR) is used to calculate its concentration profile and (iii) the component striping is done directly to resolve other components. The results of simulated and real polyprotic acid dissociation equilibria showed that the proposed combined method performs well even in situation when the successive stepwise equilibrium constants are close to each other. The applicability of method for resolving the triprotic acid system with rank deficiency due full spectral overlapping of two involved chemical species also is shown.     

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.     

Three-way partial least-squares regression for the simultaneous kinetic-enzymatic determination of xanthine and hypoxanthine in human urine

The performance of three-way principal component analysis and three-way partial least-squares regression when applied to a complex kinetic-enzymatic system is studied, in order to investigate the analytical potential of the combined use of these chemometric technologies for non-selective enzymatic systems. A enzymatic-kinetic procedure for the simultaneous determination of hypoxanthine and xanthine in spiked samples of human urine is proposed. The chemical system involves two consecutive reactions catalyzed by xanthine oxidase (EC 1.17.3.2). This enzyme catalyzes the oxidation of hypoxanthine, first to xanthine and then to uric acid, a competitive inhibitor of the reactions. The influence of uric acid during quantitative determination was considered in the design of the calibration set. The sample and enzyme solution were mixed in a stopped-flow module and the reaction was monitored using a diode array spectrophotometer. The recorded data have an intrinsical three-component structure (samples, time and wavelength). This data array was studied via three-way principal component analysis and was modeled for quantitative purposes using a three-way partial least-squares calibration procedure. Results are compared with those obtained by applying classical bilinear PLS to the previously unfolded data matrix.