Rank deficiency in spectrophotometric study of acid dissociation equilibria in mixed solvent media

A methanol/water mixture, although considered a binary system, does not conform to solute distribution theory. Moreover, similar shortcomings are encountered in studies concerning acid-base reactions in mixtures of these two components. We demonstrate that in mixtures of methanol/water and an acid-base dye as indicator, three components exist for each of the protonated and deprotonated forms. In contrast, analysis of spectrophotometric data obtained from pH-metric titration of the dye in mixtures of methanol/water, shows no more than one component for each form of the dye. This can only be interpreted as a rank deficiency in the acquired data. We used matrix augmentation strategy for eliminating rank deficiency and constructing the corresponding full rank data matrices. In addition, concentration and spectral profiles of the components were obtained using multivariate curve resolution-alternating least squares.     

An efficient method for solving the eigenvalue problem for matrices having a skew-symmetric (or skew-Hermitian) component of low rank

In the numerical modelling of mechanical systems, eigenvalue problems occur in connection with the evaluation of resonance frequencies, buckling modes and other more esoteric calculations. The matrices whose eigenvalues are sought sometimes have a skew-symmetric component and the presence of this component adds significantly to the computational effort required. In many cases where there is a skew-symmetric component, this component has a much lower rank than the symmetric component which generally has rank equal to its dimension. Examples of such cases abound in the area of rotor dynamics where the stiffness and damping matrices associated with journal bearings have significant skew-symmetric components. The solution of the eigenvalue problem for an unsymmetric matrix takes more than twice the number of operations required for the solution of the eigenvalue problem for a symmetric matrix of the same dimension. This paper puts forward a new method for the solution of the eigenvalue problem for matrices having a skew-symmetric component of low rank and shows that it is faster than established methods of comparable accuracy for the general unsymmetric NxN matrix if the rank of the skew-symmetric component is less than N/7.3.     

Application of Eigenstructure Tracking Analysis and SIMPLISMA to the Study of the Protonation Equilibria of cCMP and Several Polynucleotides

The application of eigenstructure tracking analysis (ETA) and SIMPLISMA for the investigation of the protonation equilibria of a monomer and several polynucleotides is proposed. Both approaches have been applied in the pH and in the wavelength direction to the spectroscopic data matrices obtained in the study of each equilibrum. ETA provides information about the number of components in the system, their evolution along the titration, and the local rank. SIMPLISMA is also used to obtain the number of compounds in the system, the concentration profiles, and the unit spectrum of each compound. The results obtained with SIMPLISMA and those obtained previously with the alternating least-squares approach are compared.     

Assessment of the co-elution problem in gas chromatography-mass spectrometry using non-linear optimization techniques

Multivariate curve resolution based on the minimization of an objective function (MCR-FMIN) defined directly from the non-fulfillment of constraints was applied for the first time as a deconvolution method to separate co-eluted gas chromatographic-mass spectrometric (GC-MS) signals. Simulated and real (standard real mixture and limon oil) GC-MS data were used to evaluate the feasibility of this method. The MCR-FMIN solutions have been obtained based on the rotation of principal component analysis (PCA) solutions using the non-linear optimization algorithms. Calculation of the initial values of R rotation matrix using model free analysis methods such as fixed-size moving window-evolving factor analysis (FSMW-EFA), evolving latent projective graphs (ELPGs), and heuristic evolving latent projection (HELP) was proposed for faster convergence and avoiding to be stuck in local minima in MCR-FMIN algorithm. The band boundaries of feasible solutions (MCR-BANDS) obtained using MCR-FMIN were calculated for simulated data to assess the reliability of the method. In addition, the results of this method were compared with those of two most common self-modeling curve resolution (SMCR) methods of multivariate curve resolution-alternating least square (MCR-ALS) and HELP. A reasonable result can be obtained by selecting proper constraints, such as non-negativity, unimodality, normalization, and selectivity. However, when the number of components or the level of noise in each peak cluster increase, the convergence of algorithm becomes difficult and the results are not reliable. For quick and accurate analysis of co-eluted multi-component problematic GC-MS data MCR-FMIN can be considered as an alternative method to the MCR-ALS and HELP methods.     

多变量矩阵方程异类约束解的修正共轭梯度法

基于求解线性代数方程组的共轭梯度法,通过对相关矩阵和系数的修改,建立了一种求多矩阵变量矩阵方程异类约束解的修正共轭梯度法.该算法不要求等价线性代数方程组的系数矩阵具备正定性、可逆性或者列满秩性,因此算法总是可行的.利用该算法不仅可以判断矩阵方程的异类约束解是否存在,而且在有异类约束解,不考虑舍入误差时,可在有限步计算后求得矩阵方程的一组异类约束解;选取特殊初始矩阵时,可求得矩阵方程的极小范数异类约束解.另外,还可求得指定矩阵在异类约束解集合中的最佳逼近.算例验证了该算法的有效性.     

A Method of Making Subjective Measurements Compatible with Hierarchical Matrices of Preference Ratios

A new method of making subjective measurements compatible is considered which is based on decomposing an initial inverse symmetric matrix of preference ratios of compared objects into matrices of lower dimensionality. The method is classified as a procedure for the conceptual analysis of data and makes it possible to improve the quality of adopted solutions to the problem of choosing from a set of alternatives in systems for supporting decision making.     

Teaching and learning chemometrics with MatLab

MatLab, a commercially available program for manipulation of matrices, may be used as a high-level programming language for teaching and learning multivariate chemometric procedures. Examples are given here of applications to multiwavelength spectrophotometry, iterative least-squares curve fitting by simplex optimization, evolving factor analysis, and rank annihilation.     

An information-theoretic methodology for the resolution of pure component spectra without prior information using spectroscopic measurements

The resolution of pure component spectra based on spectroscopic measurements from a reaction system is a challenging task for chemometric systems in the absence of a priori knowledge about the reaction components involved. A popular approach in the literature is based on constrained entropy minimization of the second-order derivative of the resolved pure component spectra. Using an analytical information theoretic framework, it can however be shown that minimization of this cost function is not sufficient to completely separate the underlying components from a set of mixture spectra. Instead, an augmented objective function derived from this analysis is proposed for complete minimization of the mutual information between separated components. The final optimization approach is further shown to be analog to independent component analysis (ICA), a signal processing technique successfully applied to biomedical and speech data to separate linear source mixtures in the absence of a priori information. The developed theoretical insights and proposed methodologies in this paper are illustrated in a simulation study on the separation of three component spectra based on absorbance data acquired from a first-order kinetic reaction system.     

Second-order advantage achieved by unfolded-partial least-squares/residual bilinearization modeling of excitation-emission fluorescence data presenting inner filter effects

A second-order multivariate calibration approach, based on a combination of unfolded-partial least-squares with residual bilinearization (U-PLS/RBL), has been applied to fluorescence excitation-emission matrix data for multicomponent mixtures showing inner filter effects. The employed chemometric algorithm is the most successful one regarding the prediction of analyte concentrations when significant inner filter effects occur, even in the presence of unexpected sample components, which require strict adherence to the second-order advantage. Results for simulated fluorescence excitation-emission data are described, in comparison with the classical approach based on parallel factor analysis and other second-order algorithms, including generalized rank annihilation, bilinear least squares combined with residual bilinearization and multivariate curve resolution-alternating leastsquares. A set of experimental data was also studied, in which calibration was performed with fluorescence excitation-emission matrices for samples containing mixtures of chrysene (the analyte of interest) and benzopyrene (which produced strong inner filter effect across the useful wavelength range). Prediction was made on validation samples with a qualitative composition similar to the calibration set, and also on test samples containing an unexpected component (pyrene). In this latter case, U-PLS/RBL showed a unique success for the analysis of the calibrated component chrysene, achieving the useful second-order advantage.     

Space-time measurement of indoor radio propagation

Most existing techniques for indoor radio propagation measurement do not resolve the angles from which signal components arrive at the receiving antenna. Knowledge of the angle-of-arrival is required for evaluation of evolving systems that employ smart antenna technology to provide features such as geolocation, interference cancellation, and space-division multiplexing. This paper presents a novel technique for the joint measurement of the angles, times and complex amplitudes of discrete path arrivals in an indoor propagation environment. A data acquisition system, based upon a vector network analyzer and multichannel antenna array is described, together with its use to collect channel measurement matrices. The inherent error sources present in these measurement matrices are investigated using a compact indoor anechoic range. Two signal processing algorithms are presented whereby the channel parameters may be estimated from raw measurements. In the first approach, an optimum beamformer is derived which compensates for systematic errors in the data acquisition system. This approach features very low computational complexity, and delivers modest resolution of path components. The second algorithm is based upon the maximum likelihood criterion, using the measured calibration matrices as space-time basis functions. This algorithm provides super-resolution of all path parameters, at the cost of increased computation. Several example measurements are given, and future directions of our research are indicated     

Programming Univariate and Multivariate Analysis of Variance

A formulation of analysis of variance based on a model for the subclass means is presented. The deficiency of rank in the model matrix is handled, not by restricting the parameters, but by factoring the matrix as a product of two matrices, one providing a column basis for the model and the other representing linear functions of the parameters. In terms of the column basis and a diagonal matrix of subclass or incidence numbers, a compact matrix solution is derived which provides for testing a hierarchy of hypotheses in the non-orthogonal case. Two theorems are given showing that a column basis for crossed and/or nested designs can be constructed from Kronecker products of equi-angular vectors, contrast matrices, and identity matrices. This construction can be controlled in machine computation by a symbolic representation of each degree of freedom for hypothesis in the analysis. Provision for a multivariate analysis of variance procedure for multiple response data is described. Analysis of covariance, both in the univariate and multivariate case, is shown to be most convenient computationally as part of the multivariate procedure.     

Maximum likelihood principal component analysis with correlated measurement errors: theoretical and practical considerations

Procedures to compensate for correlated measurement errors in multivariate data analysis are described. These procedures are based on the method of maximum likelihood principal component analysis (MLPCA), previously described in the literature. MLPCA is a decomposition method similar to conventional PCA, but it takes into account measurement uncertainty in the decomposition process, placing less emphasis on measurements with large variance. Although the original MLPCA algorithm can accommodate correlated measurement errors, two drawbacks have limited its practical utility in these cases: (1) an inability to handle rank deficient error covariance matrices, and (2) demanding memory and computational requirements. This paper describes two simplifications to the original algorithm that apply when errors are correlated only within the rows of a data matrix and when all of these row covariance matrices are equal. Simulated and experimental data for three-component mixtures are used to test the new methods. It was found that inclusion of error covariance information via MLPCA always gave results which were at least as good and normally better than PCA when the true error covariance matrix was available. However, when the error covariance matrix is estimated from replicates, the relative performance depends on the quality of the estimate and the degree of correlation. For experimental data consisting of mixtures of cobalt, chromium and nickel ions, maximum likelihood principal components regression showed an improvement of up to 50% in the cross-validation error when error covariance information was included.     

Sparse electrocardiogram signals recovery based on solving a row echelon‐like form of system

The study of biology and medicine in a noise environment is an evolving direction in biological data analysis. Among these studies, analysis of electrocardiogram (ECG) signals in a noise environment is a challenging direction in personalized medicine. Due to its periodic characteristic, ECG signal can be roughly regarded as sparse biomedical signals. This study proposes a two‐stage recovery algorithm for sparse biomedical signals in time domain. In the first stage, the concentration subspaces are found in advance. Then by exploiting these subspaces, the mixing matrix is estimated accurately. In the second stage, based on the number of active sources at each time point, the time points are divided into different layers. Next, by constructing some transformation matrices, these time points form a row echelon‐like system. After that, the sources at each layer can be solved out explicitly by corresponding matrix operations. It is noting that all these operations are conducted under a weak sparse condition that the number of active sources is less than the number of observations. Experimental results show that the proposed method has a better performance for sparse ECG signal recovery problem.Inspec keywords: electrocardiography, matrix algebra, medical signal processingOther keywords: sparse electrocardiogram signal recovery, row echelon‐like form of system, noise environment, biological data analysis, personalised medicine, dictionary learning algorithm, transformation matrices, sparse biomedical signal recovery     

基于鲁棒主成分分析和MFCC反复结构的歌声分离方法

针对单一传统方法对歌声分离不彻底的问题，文章提出了一种基于鲁棒主成分分析（Robust Principal Component Analysis, RPCA）和梅尔频率倒谱系数（Mel Frequency Cepstrum Coefficients, MFCC）反复结构的两步歌声伴奏分离模型。该模型有效地改善了鲁棒主成分分析对歌声分离不完全和梅尔频率倒谱系数反复结构歌声在低频处分离不佳的问题。首先使用鲁棒主成分分析将混合音乐信号分解为低秩矩阵和稀疏矩阵，然后分别对其提取梅尔频率倒谱系数特征参数并且对其进行相似运算，构建相似矩阵及建立梅尔频率倒谱系数反复结构模型并通过反复结构模型分别得到低秩矩阵和稀疏矩阵相关的掩蔽矩阵，最后根据构建的掩蔽矩阵模型以及傅里叶逆变换得到背景音乐和歌声。在公开数据集上进行了实验，实验结果表明本文算法在歌声分离性能上与比较算法相比，平均信号干扰比值最高有接近7 dB的提高。     

A fast dual boundary element method for 3D anisotropic crack problems

In the present paper a fast solver for dual boundary element analysis of 3D anisotropic crack problems is formulated, implemented and tested. The fast solver is based on the use of hierarchical matrices for the representation of the collocation matrix. The admissible low rank blocks are computed by adaptive cross approximation (ACA). The performance of ACA against the accuracy of the adopted computational scheme for the evaluation of the anisotropic kernels is investigated, focusing on the balance between the kernel representation accuracy and the accuracy required for ACA. The system solution is computed by a preconditioned GMRES and the preconditioner is built exploiting the hierarchical arithmetic and taking full advantage of the hierarchical format. The effectiveness of the proposed technique for anisotropic crack problems has been numerically demonstrated, highlighting the accuracy as well as the significant reduction in memory storage and analysis time. In particular, it has been numerically shown that the computational cost grows almost linearly with the number of degrees of freedom, obtaining up to solution speedups of order 10 for systems of order 10⁴. Moreover, the sensitivity of the performance of the numerical scheme to materials with different degrees of anisotropy has been assessed. Copyright © 2009 John Wiley & Sons, Ltd.     

Stability analysis of constraints in flexible multibody systems dynamics

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems assume that the constraint equations are linearly independent. During the motion, when the system is at a singular configuration, the constraint Jacobian matrix possesses less than full rank and hence it results in singularities. This occurs when the direction of a constraint coincides with the direction of the lost degree of freedom. In this paper the constraint equations for deformable bodies are modified for use in the neighborhood of the singular configuration to yield the system inertia matrix which is nonsingular and also to take the actual generalized constraint forces into account. The procedures developed are applicable to both the augmented approach and the coordinate reduction methods. For the modeling of the constrained flexible multibody systems, a general recursive formulation is developed using Kane's equations, finite element method and modal analysis techniques. The system may contain revolute, prismatic, spherical or other types of joints, as well as geometrical nonlinearities; the rotary inertia is also automatically included. Simulation of a two-link flexible manipulator is presented at a singular configuration to demonstrate the utility of the method.     

Supplier evaluation and selection: an augmented DEA approach

Evaluating and selecting suppliers is an essential part of effectively managing today's dynamic and global supply chains. In this paper, we propose a supplier evaluation and selection methodology based on an extension of data envelopment analysis (DEA) that can evaluate suppliers in an efficient manner. Through the incorporations of a range of virtual standards, the proposed methodology termed augmented DEA, has enhanced discriminatory power over basic DEA models to rank suppliers. In addition, weight constraints are introduced to reduce the possibility of having inappropriate input and output factor weights. We demonstrate the application of augmented DEA with comparison experiments and find that the augmented DEA model has advantages over the basic DEA model as well as the cross-efficiency and super-efficiency models. Finally, we present a case application with data obtained from a communication and aviation electronics company to demonstrate the applicability and use of augmented DEA.     

Resolution of rotational ambiguity for three-component systems

Soft modeling of multivariate data is a powerful method for the analysis of processes that cannot be described quantitatively by a chemical model. Soft modeling usually does not result in unique solutions. Thus, the determination of the range of feasible solutions is important. For two-component systems the determination of that range is well-understood; for three-component systems the task is remarkably more complex. We present a novel method that can be applied to any multivariate data set, irrespective of overlap or realistic noise level. The expansion to four components is indicated.     

A Fortran program to generate finite difference formulas

A discretization process is described by which it is possible to generate finite difference formulas for arbitrary linear two-dimensional partial differential equations. The process is based on a novel approach to finite difference analysis in which differential operators are approximated by rectangular matrices. In this approach, the discrete form of a complicated operator is obtained by performing simple numerical operations on elementary matrix factors. The analysis is augmented by a listing of a computer program based on the method for the automatic generation of finite difference formulas.     

An in vitro release study of indomethacin from nanoparticles based on methyl methacrylate/glycidyl methacrylate copolymers

Indomethacin was coupled onto some macromolecular nanostructures based on methyl methacrylate copolymers with glycidyl methacrylate and tested as a model drug. The polymeric matrices were synthesized by radical emulsion copolymerization with and without the presence of a continuous external magnetic field of 1500 Gs intensity. Mathematical analysis of the release data was performed using Higuchi, Peppas–Korsmeyer equations. NIR chemical imaging (NIR-CI) was used to provide information about the spatial distribution of the components in the studied nanostructures. This opportunity was used to visualize the spatial distribution of bioactive substances (indomethacin) into the polymeric matrix, as well as to evaluate the degree of chemical and/or physical heterogeneity of the bioactive samples. The release rate dependence on the synthesis conditions as well as on the chemical compositions of the tested polymeric systems, it was also evidenced.