Three-Mode Abstracts, Part A

INDEX

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• Abdollahi, H., & Nazari, F. (2003).
  Rank annihilation factor analysis for spectrophotometric study of complex formation equilibria. Analytica Chimica Acta, 486, 109-123.
  The use rank annihilation factor analysis (RAFA) for spectrophotometric studies of complex formation equilibria are proposed. One-step complex formation and two successive and mononuclear complex formation systems studied successfully by proposed methods. When the complex stability constant acts as an optimizing object, and simply combined with the pure spectrum of ligand, the rank of original data matrix can be reduced by one by annihilating the information of the ligand from the original data matrix. The residual standard deviation (R.S.D.) of the residual matrix after bilinearization of the background matrix is regarded as the evaluation function. The performance of the method has been evaluated by using synthetic data. For two-step successive complex formation systems, the effects of noise level and equilibrium constants K-1 and K-2 on output of algorithm are investigated. The applicability of method for resolving the two-step successive complex formation systems with full spectral overlapping of two complex species also is shown. Spectrophotometric studies of murexide-calcium, dithiazone-nickel and methyl thymol blue (MTB)-copper are used as experimental model systems with different complexation stoichiometries and spectral overlapping of involved components.
  
• Abel, R.B., Leurgans, S.E., & Ross, R.T. (1992).
  Multilinear models: Experimental design in spectrofluoroscopy. Technical Report No. 470. Department of Statistics, Ohio State University.
  Multilinear models, particularly PARAFAC models, are useful for spectrofluoroscopic data analysis. Experimental design techniques for nonlinear models can be applied to PARAFAC models to guide spectrofluoroscopic data collection. This report concentrates on the design criterion of D-optimality, the maximization of the determinant of the model's information matrix. The information matrix for a PARAFAC model has two distinguishing features. First, the information matrix is a function of inner and outer products of the spectra. Second, although the dimensions of a PARAFAC model's information matrix can be large, the determinant of the information matrix is a simple function of determinants of smaller matrices. Searches for D-optimal designs using this function were dramatically faster than searches using the entire information matrix. Spectrofluoroscopic data collection often results in missing observations. The optimality properties of some strategies for systematically avoiding observations are explored. Another design criterion examined is that of minimizing the asymptotic variance of the ratios of estimated absorption or emission intensities. This criterion often results in different optimal designs than the criterion of D-optimality, but D-optimal designs generally provide nearly minimal variance of ratio designs.
  
• Aberg, P. (2002).
  Electronic biopsies applications and data analysis. Doctoral thesis, Karolinska Institutet, Stockholm.
  Background/aims: This thesis presents two studies using electrical skin impedance, and an investigation concerning how information can be extracted from multivariate impedance spectra. The two investigations are sin property variations within volar forearms, and assessment of impedance of skin cancer and other lesions. Various numerical methods are described, specially multivariate techniques.
  Methods: Skin impedance spectra and trans epidermal water loss of volar forearms were measured on 8 sites covering approximately 4x4 cm2 the volar forearms of 27 healthy volunteers. Skin property variations were investigated using 3-way ANOVA and parallel factor analysis. Impedance spectra on lesions were measured at 258 benign pigmented cellular nevi (BEN), 34 basal cell carcinoma (BCC), 17 dermatofibroma (DER), 35 dysplastic nevi (DYS), and 26 seborrheic keratoses (SEB). Reference skin was measured ipsy-laterally to the lesions. Differences between lesions and references were analysed using non-parametric statistics.
  Results: 3-way ANOVA showed significant differences between inner and outer sides of forearms for impedance (P<0.001) and trans epidermal water loss (P<0.01). Additional significant variations (P<0.01) between left and right arm were shown using parallel factor analysis on the impedance spectra. Significant differences between lesions and references were found for BEN (P<0.001), BCC (P<0,001), DYS (P<0.01), and SEB (P<0.01).
  Conclusions: It is crucial to design skin studies carefully to avoid the baseline variations of skin to interfere with the studied phenomenon, and impedance methods might, after technical developments, be used as a diagnostic decision tool for skin lesions. However, the information distribution in the ingerent in the impedance spectra is such that, for subtle diagnostic applications, appropriate number-crunching techniques are necesssary.
  
• Aberg, P., Geladi, P., Nicander, I. & Ollmar, S. (2002).
  Variation of skin properties within human forearms demonstrated by non-invasive detection and multi-way analysis. Skinresearch and Technology, 8, 194-201.
  Background: It is known that the properties of human skin vary locally. The purpose of this study was to investigate the properties of human volar forearms even further using advanced non-invasive techniques and numerical methods. Methods: The skin properties of human volar forearms were investigated using measurements of trans epidermal water loss and multifrequency electrical impedance. Eight sites on the forearms of 27 healthy volunteers were measured. The sites were oriented as squares, four sites on both left and right forearm, approximately 40-50 mm apart.Results: Analysis of variance showed significant differences for epidermal water loss (P < 0.01) and the skin impedance (P < 0.001) between the inner and outer sides of the arms. Additionally, parallel factor analysis of the full skin impedance spectra also showed that there are systematic differences between right and left arm (P < 0.01).Conclusions: It is crucial to design skin studies carefully in order to minimise the effects of the local skin properties of human skin.
   
• Acar, E., Çamtepe, S. A., Krishnamoorthy, M. S., & Yener, B. (2005).
  Modeling and multiway analysis of chatroom tensors. In ISI 2005: IEEE International Conference on Intelligence and Security Informatics, Lecture Notes in Computer Science 3495, pp 256-268. Springer Verlag.
  This work identifies the limitations of n-way data analysis techniques in multidimensional stream data, such as Internet chatroom communications data, and establishes a link between data collection and performance of these techniques. Its contributions are twofold. First, it extends data analysis to multiple dimensions by constructing n-way data arrays known as high order tensors. Chatroom tensors are generated by a simulator which collects and models actual communication data. The accuracy of the model is determined by the Kolmogorov-Smirnov goodnessof- fit test which compares the simulation data with the observed (real) data. Second, a detailed computational comparison is performed to test several data analysis techniques including svd [1], and multiway techniques including Tucker1, Tucker3 [2], and Parafac [3].
  
• Acar, E., Bingol, C. A., Bingol, H., & Yener, B. (2006).
  Computational analysis of epileptic focus localization. In Proc. of The Fourth IASTED International Conference on Biomedical Engineering.
  Epilepsy surgergy outcome strongly depends on the localization of epileptic focus. The analysis of ictal EEG (scalp or intracranial) is a gold standard for definition of localization of epileptic focus. In order to automate visual analysis of large amounts of EEG data, we examine the correlations among electrodes captured by linear, nonlinear and multilinear data analysis techniques. We study the performance of these statistical tools to understand the complex structure of epilepsy seizure and localize seizure origin. Our analysis results on four patients with temporal lobe epilepsy reveal that multiway (Tucker3 [1]) and nonlinear multiway (Kernelized Tucker3) analysis techniques are capable of capturing epileptic focus precisely when validated with clinical findings whereas linear and nonlinear analysis techniques (SVD, Kernel PCA) fail to localize seizure origin.
  
• Achim, A., & Bouchard, S. (1997).
  Toward a dynamic topographic components model. Electroencephalography and Clinical Neurophysiology, 103, 381-385.**
  Möcks' topographic component model (tcm) (Möcks, J. Topographic components model for event-related potentials and some biophysical considerations. IEEE Trans. Biomed. Eng., 1988a, 35: 482-484; Möcks, J. Decomposing event-related potentials: A new topographic components model. Biol. Psychol., 1988b, 26: 199-215) decomposes event-related potentials into components uniquely determined by their respective amplitude profiles across replicates, assuming a constant topography and wave shape for each component. To accommodate possible changes in the component expression across conditions, a dynamic version of tcm is investigated which further admits component modulation in time scale. Twenty test problems were synthesized, incorporating two arbitrary topographies each activated with its own arbitrary wave shape modified, across two conditions, in amplitude, onset and duration. Seventeen problems were perfectly solved, with substantial success on the remaining three, confirming that component jitter or stretching can even help component identification.
  
• Achim, A., & Marcantoni, W. (1997).
  Principal component analysis of event-related potentials: Misallocation of variance revisited. Psychophysiology, 34, 597-606.**
  Misallocating variance, in event-related potential analysis, refers to attributing an experimental effect to components not actually affected. a vector interpretation of the relationship between mathematically derived and hue underlying components shows that misallocation depends exclusively on incorrect identification of the affected component. Simulations, using seven imperfect rotations, confirmed all predictions from the vector interpretation concerning the presence, direction, and size of misallocated variance. Contrary to principal component analysis (pca), Möcks's topographic component model (tcm) is not subject to rotation problems. These two methods were compared over 100 simulations in which the components had constant waveforms and topographies across participants. The group effect was always detected, but only pca and not tcm showed significance on other components, except when their random weights happened to differ between groups.
   
• Adamopoulos, J. (1982).
  The perception of interpersonal behavior: Dimensionality and importance of the social environment. Environment and Behavior, 14, 29-44.
  This study is aimed towards an analysis of individual cognitive structures with respect to behavior in the social environment. All analyses were performed on the deviation scores from the grand mean of all the observations. Each of the ten subjects produced the following data: 12 behaviors by 12 roles by 8 situations. Tucker's (1966) three-mode factor analysis was employed, both for females and males and for each of the ten subjects. After selection of the factors the eigenvector matrices were rotated to varimax simple structures.
  
• Adamopoulos, J. (1984).
  The differentiation of social behavior: Toward an explanation of universal interpersonal structures. Journal of Cross-Cultural Psychology, 15, 487-508.
  The 35 (behaviors) by 6 (resources) by 16 (informants) matrix was analyzed using Tucker's (1966) three-mode analytic procedure. All analyses were performed on the deviation scores around the grand mean of all observations. A principal component analysis of the cross-products yielded five behavior factors. The corresponding eigenvector matrix was subsequently rotated to varimax simple structure. Two informant and four resource factors were also rotated using the varimax criterion. The core matrix was transformed according to these rotations.
  
                 
  
• Algera, J.A.(1980).
  Kenmerken van werk. Doctoral thesis. Leiden, The Netherlands: author.
  As part of a larger study T3 (as implemented by Kroonenberg & De Leeuw, 1980) was performed on 25 jobs in a steel factory, 24 tasks and 10 judges to check whether the judges agreed on the relationships between jobs and tasks. No numerical details given.
  
• Allosio, N., Boivin, P., Bertrand, D., & Courcoux, P. (1997).
  Characterisaion of barley transformation into malt by three-way factor analysis of near infrared spectra. Journal of Near Infrared Spectroscopy, 5, 157-166.
  Data collected for the spectral study of time series can be presented as a three-way array in which the three modes are the batches, time and wavelengths. The parallel factor analysis (PARAFAC) model is relevant for the analysis of three-way data tables, while the malting process consists of the time transformation of barley into malt. Furthermore, samples were collected on each day of an industrial malting process and their near infrared spectra were recorded in diffuse reflectance mode from 1100 to 2500 nm. The time and the wavelength modes associated with the first component showed that the spectra intensities allowed the classification of samples according to time. A study of the other loading vectors weighted by their coefficients on the time-mode explain some phenomena taking place during malting. PARAFAC allowed us to separate batches acoording to the malting process to which they were submitted. Batches were also differentiated according to the chemical modification rate that occurred as expressed by the biochemical analyses results of the final malts. This work shows that PARAFAC can be a useful tool in the study of time series.
  
• Alsberg, B.K., & Kvalheim, O.M. (1993).
  Compression of nth-order data arrays by B-splines. Part 1: Theory. Journal of Chemometrics, 7, 61-73.
  For efficient handling of very large data arrays, pretreatment by compression is mandatory. In the present paper B-spline methods are described as good candidates for such data array compression. The mathematical relation between the maximum entropy method for compression of data tables and the B-spline of zeroth degree is described together with the generalization of B-spline compression to nth-order data array tables in matrix and tensor algebra.
  
• Alsberg, B.K.,& Kvalheim, O.M.(1994a).
  Compression of three-mode data arrays by B-splines prior to three-mode principal component analysis. Chemometrics and Intelligent Laboratory Systems, 23, 29-38.
  Three-mode PCA is very computer demanding. It requires a large amount of storage space and many floating point operations (FLOPS). By using three-mode B-spline compression of three-mode data arrays, the original data array can be replaced by a smaller coefficient array. Three-mode principal component analysis (PCA) is then performed on the much smaller coefficient array instead of on the original array. For the compression approach to be efficient the three-mode data array is assumed to be well approximated by smooth functions. The smoothness affects the dimensions of the coefficient array. It is always possible to approximate the data to any precision but the reward in reduced computation time and storage is lost when the dimensions of the coefficient array approach dimensions of the original array.
  
• Alsberg, B.K., & Kvalheim, O.M.(1994b).
  Speed improvement of multivariate algorithms by the method of postponed basis matrix multiplication. Part I: Principal component analysis. Chemometrics and Intelligent Laboratory Systems, 24, 31-42.
  Compression is one way of making analysis of large data matrices faster. Compression is here defined as the case when a alarge matrix X is replaced by a smaller coefficient C. The coefficients are obtained by least squares fitting to some compression basis. When performing, e.g., principal component analysis (PCA) of C, the results are comparable but not equal to the results from analyzing X. In this paper a solution to this problem is suggested by rewriting the PCA algorithm in terms of C and the compression basis matrices. This has been accomplished by applying a method where speed improvent is achieved by postponing basis matrix calculations in key steps of the PCA algorithm. The method suggested can also be applied to other (but not all kinds of) multivariate algorithms.
  
• Alsberg, B.K.,& Kvalheim, O.M.(1994c).
  Speed improvement of multivariate algorithms by the method of postponed basis matrix multiplication. Part II: Three-mode principal component analysis. Chemometrics and Intelligent Laboratory Systems, 24, 43-54.
  Compression is one way of making analysis of large data arrays faster. Compression is here defined as the case when a large array X is replaced by a smaller coefficient array C. The coefficients are obtained by least squares fitting to some compression basis. This paper deals with three-mode arrays. When performing, e.g., three-mode principal component analysis of C the results are comparable but not equal to the results from analyzing X. In this paper a solution is suggested to this problem by rewriting the three-mode PCA algorithm which utilizes C and the compression basis matrices. This has been accomplished by applying a method where speed improvement is achieved by postponing basis matrix calculations in key steps of the three-mode PCA algorithm.
  
• Alsberg, B.K. (1997).
  A diagram notation for N-mode array equations. Journal of Chemometrics, 11, 251-266.
  An intuitive and user-friendly notation based on a diagrammatical way to visualise and manipulate N-mode array equations is proposed. The diagrams have the appearance of graphs and make some manipulations of equations involving higher-order arrays easier.
  
• Altink, W.M.M., & Born, M.Ph. (1987).
  Achievement strategies in work organizations: Concept analysis and developement of a situation-response inventory. Social and Behavioral Science Documents, 17, ##-##.
  An SR-questionaire was applied in the context of organisational problems. The methods of analysis employed here are 1) Three-way analysis of variance and 2)three-mode principal component analysis (Kroonenberg, 1983). No numerical details given.
  
• Altink, W.M.M., Born, M.Ph., & Algera, J.A. (1987).
  De SR-vragenlijstmethode bij organisatievraagstellingen: Nieuwe mogelijkheden voor diagnostiek van werkgedrag? [The method of SR-inventory in workorganizations: New possibilities for diagnostics of work-attitude?] Gedrag en Organisatie, 1, 38-53.
  The method of the SR-(situation-response-) inventory is not yet used within workorganizations. Up until now, its use is restricted to studies within the domain of personality psychology. A unique characteristic of the SR-inventory is that respondents describe their response-styles in relation to various situations. Consequently, it is possible to investigate the relationships between situation-, response-, and person-factors. In this article the method of the SR-inventory is discussed with respect to its use in personnel selection and organizational change. Results obtained with a recently developed SR-inventory for the measurement of managerial workstyles serve as an illustration. The conclusion is that the method of the SR-inventory has several advantages over other research-techniques currently being used to analyze organizational problems. When actually using the SR-inventory within workorganizations, some aspects need special attention.
  
   
• Amato, V. (1989).
  Autoregressive process for three-way data matrices. In R. Coppi and S. Bolasco (Eds.), Multiway data analysis (pp. 383-389). Amsterdam: Elsevier.
  A structural autoregressive process of first order (Amato, 1967, 1977) is here extended to the analysis of three-way data matrices describing complex evolutive phenomena. The proposed approach consists of three steps. The first is to estimate the basic matrix that generates the process. The second performs the spectral decomposition associated to this matrix in order to evidence the so called latent environments. The third step is devoted to asymptotic behaviour study of the series. A numerical example on real data will conclude the paper.
  
• Amato, V. (1992).
  Special vec for a diagonal form of a group of matrices. Statistica Applicata, 4, 653-657.
  This article generalizes the usual method of principal components analysis of a real square matrix to principal components analysis of a group of matrices. We shall use the theory of characteristic eigenvalues and eigenvectors, but with reference to several real square matrices A₁,...,A_m of order n arranged to form a labda-matrix in the Frazer, Duncan & Collar sense. If the determinant of the polynomial of the highest degree m in l with not null matrix coefficients A_s (s=1,...,m) independent of labda is zero, then A_s is said to be a G group. If in particular m=1, we obtain the usual analysis for the only square matrix A₁, that represents in this special case an improper G group. Partitioned matric of order nm, called W, that includes A_s, is treated here. Associated with W there exists a V_m-1 special vec of nm components, denominated eigenvec of degree m-1, such that W V_m-1=V_m. Besides this paper considers for G, the Flury (1984) hypothesis which positive definite symmetric matrices A_s are simultaneously diagonalizable: B'A_sB=D_s (diagonal), with B orthogonal of order n. A rule to rearrange the nm eigenvalues of W into m diagonal matrices delta_s, is here illustrated. A diagonal form for G, in the Flury assumption, concludes the article.
  
• Amrheim, M., Srinivasan, B., Bonvin, D., & Schumacher, M. M. (1996).
  Inferring concentrations on-line from near-infrared spectra: Nonlinear calibration via mid-infrared measurements. Computers & Chemical Engineering, 20, S975-S980.
  Near-Infrared (NIR) spectroscopic methods for on-line concentration estimation are gaining popularity in chemical production. The problem with NIR data, however, is its nonlinear nature rendering standard factor-analytical (FA) techniques inapplicable to estimate the concentrations. To infer concentrations on-line from NIR spectra, the following steps are proposed: (i) measure both the NIR and Mid-Infrared (MIR) spectra of unknown reaction mixtures in the laboratory (MIR spectroscopy has the desirable property of linearity with respect to concentrations), (ii) apply FA techniques to MIR data and infer concentrations, (iii) calibrate NIR against the estimated concentrations by using standard nonlinear regression methods and (iv) use the calibration model in subsequent production runs. Experimental results are presented to illustrate the efficiency of this method.
   
• Andersen, A. H. & Rayens, W. S. (2004).
  Structure-seeking multilinear methods for the analysis of fMRI data. NeuroImage, 22, 728-739.
  In comprehensive fMRI studies of brain function, the data structures often contain higher-order ways such as trial, task condition, subject, and group in addition to the intrinsic dimensions of time and space. While multivariate bilinear methods such as principal component analysis (PCA) have been used successfully for extracting information about spatial and temporal features in data from a single fMRI run, the need to unfold higher-order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data. These additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiolog-ically meaningful. Multiway analysis of fMRI data from multiple runs of a bilateral finger-tapping paradigm was performed using the parallel factor (PARAFAC) model. A trilinear model was fitted to a data cube of dimensions voxels by time by run. Similarly, a quadri-linear model was fitted to a higher-way structure of dimensions voxels by time by trial by run. The spatial and temporal response components were extracted and validated by comparison to results from traditional SVD/PCA analyses based on scenarios of unfolding into lower-order bilinear structures.
  
• Andersen, C. M. (2003).
  New aspects of chemometrics applied to spectroscopy - examples from research in fish production. Unpublished PhD thesis, The Royal Beterinary and Agricultural University, Stockholm.
  The overall purpuse of the present Ph.D. thesis is to show how new as well as known chemometric methods can be applied to spectroscopic data. The spectroscopic measurements are obtained as multivariate or multi-way data, which due to complexity of the measured samples can be rather complex. Therefore, chemometric methods are necessary in order to resolve and fully understand the underlying structure reflected in the measurements. Fluorescence spectroscopy and nuclear magnetic resonance (NMR) measured on fish muscles are used as example to illustrate the potential use of the methods on real products. Focus has been on three main subject, which are exemplified in the six included papers. The three subjects are:
  1) Exploratory analysis of spectroscopic data
  2) Validation of models
  3) Quantifying and handling errors in instrumental measurements
  Application of spectroscopic measurements makes it possible to measute the product non-destructively. However, such measurements can be rather complex due to the composition and heterogeneity of the food product. This is illustrated by exploratory analysis of fluorescence emissions of cod and salmon, fluorescence landscapes of fish muscle extracts and low-field NMR measured on gadoid fish muscle. Using principal component analysis (PCA) and parallel factor analysis (PARAFAC) to interpret the data, it is suggested that fluorescence spectra of fish muscle are primarily due to the connective tissue, NADH, ocidattion product, lipids, pigments and amino acids. Slicing modelling of low-field NMR decays shows the distribution of water within fish fillets and the division of water into different pools. The content of water in these pools is correated to the water holding capacity (WHC) of the fish muscle, which is an important quality parameter. Furthermore, the exploratory analysis shows how three-way data can be decomposed and interpreted. Problems involved in PARAFAC modelling of fluorescence excitation-emission data such as missing values and scatter are discussed and examples of how these can be handled are illustrated. Sampling is an important aspect to consider, since fish muscle as other biological material is heterogeneous. An example is given of how sampling errors in instrumental measurements can be quantified and handled when these measurements are used for estimating a reference value by univariate or multivariate regression. Furthermore, it is shown how traditional univariate statistics can be used in multivariate situations by calculating the net analyte signal. Validation is another important factor that should be considered in all data analysis. Validation is necessary to ensure valid and reliable modelling and is included as a part of the chemometric modelling in all the papers. In the thesis it is described how the chosen validation method depends on the purpose of the data analysis and the chemometric model. Overall, the project has illustrated that spectroscopic methods such as fluorescence and NMR in combination with chemometrics can provide information about underlying chemical and physical parameters in fish. The results are promising and give indications of considerable potential for the used method, which encourages further research and development.
  
• Andersen, C. M. & Bro, R. (2003)
  Practical aspects of PARAFAC modeling of fluorescence excitation-emission data Journal of Chemometrics, 17, 200-215.
  This paper presents a dedicated investigation and practical description of how to apply PARAFAC modeling to complicated fluorescence excitation-emission measurements. The steps involved in finding the optimal PARAFAC model are described in detail based on the characteristics of fluorescence data. These steps include choosing the right number of components, handling problems with missing values and scatter, detecting variables influenced by noise and identifying outliers. Various validation methods are applied in order to ensure that the optimal model has been found and several common data-specific problems and their solutions are explained. Finally, interpretations of the specific models are given. The paper can be used as a tutorial for investigating fluorescence landscapes with multi-way analysis.
  
• Anderson, C.J. (1996).
  The analysis of three-way contingency tables by three-mode association models. Psychometrika, 61, 465-483.
  The RC(M) association model (Goodman, 1979, 1985, 1991) is useful for analyzing the relationship between the variables of a 2-way cross-classification. The models presented here are generalizations of the RC(M) association model for 3-way tables. The family of models proposed here, "3-mode association" models, use Tucker's 3-mode components model (Tucker, 1964, 1966; Kroonenberg, 1983) to represent either the three factor interaction or the combined effects of two and three factor interactions. An example from a study in developmental psychology (Kramer & Gottman, 1992) is provided to illustrate the usefulness of the proposed models.
  
  
• Andersson, C.A. (1999).
  Multiway data in chemometrics. Exploratory data analysis in chemistry with soft multi-linear modelling. Bulletin of the International Statistical Institute, 58, 215-218.
  The most popular models for the analysis of multi-way data structures, i.e., the Tucker and CANDECOMP-PARAFAC (CP) models, were developed in the domain of numerical psychology. However, data collected by many modern analytical chemical instruments are multi-way data structures per definition, and many applications have shown that data analysis gains in robustness and information outcome by deriving meaningful parameters from appropriately chosen multi-linear models of such measurements. The current presentation covers the two basic models of widest general interest and two applications from the sugar industry. Both applications are based on fluorescence measurements of aqeous samples.
  
• Andersson, C.A. (2000).
  Exploratory multivariate data analysis with applications in food technology. Kopenhagen, Denmark: DSR Grafik.
  Section I: Exploratory multivariate data analysis
  Section II: Algorithms, models and applications: 1, direct orthogonalization; 2, A new criterion for simple structure transformations of core arrays in N-way PCA with application to fluorometric data; 3, A general algorithm for obtaining simple structure of core arrays in N-way PCA with application to fluorometric data; 4, Improving the speed of multiway algorithms. Part I: Tucker3; 5, Improving the speed of multiway algorithms. Part II: Compression; 6, Further improvements of the speed of the three-way Tucker3 algorithm; 7, Chemometrics in food science; 8, Multi-way chemometrics for mathematical separation of fluorescent colorants and colour precursors from spectrofluorimetry of beet sugar and beet sugar thick juice as validated by HPLC analysis; 9, Analysis of N-dimensional data arrays from fluorescence spectroscopy of an intermediate sugar product; 10, PARAFAC2- Part II. Modeling chromatographic data with retention time shifts.
  
• Andersson, C.A., & Bro, R. (1998).
  Improving the speed of multi-way algorithms: Part I. Tucker3. Chemometrics and Intelligent Laboratory Systems, 42, 93-103.
  In an attempt to improve the speed of multi-way algorithms, this paper investigates several different implementations of the Tucker3 algorithm. The interest is specifically aimed at developing a fast algorithm in the MATLAB (TM) environment that is suitable for large data arrays. Nine different implementations are developed and tested on real and simulated data. In a subsequent paper, it will be demonstrated that a fast algorithm for the Tucker3 model provides a perfect basis for improving the speed of other multi-way algorithms. From the Internet address http:\\www.models.kvl.dk\source\, the developed algorithms can be downloaded.
  
• Andersson, C. A., & Bro, R. (2000).
  The N-way Toolbox for MATLAB. Chemometrics and Intelligent Laboratory Systems, 52, 1-4.
  This communication describes a free toolbox for MATLABw for analysis of multiway data. The toolbox is called ‘‘The N-way Toolbox for MATLAB’’ and is available on the internet at http:rrwww.models.kvl.dkrsourcer. This communication is by no means an attempt to summarize or review the extensive work done in multiway data analysis but is intended solely for informing the reader of the existence, functionality, and applicability of the N-way Toolbox for MATLAB.
  
• Andersson, C.A., & Henrion, R. (1999).
  A general algorithm for obtaining simple structure of core arrays in N- way PCA with application to fluorometric data. Computational Statistics & Data Analysis, 31, 255-278.
  Simplifying the structure of core arrays from N-way PCA or Tucker3 models is desirable to allow for easy interpretation of the factor estimates. In the present paper, first a general algorithm for maximizing a differentiable goal function depending on a set of orthogonal matrices is formulated and then specified to the problem of estimating orthonormal transformation matrices for rotating core arrays to simpler structure. The generality of the chosen approach allows to cope with all possible transformation criteria by just changing one command in the implementation. In particular, the classical body-and slice-wise diagonalization of core arrays as well as the recently proposed maximization of the variance of squared entries are covered. The stability of the algorithm is addressed by a simulation study using 120 three-way core arrays of dimension (4,4,4). Each core array instantiates a class of 50 equivalent cores by random orthonormal transformations. Theoretically, each core within a given class has the same optimum with respect to the chosen criterion, and the ability of the algorithm to provide that result has been investigated. The algorithm proves to work with a high degree of stability and consistency in optimizing the three discussed goal functions. In addition, theoretical convergence results of the algorithm are provided. In particular, monotonic convergence of functional values and convergence of iterates towards a stationary solution an proven. To illustrate the effect of maximizing the variance-of-squares and the functionality of the algorithm, the proposed method is applied to a three-way data array from fluorometric analysis of fractions obtained from low-pressure chromatographic separation of a preliminary sugar product, thick juice. A significant gain in simplicity is achieved, and in particular optimizing variance-of-squares provides a simple core structure for the data under investigation. The proposed algorithms for maximizing variance-of-squares, body diagonality and slice-wise diagonality have been implemented in MATLAB and are available by contact to the authors.
  
• Andersson, C.A., Munck, L., Henrion, R., & Henrion, G. (1997).
  Analysis of N-dimensional data arrays from fluorescence spectroscopy of an intermediary sugar product. Fresenius Journal of Analytical Chemistry, 2, 2.
  Unwanted formation of colour takes place during the production of crystalline sugar. The degree of colouration depends partly on the necessary processing conditions, e.g. Heating and pH, and partly on the initial composition and condition of the sugar beets used as raw material. Reducing sugars are formed during the process. These are reactive compounds forming a variety of coloured complexes and strong precursors to further formation of colour and many of these compounds contain fluorophores. In the present work it is discussed if spectrofluorometric screening of intermediary sugar products prior to the final heating stages combined with a multiway chemometric approach can provide information that significantly reflects the condition of the process and the beets. The model used is the n-way pca (principal component analysis) which is an exploratory model, not necessitating explicit modelling of single parameters nor any assumptions towards parameter interaction. By use of a 4-way pca of order (3,2,3,3) satisfactory classification of 47 thick juice samples belonging to 5 factories has been obtained from a spectrofluorometric screening method. Also, a temporal trend has been found to evolve during the time of production. The investigation substantiates the use of modern models from data analysis for extracting significant information from large and complex data sets.
  
• Andersson, G.G., Dable, B.K. & Booksh, K.S. (1999).
  Weighted parallel factor analysis for calibration of HPLC-UV/Vis spectrometers in the presence of Beer's law deviations. Chemometrics and Intelligent Laboratory Systems, 49, 195-213.
  An extension of the parallel factor analysis (PARAFAC) methodology is presented to allow accurate and reliable quantitative and qualitative analysis of nonlinear data collected from hyphenated instrumentation. The weighted PARAFAC method is applied to high-performance liquid chromatography-ultraviolet/visible (HPLC-UV/Vis) diode array spectrometry analysis. It is demonstrated that this method improves the quantitative errors when spectroscopic nonlinearities from solvent-solute interactions or detector saturation are introduced. As much as 50% improvements in the root mean squared errors of estimation are realized for test samples. This weighted PARAFAC algorithm implicitly treats nonlinear data as missing values. A method requiring no a priori information is presented, that facilitates determination of the nonlinear regions and optimal application of the weighted parafac algorithm.
  
• Andersson, M., Josefson, M., Langkilde, F., W., & Wahlund, K. G. (1999).
  Monitoring of a film coating process for tablets using near infrared reflectance spectrometry. Journal of Pharmaceutical and Biomedical Analysis, 20, 27-37.
  A process analytical chemical method using near infrared diffuse reflectance spectrometry was developed for the determination of the amount of tablet coating on single tablets. This method is based on calibration of the spectra versus the added mass of coating solution. The tablet core was composed of two halves of different chemical composition and spectra were recorded from both sides of the tablets. The calibration was carried out using the chemometric methods principal component analysis (PCA), partial least squares (PLS), and multiplicative signal correction (MSC). The PLS-model utilised spectra obtained from both sides, pretreated with MSG, and ordered into one object. This method can be used in process analytical chemistry at-line. Additional characterisation of the measurements was obtained by calibrating the spectra versus coating thicknesses obtained from optical microscopy. Using PCA, it was possible to roughly estimate the maximum depth in the coating material that returns chemical information, the 'information depth', which was 0.1-0.2 mm.
  
• Andersson, M., & Knuuttila, K. G. (2002).
  The multivariate use of vibrational spectroscopy for chemical characterisation of chromotography media. Vibrational Spectroscopy, 29, 133-138.
  Chromatography media used in purification of biomolecules are extensively tested with various methods to ensure high quality products. The spherical and porous polymer particles are derivatised to give different chromatographic properties. The variations in, for instance, ligand content and chromatographic function are generally tested with a number of conventional analytical methods (mainly wet-chemical and functional methods) and these are time consuming. However, by using vibrational spectroscopy, chemical information can be efficiently obtained from the spectral data. In combination with traditional analytical data, multivariate models can be constructed. These models can then be used as a tool for determination (prediction) of, for example, ligand content. In the present work, prediction of ionic capacity and content of allyl groups are exemplified by Raman spectroscopy of two different types of agarose-based media (Sepharose(TM) Fast Flow Sepharose(TM)). The conclusion is that vibrational spectroscopy is a simple, fast and highly informative tool for both qualitative and quantitative characterisation of adsorbents used in chromatography of biomolecules.
  
• Andrews, D. T., & Wentzell, P. D. (1997).
  Applications of maximum likelihood principal component analysis: incomplete data sets and calibration transfer. Analytica Chimica Acta, 350, 341-352.
  The application of a new method to the multivariate analysis of incomplete data sets is described. The new method, called maximum likelihood principal component analysis (MLPCA), is analogous to conventional principal component analysis (PCA), but incorporates measurement error variance information in the decomposition of multivariate data. Missing measurements can be handled in a reliable and simple manner by assigning large measurement uncertainties to them. The problem of missing data is pervasive in chemistry, and MLPCA is applied to three sets of experimental data to illustrate its utility. For exploratory data analysis, a data set from the analysis of archeological artifacts is used to show that the principal components extracted by MLPCA retain much of the original information even when a significant number of measurements are missing. Maximum likelihood projections of censored data can often preserve original clusters among the samples and can, through the propagation of error, indicate which samples are likely to be projected erroneously. To demonstrate its utility in modeling applications, MLPCA is also applied in the development of a model for chromatographic retention based on a data set which is only 80% complete. MLPCA can predict missing values and assign error estimates to these points. Finally, the problem of calibration transfer between instruments can be regarded as a missing data problem in which entire spectra are missing on the 'slave' instrument. Using NIR spectra obtained from two instruments, it is shown that spectra on the slave instrument can be predicted from a small subset of calibration transfer samples even if a different wavelength range is employed. Concentration prediction errors obtained by this approach were comparable to cross-validation errors obtained for the slave instrument when all spectra were available.
  
• Antunes, A. M., Ferreira, M. M. C., Melgo, M. S., & Volpe, P. L. O. (1999).
  Potassium transport through liquid membranes using spectral and chemometrics methods. Journal of Molecular Structure, 481, 563-567.
  Chemometric techniques were applied to follow the transport of potassium through a chloroform membrane, using ternary mixtures of the isomeric anions, 2,4-dinitrophenolate (2,4-dnp) and 2,5-dinitrophenolate (2,5-dnp) with also 2-nitrophenolate (ortho-). Some spectral differences can be observed in the uv-vis region, although 2,4- and 2,5-dnp have very similar spectra. Using the trilinear decomposition method (tld), it is possible to estimate the spectra of the pure components and simultaneously their kinetic profiles. The estimated spectral profiles are in excellent agreement with experimental results and the kinetic profiles agree with those obtained by ordinary least squares (ols). The transport rates are calculated and compared. The ortho- isomer has a very small rate of transport.
     
• Appellof, C.J., & Davidson, E.R. (1981).
  Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents. Analytical Chemistry, 53, 2053-2056.
  In this paper, the Tucker method is used for qualitative analysis of a multicomponent fluorescent mixture.
  
• Appellof, C.J., & Davidson, E.R. (1983).
  Three-dimensional rank annihilation for multi-component determinations. Analytica Chimica Acta, 146, 9-14.
  The method of rank annihilation for multi-component determinations is extended to a three-dimensional data array. The possibility of improved sensitivity over the two-dimensional method is shown. An illustration using data of the type expected from a liquid chromatograph with a video-fluorimeter as detector is presented.
  
     
• Arabie, P., & Boorman, S.A. (1973).
  Multidimensional scaling of measures of distance between partitions. Journal of Mathematical Psychology, 10, 148-203.
  The techniques of multidimensional scaling were used to study the numerical behavior of twelve measures of distance between partition lattices of four different sizes. The results offer additional support for a system of classifying partition metrics, as proposed by Boorman (1970) and Boorman and Arabie (1972). While the scaling solutions illuminated differences between the measures, at the same time the particular data with wich the measures were concerned offered a basis both for counterexamples to some common assumptions about multidimensional scaling and for some conjectures as to the nature of scaling solutions. The implications of the latter findings for selected examples from the literature are considered. In addition, the methods of partition data analysis discussed here are applied to an example using sociobiological data. Finally, an argument is amde against undue emphasis upon interpreting dimensions in nonmetric scaling solutions.
  
• Arabie, P., & Carroll, J.D. (1980b).
  MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika, 45, 211-235.
  We present a new algorithm, MAPCLUS (MAthematical Programming CLUStering), for fitting the Shepard-Arabie ADCLUS (for ADditive CLUStering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS (INdividual Differences CLUStering).
  
• Arabie, P., Carroll, J.D., DeSarbo, W.S., & Wind, J. (1981).
  Overlapping clustering: A new method for product positioning. Journal of Marketing Research, 18, 310-317.
  Most clustering techniques used in product positioning and market segmention studies render mutually exclusive equivalence classes of the relevant products or subject space. Such classificatory techniques are thus restricted to the extent that they preclude overlap between subsets or equivalence classes. An overlapping clustering model, ADCLUS, is described which can be used in marketing studies involving products/subjects that can belong to more than one group or cluster simultaneously. The authors provide theoretical justification for and an application of the approach, using the MAPCLUS algorithm for fitting the ADCLUS model.
  
• Arabie, P., & Maschmeyer, C. (1988).
  Some current models for the perception and judgment of risk. Organizational Behavior and Human Decision Processes, 41, 300-329.
  We survey some of the models more recently used to portray panelists' perceptions of risk, viewed as a complex psychological response. These models are compared (a) as continuous versus discrete, (b) with regard to type of data and tasks required of panelists, and (c) by facility for portraying different patterns of judgments among panelists. Substantive results from applying different models to the same risks and/or data are presented. Finally, we consider possible future directions for research in the perception of risk, oriented toward use of the models presented here.
  
  
• Arabie, P., Mashmeyer, C.J., & Carroll, J.D. (1987).
  Impact scaling: Method and application. Technological Forecasting and Social Change, 32, 245-272.
  Impact scaling is a method of forecasting, using expert panelists' judgment of conditional and unconditional probabilities of selected possible future events, as well as the events' likely effects on such indicators as business parameters. Given the recent availability of a discrete psychological model (INDCLUS) for representing structure of the events as well as differences among the panelists, a more elaborate and versatile set of models for implementing impact scaling is demonstrated. An illustration is provided using panelists of varied professional status to give judgments about impacts of possible future events on various indices of stock market performance. Finally, further extensions of impact scaling to enhance the utility and precision as a model of judgment and method of forecasting is considered.
  
• Arancibia, J. A., Olivieri, A. C., & Escandar, G. M.(2002).
  First- and second-order multivariate calibration applied to biological samples: determination of anti-inflammatories in serum and urine. Analytical and Bioanalytical Chemistry, 374, 451-459.
  Two different spectrofluorimetric methods for the determination of piroxicam (PX) in serum are presented and discussed. One of them is based on the use of three-way fluorescence data and multivariate calibration performed with parallel factor analysis (PARAFAC) and self-weighted alternating trilinear decomposition (SWATLD). This methodology exploits the so-called second-order advantage of the three-way data, allowing to obtain the concentration of the studied analyte in the presence of any number of uncalibrated (serum) components. The method was developed following two different procedures: internal standard addition and external calibration with standard solutions, which were compared and discussed. The second approach investigated is based on the combination of solid-phase extraction (SPE) and room temperature fluorimetry. Both methods here presented yield satisfactory results. The concentration range in which PX could be determined in serum was 1-10 mug ml(-1). The limits of quantification for the experimental solutions using the chemometric approach were 0.09 mug ml(-1) for the standard addition mode and 0.12 mug ml(-1) using external calibration (both for PARAFAC and SWATLD algorithms). In the solid-surface fluorimetric method, the calibration graph was linear up to 0.22 mug ml(-1) and the limit of quantification was 0.02 mug ml(-1).
  
• Arancibia, J. A., & Escandar, G. M.(2003).
  Two different strategies for the fluorimetric determination of piroxicam in serum. Talanta, 60, 1113-1121.
  First- and second-order multivariate calibration of fluorescence data have been compared as regards the determination of anti-inflammatories and metabolites in the biological fluids serum and urine. The simultaneous resolution of naproxen-salicylic acid mixtures in serum and naproxen-salicylic acid-salicyluric acid mixtures in urine was accomplished and employed for a discussion of the relative advantages of the applied chemometric tools. The analysis of second-order fluorescence excitation-emission matrices was performed using iteratively reweighted generalized rank annihilation method (IRGRAM), parallel factor analysis (PARAFAC), and self-weighted alternating trilinear decomposition (SWATLD). The results were compared with first-order fluorescence emission data analyzed with partial least-squares regression (PLS). In all cases, the performance of the methods was improved through the formation of inclusion complexes of the analytes with beta-cyclodextrin. The concentration ranges in which the analytes could be determined were as follows: naproxen, 0-250 ng mL(-1) in serum and 0-200 ng mL(-1) in urine; salicylic acid, 0-500 ng mL(-1) in serum and 0-300 ng mL(-1) in urine, and salicyluric acid, 0-300 ng mL(-1) in urine.
  
• Arcelay, A.R., Ross, R.T., & Ezzeddine, B.M. (1988).
  Photosystem I generates a free-energy change of 0.7 electron volts or less. Biochimica et Biophysica Acta - Bioenergetics, 936, 199-207.
  The intensity of delayed luminescence from wild-type Scenedesmus obliquus was used to refine a previous determination (Marchiarullo, M.A. and Ross, R.T. (1981) Biochim. Biophys. Acta 636, 254-257) of the free-energy change in Photosystem II, which we find to be 0,99 eV under physiological conditions. Similar measurements were made of the very weak luminescence from a mutant deficient in plastoquinone A, originally with the expectation that this luminescence would be from Photosystem I. However, the action spectrum for excitation of mutant luminescence is that of Photosystem II. There is no measurable change in the properties of the mutant luminescence with changes in excitation wavelength extending to 708 nm, leading us to conclude that Photosystem I contributes less that 5% of the emission 10ms after excitation. This limit on the intensity of emision from Photosystem I places an upper limit of 0.69 eV on its free- energy difference.
  
• Artyushkova, K., & Fulghum, J. E.(2001).
  Identification of chemical components in XPS spectra and images using multivariate statistical analysis methods. Jounral of Electron Spectroscopy and Related Phenomena, 121, 33-55.
  A variety of data analysis methods can be used to enhance the information obtained from a measurement, or to simplify extraction of significant components from large data sets. Much work is needed to improve the quantification and interpretation of XPS spectra and images from complex organics. Multivariate analysis (MVA) is increasingly used for applications in electron spectroscopy to aid the analyst in interpreting the vast amount of information yielded by spectroscopic techniques. In general, the goals of MVA are to determine the number of components present, identify the chemical components, and quantify component concentrations in the mixture. Principal component analysis (PCA) is frequently used to determine the number of mathematical components which describe the data set. These mathematical components must then be related to chemically meaningful components. Various approaches to solve rotational ambiguities of spectral resolution, including local rank method (EFA), pure variables method (Simplisma) and multivariate curve resolution (MCR), are tested in the determination of chemical components from XPS data. Limitations associated with the resolution of a single matrix are shown to be partially or completely overcome when several related matrices are treated together. The test data sets contain XPS images or spectra acquired from blends of poly(vinyl chloride), PVC, and poly(methyl methacrylate), PMMA. The PVC degrades rapidly upon exposure to the X-ray beam. Spectra and images from the blend, acquired as a function of time, provide the multi-dimensional data sets for algorithm evaluation. In addition to spectral resolution, multivariate image analysis methods, such as principal component analysis, are used to extract maps of the pure components from an images-to-spectra data set.
  
• Artyushkova, K., & Fulghum, J. E.(2002).
  Multivariate image analysis methods applied to XPS imaging data sets. Surface and Interface Analysis, 33,185-195.
  Recent improvements in imaging photoelectron spectroscopy enhance lateral and vertical characterization of heterogeneous samples at the cost of increasing complexity in the XPS data sets acquired. These imaging data sets require more sophisticated analysis methods than visual inspection if the data are to be interpreted effectively. Multivariate analysis (MVA) methods are increasingly utilized in surface spectroscopies to aid the analyst in interpreting the vast amount of information resulting from these multidimensional data set acquisitions. In this work, image processing analysis methods are tested on XPS data sets acquired from polymer blends. Images from the blends, acquired as a function of composition, time or energy, provide multidimensional data sets for algorithm evaluation. Multivariate image analysis (MIA) methods such as scatter diagrams, principal component analysis (PCA) and classification methods are used to extract maps of pure components from degradation and images-to-spectra data sets. In some cases the MVA results can be compared directly with the XPS spectra or images, which provide a critical reference point. This work will demonstrate that additional information can result from the application of MIA methods, even when direct spectral or image interpretation is possible.
  
• Aruga, R., Gastaldi, D., Negro, G., & Ostacoli, G. (1995).
  Pollution of a river basin and its evolution with time studied by multivariate statistical analysis. Analytica Chimica Acta, 310, 15-25.
  A set of quantitative analytical data for 13 rivers belonging to the basin of the Tanaro (Piedmont region, north-western Italy) has been processed by multivariate statistical techniques. The original matrix, which refers to the year 1990, consisted of 20 chemical and physico-chemical variables, determined at 44 sampling sites. The following methods have been used for the treatment of the data: Cluster analysis (unsupervised pattern recognition), factor analysis, variable selection by fisher weights. A comparison has also been made between the data referring to 1990 and to 1978 in order to investigate the evolution of the environmental situation for the Tanaro basin after such a lapse of time. This comparison has been carried out by processing the 3-d matrix which collects the data of the two different periods.
     
• Atkinson, A.C., & Riani, M.(2004).
  The forward search and data visualisation. Computational Statistics, 19, 29-54.
  A statistical analysis using the forward search produces many graphs. For multivariate data an appreciable proportion of these are a variety of plots of the Mahalanobis distances of the individual observations during the search. Each unit, originally a point in v-dimensional space, is then represented by a curve in two dimensions connecting the almost n values of the distance for each unit calculated during the search. Our task is now to recognise and classify these curves: we may find several clusters of data, or outliers or some unexpected, non-normal, structure. We look at the plots from five data sets. Statistical techniques include cluster analysis and transformations to multivariate normality.
             
• Azubel, M., Fernández, F.M., Tudino, M.B. & Troccoli, O.E. (1999).
  Novel application and comparison of multivariate calibration for the simultaneous determination of Cu, Zn and Mn at trace levels using flow injection diode array spectrophotometry. Analytica Chimica Acta, 398, 93-102.
  Three different calibration approaches: Partial Least Squares, Unfold Partial Least Squares and n-way partial least squares are compared in terms of explained variance, root mean square error of calibration and root mean square error of cross-validation. Attention is also focused on the application of genetic algorithms to spectral data as a way to obtain an improvement in calibration accuracy. Influence of initial starting conditions for the genetic algorithms (population size, mutation probability, % initial terms etc.) was investigated by means of factorial experimental designs.
  

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