Three-Mode Abstracts, Part D

INDEX

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• Dahl, K.S., Piovoso, M.J. & Kosanovich, K.A. (1999).
  Translating third-order data analysis methods to chemical batch processes. Chemometrics and Intelligent Laboratory Systems, 46, 161- 180.
  Measurements collected from batch processes naturally produce a third-order or three-dimensional data form. The same structure also results when multiple samples are measured using hyphenated analysis techniques such as liquid chromatography with diode array detection. Analysis of third-order data by principal components analysis (PCA) is achieved by a nonunique rearrangement that produces a two-dimensional array. This preferentially models only one of the three orders present. In contrast, methods such as parallel factor analysis (PARAFAC) apply a particular decomposition that accounts for all three orders explicitly. The results from either approach should be related if data are to be interpreted reliably for applications to batch processes such as on-line monitoring and control. This work compares these two approaches from an applied point of view. To accomplish this objective, exemplary methods are selected from each type of analysis, parallel factor analysis (PARAFAC) and multiway principal components analysis (MPCA). These are employed to analyze data obtained during the manufacture of a condensation polymer in an industrial batch reactor.

• Dahl, T., & Naes, T. (2004).
  Outlier and group detection in sensory panels using hierarchical cluster analysis with the Procrustes distance. Food Quality and Preference, 15, 195-208.
  Generalised Procrustes analysis (GPA) is a much-used method for analysing sensory profile data. In this paper, hierarchical clustering using the Procrustes distance is proposed for situations where the data profiles are believed to come from a non-homogeneous group. This new approach to sensory panel analysis may be used at an exploratory stage, in combination with GPA, to gain insight into the structures of the data set. It can help the researcher detect outliers and subgroups, help him/her make decisions regarding further analysis, and reduce the risk of erroneous inference about the data.

• Dai, K. (1982).
  Application of the three-mode factor analysis to industrial design of chair styles. Japanese Psychological Review, 25,91-103. This paper presents a modified vrsion of Tucker's (1966) method, which was applied to a practical problem of chair style image. One hundred and fifty subjects rated photographs of twenty styles of chairs in market, with 28 scales.

  

• Dai, K. (1985).
  Application of a three-mode factor analysis to brand images of whisky. Reports of Statistical Application Research, 32, 11-22.
  New interpretation and calculation methods of Tucker's three-mode factor analysis are proposed. The most marked modification is the option, which mode is the theme of factor analysis and which is not, depending on the purpose of the survey. The score matrix, in which one mode is summarized into factors, is defined as "one-mode factor score matrix". Two-mode factor score and three-mode factor score are also defined. This generalized three-mode factor analysis was applied to a survey, subjects numbering 2762, on brand images of whisky. An example of an interpretation for one-mode factor score and two-mode factor score was presented.

• Damiani, P. C., Nepote, A. J., Bearzotti, M., & Olivieri, A. C. (2004).
  A test field for the second-order advantage in bilinear least-squares and parallel factor analyses: Fluorescence determination of ciprofloxacin in human urine. Analytical Chemistry, 76, 2798-2806.
  The analytical performances of two algorithms, the recently introduced bilinear least-squares (BLLS) and the popular parallel factor analysis (PARAFAC), are compared as regards second-order fluorescence data recorded for the determination of the fluoroquinolone antibiotic ciprofloxacin in human urine samples. The applied chemometric methodologies employ different strategies for exploiting the so-called second-order advantage, which allows one to obtain individual concentrations of calibrated analytes in the presence of any number of uncalibrated (urine) components. Analysis of a spiked urine test set (in the analyte concentration range 0-200 mg L-1) showed that BLLS provides results of slightly better quality than PARAFAC. Satisfactory results have been obtained on comparing the concentrations predicted for a series of real urine samples with those furnished by liquid chromatography. The limit of detection of the fluorescence-based methods is similar to5 mg L-1.

• D'Ambra, L. (1989). Least squares criterion for asymmetric dependence models in three-way contingency tables. In E. Diday (Ed.), Data analysis, learning symbolic and numeric knowledge (pp. 15-20). New York: Nova Science.
  This paper deals with logbilinear and logtrilinear models proposed by Goodman for three-way contingency tables. The proposed approach allows to study the dependence or association structure between two or three qualitative varibales. The parameters of the models are estimated by means of the least squares methods.

• D'Ambra, L., & Kiers, H. (1990).
  Analysis of log-trilinear models for a three-way contingency table using PARAFAC/CANDECOMP. In the Proceedings of the Study Days of the Italian Statistical Society, 101-114, Pescara, 11-12 October.
  This paper deals with log-trilinear models derived from the models proposed by Goodman (1986) for three-way contingency tables. The proposed approach allows to study the association structure between three qualitative variables. The parameters of the models are estimated by means of PARAFAC/CANDECOMP and the results are compared to those of a generalized singular value decomposition approach.

  

• D'Ambra, L., & Lauro, N. (1989).
  Non symmetrical analysis of three-way contingency tables. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 301-315). Amsterdam: Elsevier.
  This paper deals with a non symmetrical analysis of three-way contingency tables in order to study the structure of dependence among qualitative variables that play a different role in the analysis. The proposed approach allows the decomposition of non symmetric association indexes such as the Goodman-Kruskal tau and its extension dues to Gray-Williams.

• Da Silva, J. C. G. E., & Novais, S. A. G. (1998).
  Trilinear PARAFAC decomposition of synchronous fluorescence spectra of mixtures of the major metabolites of acetylsalicylic acid. Analyst, 123, 2067-2070.
  Mixtures of the three major metabolites of acetylsalicylic acid (salicylic, gentisic and salicyluric acid) were analyzed by synchronous molecular fluorescence spectroscopy. Three-way data matrices were generated by acquisition of spectra as a function bf the pH (between 2 and 11) and of different relative concentrations of the three components. The PARAFAC trilinear model, without restrictions and using one factor per metabolite, was used in the data analysis. A full decomposition of the data matrices into the spectra, concentration and pH profiles was obtained. This result shows that molecular fluorescence spectroscopy can be used for the development of robust analytical methods for th simultaneous determination of the three major metabolites of acetylsalicylic acid in complex background samples.

• Da Silva, J. C. G. E., & Oliveira, C. J. S. (1999).
  Parafac decomposition of three-way kinetic-spectrophotometric spectral matrices corresponding to mixtures of heavy metal ions. Talanta, 49, 889-897.
  Binary and ternary mixtures of some of the following heavy metal ions Zn(II), NI(II), Pb(II), Co(II) and Cd(II) were analyzed by a ligand substitution kinetic method. Three-way data matrices were generated by acquisition of W-Vis spectra (332-580 nm) as a function of the time of a substitution reaction observed between the complex of the heavy metal ions with the non selective metallochromic indicator 4-(2-pyridylazo) resorcinol (PAR) and EDTA, and of different relative concentration of the metal ions (1-6 mM). The PARAFAC trilinear model, without restrictions, was used in the data analysis. A full decomposition of the data matrices was obtained (spectra, concentration and time profiles). It was shown that ligand substitution kinetic methods coupled to three-way chemometric analytical methods can be used for the development of robust sensors for the analysis of binary [Zn(II) + Ni(II), Pb(II) + Cd(II), Zn(II) + Pb(II)] or ternary [Zn(II) + Pb(II) + Co(II)] mixtures of metal ions in the micromolar concentration range.

• Da Silva, J. C. G. E., Leitao, J. M. M., Costa, F. S. & Ribeiro, J. L. A. (2002).
  Detection of verapamil drug by fluorescence and trilinear decompositim techniques Analytica Chimica Acta, 453, 105-115.
  A three-way analytical methodology experimentally based on fluorescence excitation emission matrix (EEM) and in PARAFAC and TLD chemometric analysis was assessed for the quantification of verapamil drug in a tablet formulation. A standard addition procedure generates experimental information compatible with the chemometric data analysis model allowing the estimation of verapamil with a detection limit of about 0.04 mg/l using methanol as solvent. The structure of the verapamil EEM follows a trilinear model, but background signals (first- and second-order scatter bands) did not-a trilinear three-factor model is necessary to describe experimental datasets. The comparison of a three-factor PARAFAC model with a United States Pharmacopoeia (USP) standard chromatographic method showed similar results. (C) 2002 Elsevier Science B.V. All rights reserved.

• D'Aubigny, G., & Polit, E. (1989).
  Some optimality properties of the generalization of the Tucker method to the analysis of n-way tables with specified metrics. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 39-52). Amsterdam: Elsevier.
  A tensor space formulation of the TUCKER-3 method results in the extension of this approach to n-way n-mode data sets. The introduction of non identity metrics attached to each mode yields a statistical modelling enrichment of this multivariate data analysis method. The evaluation of extended forms of the usual optimality criteria, worked out by OKAMOTO (1969) in the 2-way PCA case, is presented in the general setting of the n-PCA of the tensor structure (X,Q) of order n, where X is derived from the observation and Q defines a Euclidean geometry on the representation space. The lack of pairwise duality relations between modes that characterizes the general (i.e. n>2) situation is shown to be responsible for the falsity of the extension of several optimality properties in that case, while the introduction of non identity metrics is not influential. The representation of a tensor of order n by one matrix is a combinatorial problem. So it is of interest that representations induced by a bipartition of the modes cross-tabulating one mode with any combination of the others result in an intermediate situation, preserving some optimality properties of two-way PCA.

• Davis, E.E. & Grobstein, N.N. (1966).
  Multimode factor analysis of interpersonal perceptions. (Technical Report No. 36), Urbana-Champaign: University of Illinois, Department of Psychology.
  88 white/black, male/female students scored 28 complex person stimuli designated mainly in terms of race, sex, religion and occupation on 15 behavioral differential scales. The T3 solution was varimaxed for scales and students. The stimuli components were rotated to agree closely with the discriminant functions, emerging from an analysis on the components. The counter-rotated core matrix was interpreted but did not provide obvious subject component interpretations.

• Davis, J. W. & Gao, H. (2003).
  An expressive three-mode principal components model of human action style Image and Vision Computing, 21, 1001-1016. We present a three-mode expressive-feature model for representing and recognizing performance styles of human actions. A set of style variations for an action are initially arranged into a three-mode data representation (body pose, time, style) and factored into its three-mode principal components to reduce the data dimensionality. We next embed tunable weights on trajectories within the sub-space model to enable different context-based style estimations. We outline physical and perceptual parameterization methods for choosing style labels for the training data, from which we automatically learn the necessary expressive weights using a gradient descent procedure. Experiments are presented examining several motion-capture walking variations corresponding to carrying load, gender, and pace. Results demonstrate a greater flexibility of the expressive three-mode model, over standard squared-error style estimation, to adapt to different style matching criteria.

• Davis, J. W. & Gao, H. (2004).
  An expressive three-mode principal components model for gender recognition. Journal of Vision, 4, 363-377.
  We present a three-mode expressive-feature model for recognizing gender (female, male) from point-light displays of walking people. Prototype female and male walkers are initially decomposed into a subspace of their three-mode components (posture, time, and gender). We then apply a weight factor to each point-light trajectory in the basis representation to enable adaptive, context-based gender estimations. The weight values are automatically learned from labeled training data. We present experiments using physical (actual) and perceived (from perceptual experiments) gender labels to train and test the system. Results with 40 walkers demonstrate greater than 90% recognition for both physically and perceptually labeled training examples. The approach has a greater flexibility over standard squared-error gender estimation to successfully adapt to different matching contexts.

• Daws, J.T. (1996).
  The analysis of free-sorting data: Beyond pairwise cooccurrences. Journal of Classification, 13, 57-80.
  Free-sorting data are obtained when subjects are given a set of objects and are asked to divide them into subsets. Such data are usually reduced by counting, for each pair of objects, how many subjects placed both of them into the same subset. The present study examines the utility of a group of additional statistics, the cooccurrences of sets of three objects. Because there are dependencies among the pair and triple cooccurrences, adjusted triple similarity statistics are developed. Multidimensional scaling and cluster analysis - which usually use pair similarities as their input data - can be modified to operate on three-way similarities to create representations of the set of objects. Such methods are applied to a set of empirical sorting data: Rosenberg and Kim's (1975) fifteen kinship terms.

  

• Dawson, M.R.W., & Harshman, R.A. (1986).
  The multidimensional analysis of asymmetries in alphabetic confusion matrices: Evidence for global-to-local and local-to-global processing. Perception & Psychophysics, 40, 370-383.
  This study examined the ability of an asymmetric multidimensional scaling program (DEDICOM) to reveal information about letter-perception processes. To demonstrate its potential, we applied it to the controversy concerning local-to-global versus global-to-local letter perception. These two theories lead to different predictions about stimulus confusion asymmetries. Since DEDICOM is capable of recovering the structure of asymmetric or directional patterns, it should reveal whether a stimulus-response confusion matrix contains patterns of asymmetry more consistent with one or the other perceptual theory. This was tested using two data sets. The first (from Lupker, 1979) revealed an additive hierarchy of asymmetry strongly consistent with global-to-local processing, although unexpected additional structure and reliable anomalies indicated the need for a more refined theoretical account. The second (a full alphabetic confusion matrix combining data from Gilmore et al., 1979; Loomis, 1982; and Townsend, 1971) revealed five distinct patterns, each consisting of transformations attributable to the failure to detect specific local letter features. This solution strengthened support for local-to-global processing, in sharp contrast to the first analysis. Possible reasons for this divergence are discussed, including differences in the stimuli, exposure durations, and a hypothetical two-stage process of perception. Despite their differences, both solutions demonstrated how asymmetric scaling can reveal structure in asymmetries, which is relevant to perceptual theory and which would have been difficult to recover by other means.

  
         

• De Belie, N., Sivertsvik, M., & De Baerdemaeker, J. (2003).
  Differences in chewing sounds of dry-crisp snacks by multivariate data analysis. Journal of Sound and Vribration, 266, 625-643.
  Chewing sounds of different types of dry-crisp snacks (two types of potato chips, prawn crackers, cornflakes and low calorie snacks from extruded starch) were analysed to assess differences in sound emission patterns. The emitted sounds were recorded by a microphone placed over the ear canal. The first bite and the first subsequent chew were selected from the time signal and a fast Fourier transformation provided the power spectra. Different multivariate analysis techniques were used for classification of the snack groups. This included principal component analysis (PCA) and unfold partial least-squares (PLS) algorithms, as well as multi-way techniques such as three-way PLS, three-way PCA (Tucker3), and parallel factor analysis (PARAFAC) on the first bite and subsequent chew. The models were evaluated by calculating the classification errors and the root mean square error of prediction (RMSEP) for independent validation sets. It appeared that the logarithm of the power spectra obtained from the chewing sounds could be used successfully to distinguish the different snack groups. When different chewers were used, recalibration of the models was necessary. Multi-way models distinguished better between chewing sounds of different snack groups than PCA on bite or chew separately and than unfold PLS. From all three-way models applied, N-PLS with three components showed the best classification capabilities, resulting in classification errors of 14-18%. The major amount of incorrect classifications was due to one type of potato chips that had a very irregular shape, resulting in a wide variation of the emitted sounds.

• Defays, D. (1979).
  Tree representation of ternary relations. Journal of Mathematical Psychology, 19, 208-218.
  In this paper, trees are constructed from ternary relations; the model represents each object of an empirical set by a node of a tree so that a betweenness relation among the nodes (the node b is on the path from the node a to the node c) in the graph reflects a ternary relation among the objects. Various systems of formal properties that lead to three different kinds of tree representation are investigated.

• De Jong, S. (1998).
  Short communication regression coefficients in multipilinear PLS. Journal of Chemometrics, 12, 77-81.
  Three alternative approaches are discussed for finding the final calibration model (regression coefficients) in PLS regression of k-way Y on N-way X. The simplest approach is to skip the deflation of the X-data. From the observation that the specific deflation used in multiway PLS is inconsequential, it also follows that Bro's tri-PLS is equivalent to Ståhle's linear three-way decomposition (LTD).

  
  

• De Jong, S., & Farebrother, R. W. (1994).
  Extending the relationship between ridge-regression and continuum regression. Chemometrics and Intelligent Laboratory Systems, 25, 179-181.
  Recently, a close relationship has been established between ridge regression (RR) and a special case of continuum regression (CR). Attention was restricted to the usual positive range of values for the ridge parameter. This restriction identifies the trajectory lying between ordinary least squares (OLS) and partial least squares (PLS) regressions, leaving the trajectory between PLS and principal component regression (PCR) untouched. In this note we demonstrate that the relationship between CR and RR can be extended to the full range of methods, OLS <-> PLS <-> PCR, identified by the CR technique. For this purpose one has to admit a nonstandard variant of the RR technique in which the ridge parameter becomes negative.

  
  

• De Juan, A., & Rutan, S. C., & Tauler, R.,   Massart, D. L. (1998).
  Comparison between the direct trilinear decomposition and the multivariate curve resolution-alternating least squares methods for the resolution of three-way data sets. Chemometrics and Intelligent Laboratory Systems, 40, 19-32. Direct trilinear decomposition (DTD) and multivariate curve resolution-alternating least squares (MCR-ALS) methods are two of the most representative three-way resolution procedures. The former, non-iterative, is based on the resolution of the generalized eigenvector/eigenvalue problem and the latter, iterative, is focused on the optimization of initial estimates by using data structure and chemical constraints. DTD and MCR-ALS have been tested on a variety of three-way simulated data sets having common sources of variation in real response profiles, such as signal shift, broadening or shape distortions caused by noise. The effect of these factors on the resolution results has been evaluated through the analysis of several parameters related to the recovery of both qualitative and quantitative information and to the quality of the overall data description. Conclusions inferred from the simulated examples help to clarify the performance of both methods on a real example and to provide some general guidelines to understand better the potential of each method.

• De Juan, A., & Tauler, R. (2001).
  Comparison of 3-way resolution methods for non-trilinear data sets. Journal of Chemometrics, 15, 749-772.
  Resolution of three-way chemical data sets can be tackled using two families of chemometric methods: those assuming trilinear structure in the data set, such as direct trilinear decomposition (DTD) or parallel factor analysis (PARAFAC); and those which decompose the three-way data set according to a model lacking this structure, such as TUCKER3 or multivariate curve resolution-alternating least squares (MCR-ALS). The first group of methods provides unique solutions, whereas the second group gives solutions subject to rotational ambiguities. DTD and PARAFAC are thus the choice to deal with chemical data sets with trilinear structure. However, in the analysis of chemical data with non-trilinear structure, which is most commonly found in practice, the more ambiguous solutions given by TUCKER3 and MCR-ALS could be balanced by the major flexibility in the modelling of profiles. To assess this possibility, three-way resolution methods from the two mentioned families are applied to simulated and real data sets designed to show typical non-trilinear chemical situations, caused by shifts and shape changes in profiles.

• De Juan, A., & Tauler, R. (2003).
  Chemometrics applied to unravel multicomponent processes and mixtures - Revisiting latest trends in multivariate resolution. Analytica Chimica Acta, 500, 195-210.
  Progress in the analysis of multicomponent processes and mixtures relies on the combination of sophisticated instrumental techniques and suitable data analysis tools focused on the interpretation of the multivariate responses obtained. Despite the differences in compositional variation, complexity and origin, the raw measurements recorded in a multicomponent chemical system can be very often described with a simple model consisting of the composition-weighted sum of the signals of their pure compounds.Multivariate resolution methods have been the tools designed to unravel this pure compound information from the non-selective mixed original experimental output. The evolution of these chemometric approaches through the improvement of exploratory tools, the adaptation to work with complex data structures, the ability to introduce chemical and mathematical information in the algorithms and the better quality assessment of the results obtained is revisited. The active research on these chemometric area has allowed the successful application of these methodologies to chemical problems as complex and diverse as the interpretation of protein folding processes or the resolution of spectroscopic images.

• De Juan, A., Tauler, R., Dyson, R., Marcolli, C., Rault, M. & Maeder, M. (2004).
  Spectroscopic imaging and chemometrics: a powerful combination for global and local sample analysis. Tra-trends in Analytical Chemistry, 23, 70-79.
  Merging spectroscopic imaging and chemometrics enhances the outcomes of instrumental technology and data analysis. Multivariate exploratory and resolution methods can be adapted to image analysis and provide global and local information about pure compounds in an imaged sample. Knowing in detail how the chemical compounds are distributed over the scanned surface gives valuable information about essential issues in the manufacture and the characterization of products, such as evenness of composition and, therefore, homogeneity of the sample. The power to detect and to locate impurities is also greatly enhanced because these unwanted compounds could show locally large concentrations (and signals), even though their abundance on the surface is very low. The capabilities of this combination are shown in an example of pharmaceutical product control, where analysis of the end product requires chemical characterization and quantitative information at global and local levels. The approach used and the kind of information obtained is general and can be applied to the analysis of images in other fields.

• De Lathauwer, L.(2004a).
  First-order perturbation analysis of the best rank-(R1, R2, R3) approximation in multilinear algebra. Journal of Chemometrics, 18, 2-11.
  In this paper we perform a first-order perturbation analysis of the least squares approximation of a given higher-order tensor by a tensor having prespecified n-mode ranks. This work generalizes the classical first-order perturbation analysis of the matrix singular value decomposition. We will show that there are important differences between the matrix and the higher-order tensor case. We subsequently address (1) the best rank-1 approximation of supersymmetric tensors, (2) the best rank-(R1, R2, R3) approximation of arbitrary tensors and (3) the best rank-(R1, R2, R3) approximation of arbitrary tensors.

• De Lathauwer, L. (2006).
  A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization. SIAM Journal on Matrix Analysis and Applications, Vol28, 642-666.
  Canonical decomposition is a key concept in multilinear algebra. In this paper we consider the decomposition of higher-order tensors which have the property that the rank is smaller than the greatest dimension. We derive a new and relatively weak deterministic sufficient condition for uniqueness. The proof is constructive. It shows that the canonical components can be obtained from a simultaneous matrix diagonalization by congruence, yielding a new algorithm. From the deterministic condition we derive an easy-to-check dimensionality condition that guarantees generic uniqueness.

• De Lathauwer, De Moor, B, & Vandewalle, J. (2000a).
  An introduction to independent component analysis. Journal of Chemometrics, 14, 123-149.
  This paper is an introduction to the concept of independent component analysis (ICA) which has recently been developed in the area of signal processing. ICA is a variant of principal component analysis (PCA) in which the components are assumed to be mutually statistically independent instead of merely uncorrelated. The stronger condition allows one to remove the rotational invariance of PCA, i.e. ICA provides a meaningful unique bilinear decomposition of two-way data that can be considered as a linear mixture of a number of independent source signals. The discipline of multilinear algebra offers some means to solve the ICA problem. In this paper we briefly discuss four orthogonal tensor decompositions that can be interpreted in terms of higher-order generalizations of the symmetric eigenvalue decomposition.

• De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000b).
  A multilinear singular value decomposition. Siam Journal on Matrix Analysis and Applications, 21, 1253- 1278.
  A multilinear generalization of the singular value decomposition is discussed. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. It is then investigated how tensor symmetries affect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.

• De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000c).
  On the best rank-1 and rank-(R₁, R₂,…, R_N) approximation of higher-order tensors. SIAM Journal on Matrix Analysis and Applications, 21, 1324-1342.
  In this paper we discuss a multilinear generalization of the best rank-R approximation problem for matrices, namely, the approximation of a given higher- order tensor, in an optimal least-squares sense, by a tensor that has pre-specified column rank, row rank value, etc. For matrices, the solution is conceptually obtained by truncation of the singular value decomposition (SVD); however, this approach does not have a straightforward multilinear counterpart. We discuss higher-order generalizations of the power method and the orthogonal iteration method.

• De Lathauwer, L., De Moor, B., & Vandewalle, J. (2000d).
  A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21, 1253-1278.
  We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, .rst-order perturbation e.ects, etc., are analyzed. We investigate how tensor symmetries a.ect the decomposition and propose a multilinear generalization of the symmetric eigenvalue decomposition for pair-wise symmetric tensors.

• De Lathauwer, L., & Vandewalle, J. (2004b).
  Dimensionality reduction in higher-order signal processing and rank-(R-1, R-2, ..., R-N) reduction in multilinear algebra. Linear Algebra and its Applications, 391, 31-55.
  In this paper we review a multilinear generalization of the singular value decomposition and the best rank-(R-1, R-2,..., R-N) approximation of higher-order tensors. We show that they are important tools for dimensionality reduction in higher-order signal processing. We discuss applications in independent component analysis, simultaneous matrix diagonalization and subspace variants of algorithms based on higher-order statistics.

• De Lathauwer, L., & Vandewalle, J. (2004c).
  Dimensionality reduction in ICA and rank-(R-1, R-2, ..., R-N) reduction in multilinear algebra. Independent Component Analysis and Blind Signal Separation, 3915, 295-302.
  We show that the best rank-(R-1, R-2,..., R-N) approximation in multilinear algebra is a powerful tool for dimensionality reduction in ICA schemes without prewhitening. We consider the application to different classes of ICA algorithms.

• De la Pena, A. M., Mansilla, A. E. Gomez, D. G., Olivieri, A. C., & Goicoechea, H. C. (2003).
  Interference-free analysis using three-way fluorescence data and the parallel factor model. Determination of fluoroquinolone antibiotics in human serum. Analytical Chemistry, 75, 2640-2646.
  Three-way fluorescence data and multivariate calibration based on parallel factor analysis (PARAFAC) are combined for the simultaneous quantitation of three fluoroquinolone anitibiotics (norfloxacin, enoxacin, and ofloxacin) in human serum samples. The three analytes can be adequately determined with limits of detection of 0.2, 3.0, and 0.5 mug L-1, respectively, with minimum experimental effort. The selected analytical methodology fully exploits the so-called second-order advantage of the employed three-way data, allowing obtaining individual concentrations of calibrated analytes in the presence of any number of uncalibrated (serum) components. In contrast to PARAFAC, less satisfactory results were obtained with a multidimensional partial least-squares (nPLS) model trained with the same calibration set.

• De la Pena, A. M., Mansilla, A. E., Valenzuela, M. I. A., Goicoechea, H. C., & Olivieri, A. C. (2002).
  Comparative study of net analyte signal-based methods and partial least squares for the simultaneous determination of amoxycillin and clavulanic acid by stopped-flow kinetic analysis. Analytica Chimica Acta, 463, 75-88.
  A comparative study about advantages and limitations of net analyte signal (NAS)-based methods (NBMs) and partial least squares (PLS) calibration in kinetic analysis has been performed. The different multivariate calibration methods were applied to the determination of binary mixtures of amoxycillin and clavulauic acid, by stopped-flow kinetic analysis. The reactions of oxidation of these compounds with cerium(IV), in sulphuric acid medium, were monitored by following the changes on the fluorescence of the oxidation products, in stopped-flow mode. The differences on the kinetic profiles obtained at lambda(ex) = 256 nm and lambda(em) = 351 nm, were used to determine mixtures of both compounds by multivariate calibration of the kinetic data, using PLS-1, a modification of hybrid linear analysis (HLA) and net analyte pre-processing combined with classical least squares (NAP/CLS) methods. The NBMs allowed the selection of optimal time data regions by calculating the minimum error indicator function (EIF), improving the results and making NBMs very convenient for the analysis. In addition, the use of the net analyte signal concept allows the calculation of the analytical figures of merit, limit of detection (LOD), sensitivity and selectivity, for each component.

• De la Torre, F., & Black, M. J. (2003).
  A framework for robust subspace learning. International Journal of Computer Vision, 54, 117-142.
  Many computer vision, signal processing and statistical problems can be posed as problems of learning low dimensional linear or multi-linear models. These models have been widely used for the representation of shape, appearance, motion, etc., in computer vision applications. Methods for learning linear models can be seen as a special case of subspace fitting. One draw-back of previous learning methods is that they are based on least squares estimation techniques and hence fail to account for "outliers" which are common in realistic training sets. We review previous approaches for making linear learning methods robust to outliers and present a new method that uses an intra-sample outlier process to account for pixel outliers. We develop the theory of Robust Subspace Learning (RSL) for linear models within a continuous optimization framework based on robust M-estimation. The framework applies to a variety of linear learning problems in computer vision including eigen-analysis and structure from motion. Several synthetic and natural examples are used to develop and illustrate the theory and applications of robust subspace learning in computer vision.

• De la Vega, A.J., & Chapman, S.C. (2001).
  Genotype by environment interaction and indirect selection for yield in sunflower. II. Three-mode principal component analysis of oil and biomass yield across environments in Argentina. Field Crops Research, 72, 39-50.
  The genotype by environment (G x E) interactions observed for sunflower oil yield in different regions of Argentina can be analyzed in terms of differences among genotypes in individual environments for its components grain number, grain weight, and oil content (yield analysis). Similarly, G x E interactions observed for oil-corrected grain yield can be analyzed in terms of its determinants total biomass and harvest index (physiological analysis). Three-mode (genotypes x environments x attributes) principal component analysis was applied to 10 x 21 x 4 and 10 x I I x 3 matrices, for each of the first and the second analyses, respectively, to collectively interpret the changes in these attributes in a sunflower genotype-environment system, and to assess the relative importance of each trait as underlying determinant of the observed G x E interaction for oil yield. The 3 x 2 x 3 and 4 x 4 x 2 (genotypes x environments x attributes) principal component models explained about 65% of the variation computed for first and second approaches, respectively. For the yield analysis, the first environment component (54% of the variation) explained the common pattern of oil yield over environments and showed that oil content was highly positively correlated to oil yield, while grain number and grain weight showed lack of association with oil yield and were negatively correlated. The second environment component (11% of the variation) contrasted northern and central environments and showed that grain number is the main underlying determinant of the observed G x E interactions between these two mega-environments for oil yield. In the physiological analysis, the first environment component (29% of the variation) explained the common pattern of oil-corrected grain yield over environments and showed that harvest index was more strongly positively correlated to oil-corrected grain yield, but not to total oil-corrected biomass. The second environmental component (19% of the variation) contrasted northern and central environments and showed that oil-corrected biomass is the physiological attribute that is largely responsible for the G x E interactions for oil-corrected grain yield.

• De la Vega, A. J., Hall, A. J., & Kroonenberg, P. M. (2002).
  Investigating the physiological bases of predictable and unpredictable genotype by environment interactions using three-mode pattern analysis. Field Crops Research, 78, 165-183.
  Understanding of the underlying physiology of the genotype-specific responses to predictable and unpredictable environmental variation would improve the efficiency of selection within a complex target population of environments. Three-mode principal component analysis (PCA) can be used for interpreting the complex three-way (genotypes, environments, attributes) trial datasets from which this understanding should emerge. The efficiency of this method largely depends on the right combination between the biological and statistical models used, especially on the attributes selected to describe the genotypic responses and the centring of the three-way input data. In this study, we assessed the scope of yield determination models and double-centring of input data for generating some physiological understanding of the genotype x environment (G x E) interactions observed in a sunflower genotype-environment system and for developing ideotype-based breeding strategies. Double-centring of the three-way arrays permitted the separation of predictable and unpredictable G x E interactions. This, in combination with the use of models that explain the physiological bases of yield variation among genotypes, has served to identify three relevant sources of genotypic variation for use in a breeding program, namely: (i) attributes that can be selected to achieve specific adaptation to the target environment by emphasising predictable interactions (e.g. duration of grain filling, a trait associated with canopy stay green); (ii) attributes that allow the unpredictable G x E interactions to be accommodated, improving the linkage between managed-environments and target production environments (e.g. grain set); and (iii) genotypes of similar response pattern for yield but contrasting relative behaviour for the primary and secondary yield determinants. Breeding projects involving crosses between these genotypes could generate better opportunities for yield improvements for individual mega-environments.

• De Leeuw, J. & Pruzansky, S. (1978).
  A new computational method to fit the weighted Euclidean distance model. Psychometrika, 43, 479-490.
  Deals mainly with fitting the weighted Euclidean distance model, (INDSCAL), but also includes a discussion of three-mode scaling, and ways to rotate the results of that procedure to INDSCAL form (see also MacCallum, 1976; Cohen, 1974, 1975)

  

• De Ligny, C.L., Spanjer, M.C., Van Houwelingen, J.C, & Weesie, H.M. (1984).
  Three-mode factor analysis of data on retention in normal-phase high-performance liquid chromatography. Journal of Chromatography, 301, 311-324.
  It is shown that the Snyder equation is not quite satisfactory for fitting retention data in normal-phase high-performance liquid chromatography (HPLC) on chemically bonded phases. This equation is a special case of the mathematical-statistical three-mode factor analysis model. This model, in its general form, has been used to fit two sets of literature data on the retention in normal-phase HPLC for 19 solutes on six adsorbents with two eluents, and for 39 solutes on three adsorbents with two eluents, respectively. This study represents the first application of three-mode factor analysis with missing data, and also the first application of three-mode factor analysis in the field of the natural sciences. The accuracy of the fit of the observations and of the prediction of the missing data, for various numbers of extracted factors, is discussed.

• Demir, C., & Brereton, R.G. (1997).
  Calibration of gas chromatography-mass spectrometry of two-component mixtures using univariate regression and two- and three-way partial least squares. Analyst, 122, 631-638.
  Univariate calibration and two-way and three-way partial least squares (PLS) were applied to a series of GC-MS results for 21 mixtures of two closely eluting compounds, salbutamol and clenbuterol. Steps in the analysis, including baseline correction, alignment of chromatograms, mass selection, unfolding (for three-way data), standardizing and centring, are described, appropriately modified for the problem in hand. Both mass spectral and, for three-way data, time dependent loadings can be calculated. The quality of quantitative predictions was determined using a leave one out cross validation method. For PLS slightly better predictions were obtained compared with the best predictions for univariate single ion monitoring. Three-way PLS provides a wealth of extra information.

• Denis, J.B., & Dhorne, T. (1989).
  Orthogonal tensor decomposition of 3-way tables. In R. Coppi and S. Bolasco (Eds.), Multiway data analysis (pp. 31-37). Amsterdam: Elsevier.
  The authors' interest in 3-way tables is concerned when all of the three entries are of equal consideration from the view point of exploratory data analysis. Moreover they look for an exhaustive and 3-way symmetrical decomposition. In a first part, counterexamples of natural extensions of good properties concerning singular value decomposition of matrices are shown. In a second part, a reduced form of 3-way tables named "rocket form" is proposed, it can be seen as an extension of the singular value decomposition of matrices.

  

• De Rooij, M. (2001).
  Distance models for transition frequency data. Unpublished doctoral thesis, Leiden University, Leiden The Netherlands.
  1. Transition frequency data
  Contingency tables; Log-linear analysis of transition frequency data; Distance models; Overview of this monograph
  2. Multinomial logit models: with and without weights
  Review of sequential data analysis; Imagery play therapy; Regression models for contingency tables; Data analysis: Lag-1 and lag-2 transitions; Forecasting new behavior through the use of weights; Data analysis: Forecasting new behavior; Discussion
  3. Distance association models for two-way data
  Distance association models; The odds ratio; Identification of degrees of freedom; Data analysis; Comparison to RC(M)-model and discussion
  4. Asymmetric triadic distance models
  Asymmetric triadic distance models; Data analysis; Comparison and discussion
  5. Distance models and three-way association
  Models for three-way transition tables and distance restrictions; Triadic distance models; Three-way distance models; Discussion
  6. Distance association models for K two-way transition tables
  The model; ML-estimation; Data analysis; Discussion
  7. Conclusion
  The answer summarized; Some considerations
  

• De Rooij, M., (2001).
  Distance association models for the analysis of repeated transition frequency tables. Statistica Neerlandica, 55, 157-181.
  The present paper is concerned with the analysis of repeated transition frequency tables, for example, occupational mobility data measured in different cohorts. The association present in such a table will be modeled by a distance in Euclidean space. A large distance corresponds to a small association; a small distance corresponds to a large association. A more direct interpretation is that more transitions occur between categories that are close together in a social space. It is assumed that the same social structure (space) exists for the different slices (cohorts/time points) of a table, but that the dimensions of this space are weighted for the different slices, i.e., for each slice the dimensions are stretched or squeezed. We will propose a model, discuss an algorithm to obtain maximum likelihood estimates and apply the model to an empirical data set.

• De Rooij, M., (2002).
  Distance models for three-way tables and three-way association. Journal of classification, 19, 161-178.
  In the present paper we study distance models for the analysis of three-way contingency tables. Specifically, we will study three-way association under these models measured by the second order odds ratio. Two kinds of distance models will be studied: (a) Models for three-way tables where each way is treated on an equal footing; (b) Models for multiple two-way tables, where one of the three ways has a special importance. For the first kind of models, called triadic distance models, we will show that there exists a natural conjugacy between the Exponential-p similarity function, the L-p-transform and the Minkowski-p distance. For triadic distance models defined by the L-p-transform we will prove that they do not model three-way association. Moreover, triadic distance models defined by the L-p-transform are restricted multiple dyadic distances, where each dyadic distance is defined for a two-way margin of the three-way table. Distance models for three-way two-mode data, called three-way distance models, do succeed in modeling three-way association.

• De Rooij, M., & Heiser, W.J. (2000).
  Triadic distance models for the analysis of asymmetric three-way proximity data. British Journal of Mathematical and Statistical Psychology, 53, 99-119.
  Triadic distance models can be used to analyse proximity data defined on triples of objects. Three-way symmetry is a common assumption for triadic distance models. In the present study three-way symmetry is not assumed. Triadic distance models are presented for the analysis of asymmetric three-way proximity data that result in a simultaneous representation of symmetry and asymmetry in a low- dimensional configuration. An iterative majorization algorithm is developed for obtaining the coordinates and the representation of the asymmetry. The models are illustrated by an example using longitudinal categorical data.

• De Rooij, M., & Gower, J. C. (2004).
  The geometry of triadic distances. Journal of Classification, 20, 181-220.
  Triadic distances t defined as functions of the Euclidean (dyadic) distances a(1), a(2), a(3) between three points are studied. Special attention is paid to the contours of all points giving the same value of t when a(3) is kept constant. These isocontours allow some general comments to be made about the suitability, or not, for practical investigations of certain definitions of triadic distance. We are especially interested in those definitions of triadic distance, designated as canonical, that have optimal properties. An appendix gives some results we have found useful.

• DeSarbo, W.S., & Carroll, J.D. (1985).
  Three-way metric unfolding via alternating weighted least squares. Psychometrika,50, 275-300.
  Three-way unfolding was developed to accommodate the analysis of two-mode, three-way data (e.g., nonsymmetric proximities for stimulus objects collected over time) and three-mode, three-way data (e.g., Ss rendering preference judgments for various stimuli in different usage situations). The authors present a revised objective function and new algorithm to prevent the common type of degenerate solutions encountered in typical unfolding analysis. The three-way unfolding model, weighted objective function, and new algorithm are presented, and Monte Carlo work investigating the effect of several data and model factors on overall algorithm performance is described. Three applications of the methodology are reported.

• DeSarbo, W.S., Carroll, J.D., Clark, L.A., & Green, P.E. (1984).
  Synthesized clustering: A method for amalgamating alternative clustering bases with differential weighting of variables. Psychometrika, 49, 57-78.
  In the application of clustering methods to real world data sets, two problems frequently arise: (a) how can the various contributory variables in a specific battery be weighted so as to enhance some cluster structure that may be present, and (b) how can various alternative batteries be combined to produce a single clustering that "best" incorporates each contributory set. A new method is proposed (SYNCLUS, SYNthesized CLUStering) for dealing with these two problems.

• DeSarbo, W.S., Carroll, J.D., Lehmann, D.R., & O'Shaughnessy, J. (1982).
  Three-way multivariate conjoint analysis. Marketing Science, 1, 323-350.
  Three-way multivariate conjoint analysis is developed as an extension of traditional metric conjoint analysis allowing one to examine several dependent variables simultaneously, as well as individual differences in response. Four nested models are developed to examine the effects of experimental design, the dependent variables, and individual differences. An illustration concerning the relationship of product characteristics to the importance of various decision- making criteria for industrial purchasing is provided. Finally, extensions of the model(s) to other marketing applications and non-metric analyses are discussed.

  

• DeSarbo, W.S., & Harshman, R.A. (1985).
  Celebrity-brand congruence analysis. Journal of Current Issues & Research in Advertising, 1, 17-52.
  This paper proposes a method for uncovering the perceptual-cognitive overtones of product and possible commercial spokesmen, and then examines their relationships so as to establish a basis for "optimizing" potentially desirable source effects. As a demonstration of this method, the PARAFAC three-way factor analysis procedure is applied to individuals' associative judgements (measured on a set of semantic differential scales) concerning twelve automobile makes and twelve celebrities (commercial spokesmen).

• DeSarbo, W.S., Libby, R., & Jedidi, K. (1994).
  CATSCALE: A stochastic multidimensional scaling methodology for the spatial analysis of sorting data and the study of stimulus categorization. Computational Statistics & Data Analysis, 18, 165-184.
  Sorting tasks have provided researchers with valuable alternatives to traditional paired-comparison similarity judgments. They are particularly well-suited to studies involving large stimulus sets where exhaustive paired-comparison judgments become infeasible, especially in psychological studies investigating stimulus categorization. This paper presents a new stochastic multidimensional scaling procedure called CATSCALE for the analysis of three-way sorting data (as typicaly collected in categorization studies). We briefly present a review of the role of sorting tasks, especially in categorization studies, as well as a description of several traditional modes of analysis. The CATSCALE model and maximum likelihood based estimation procedure are described. An application of CATSCALE is presented with respect to a behavioral accounting study examining auditors' categorization processes with respect to various types of errors found in typical financial statements.

• De Sena, M. M., Poppi, R. J., Frighetto, R. T. S., & Valarini, P. J. (2000).
  Evaluation of the use of chemometric methods in soil analysis. Quimica Nova, 23, 547-556.
  One of the major interests in soil analysis is the evaluation of its chemical, physical and biological parameters, which are indicators of soil quality (the most important is the organic matter). Besides there is a great interest in the study of humic substances and on the assessment of pollutants, such as pesticides and heavy metals, in soils. Chemometrics is a powerful tool to deal with these problems and can help soil researchers to extract much more information from their data. In spite of this, the presence of these kinds of strategies in the literature has obtained projection only recently. The utilization of chemometric methods in soil analysis is evaluated in this article. The applications will be divided in four parts (with emphasis in the first two): (i) descriptive and exploratory methods based on Principal Component Analysis (PCA); (ii) multivariate calibration methods (MLR, PCR and PLS); (iii) methods such as Evolving Factor Analysis and SIMPLISMA; and (iv) artificial intelligence methods, such as Artificial Neural Networks.

• De Soete, G., & Carroll, J.D. (1988).
  Optimal weighting for one-mode and two-mode ultrametric tree representations of three-way three-mode data. In M.G.H. Jansen & W.H. van Schuur (Eds.), The many faces of multivariate analysis. Vol. I. Groningen: RION.
  Two different approaches are proposed for representing a three-way three-mode array by a single ultrametric tree structure. In the first approach, the elements corresponding to the levels of one mode (say, the first) are represented by the terminal nodes of a one-mode ultrametric tree. Dissimilarities between the levels of that mode are computed from the data, weighting the levels of the other two modes. These weights are estimated such that the resulting dissimilarities satisfy the ultrametric inequality as well as possible. In the second approach, the objects of two modes (say, the first two) are jointly represented by the terminal nodes of a two-mode (or two- class) ultrametric tree, From the data, a rectangular proximity matrix is calculated weighting the levels of the third way optimally.

• De Soete, G., & Carroll, J.D. (1989).
  Ultrametric tree representations of three-way three-mode data. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 415-426). Amsterdam: Elsevier.
  Three different procedures are developed for representing three-way three-mode data by one or more ultrametric trees. In the first approach, the levels of one mode (say, the first) are represented by the terminal nodes of an ultrametric tree, using proximities that are computed from the complete data array weighting the levels of the second and third ways optimally. In the second approach, the levels of two modes (say, the first two) are jointly represented by the terminal nodes of a two-class ultrametric tree. The data are first optimally aggregated over the third way. Finally, in the third approach, the first two modes are represented for each level of the third mode by a two-class ultrametric tree subject to the constraint that all trees have the same topology.

• De Soete, G., & Carroll, J.D. (1996).
  Tree and other network models for representing proximity data. In P. Arabie, L.J. Hubert, & G. De Soete (Eds.), Clustering and Classification (pp. 157-197). River Edge, NJ: World Scientific Publishing.
  In this paper tree and other network models for analyzing different kinds of proximity data are discussed. In the section on three-way data techniques for both one, two and three different modes are considered.

  

• De Soete, G., Carroll, J.D., & Chaturvedi, A.D. (1993).
  A modified CANDECOMP method for fitting the extended INDSCAL model. Journal of Classification, 10, 75-91.
  A modified CANDECOMP algorithm is presented for fitting the metric version of the Extended INDSCAL model to three-way proximity data. The Extended INDSCAL model assumes, in addition to the common dimensions, a unique dimension for each object. The modified CANDECOMP algorithm fits the Extended INDSCAL model in a dimension-wise fashion and ensures that the subject weights for the common and the unique dimensions are non- negative. A Monte Carlo study is reported to illustrate that the method is fairly intensive to the choice of the initial parameter estimates. A second Monte Carlo study shows that the method is able to recover an underlying Extended INDSCAL structure if present in the data. Finally, the method is applied for illustrative purposes to some empirical data on pain relievers. In the final section, some other possible uses of the new method are discussed.

• Devaux, M. F., Courcoux, P., Vigneau, E., & Novales, B. (1998).
  Generalised Canonical Correlation Analysis for the interpretation of fluorescence spectral data. Analusis, 26, 310-316.
  The paper reports an application of Generalised Canonical Correlation Analysis to fluorescence spectral data. Emission fluorescence spectra can be recorded for several excitation wavelengths and can be presented as 3-way data tables. The objectives of the data treatment are to describe and compare the samples by taking into account all the emission spectra, and to reveal characteristic excitation and emission wavelengths. Generalised Canonical Correlation Analysis has been tested on fluorescence emission spectra acquired for binary mixtures of raw materials in the food domain. The application of the method within the context of spectral data is presented.

  
         

• Di Ciaccio, A., & Bove, G. (1987).
  A factorial method for the analysis of three-way data in tensor spaces. In the Fifth International Symposium on Data Analysis and Informatics, 2, 137-145.
  A three-mode data matrix X can be decomposed in three different ways according to each of its modes and several relevant matrices can be defined following the three scheme of decomposition. These matrices are represented as tensors, according to the definition, in RKxRK or in RJxRJ or else in RIxRI and their properties and relationships are investigated. Useful graphical representations of these tensors are proposed to support standard results of other three-way methods. An application to the Languedoc-Roussillon data is presented.

  

• Dien, J. (1998).
  Adressing Misallocation of Variance in Principal Components Analysis of Event-Related Potentials. Brain Topography, 11 (pp. 43-55).
  Interpretation of evoked response potentials is complicated by the extensive superposition of multiple electrical events. The most common approach to disentangling these features is principal components analysis (PCA). Critics have demonstrated a number of caveats that complicate interpretation, notably misallocation of variance and latency jitter. This paper describes some further caveats to PCA as well as using simulations to evaluate three potential methods for addressing them: parallel analysis, oblique rotations, and spatial PCA. An improved simulation model is introduced for examining these issues. It is concluded that PCA is an essential statistical tool for event-related potential analysis, but only if applied appropriately.

• Diewok, J., De Juan, A., Tauler, R., & Lendl, B. (2002).
  Quantitation of mixtures of diprotic organic acids by FT-IR flow titrations and multivariate curve resolution. Applied Spectroscopy, 56, 40-50.
  The combination of FT-IR flow titrations and second-order calibration methods is assessed for the first time as a potential method for quantifying mixtures of organic acids. FT-IR spectral information is richer and more specific than that provided by other spectrometric techniques. Also, the flow titration of the sample allows the use of second-order data analysis methods, such as multivariate curve resolution, which can quantify multianalyte systems using pure analyte standards in the presence of unknown interferents. Problems linked to FT-IR flows titrations, such as the impossibility of using buffer substances and the presence of major baseline contributions in the measurements due to changes in the water structureless absorption, have been overcome through the design of an adapted experimental setup and the use of second derivative spectra. Malic and/or tartaric acid are the model substances chosen to be quantified in the presence or absence of interferents. Both substances are diprotic acids with similar pK(a) values and, therefore, similar pH-dependent evolution. Samples with one or both acids with their related standards in the presence or absence of an inert interferent (sugar) are successfully resolved and quantified for a wide range of [analyte] : [interferent] ratios with no need for pH control in the process. These results suggest this methodology as an optimal and robust alternative for routine analysis.

• Diewok, J., De Juan, A., Maeder, M., Tauler, R., & Lendl, B. (2003).
  Application of a combination of hard and soft modeling for equilibrium systems to the quantitative analysis of pH-modulated mixture samples. Analytical Chemistry, 75, 641-647.
  pH modulation of aqueous mixture samples combined with FT-IR detection and a powerful second-order resolution method is proposed for both resolution and quantitation of acid analytes in the presence of similarly behaving interferences. The proposed method allows for the analyte determination in mixtures using a single standard sample per analyte. Due to the very similar pK(a) values of the investigated analytes and interferences, the highly correlated concentration profiles of these compounds cannot be successfully resolved with pure soft-modeling second-order approaches. The inclusion of a hard-modeling constraint based on the acid-base equilibrium model in the soft-modeling curve resolution method has allowed for the unambiguous resolution of the analyte profiles and, as a consequence, for the correct quantitation of this compound in the mixture sample. A detailed discussion of the combined hard-soft-modeling approach as well as the analytical problem and the quantitation results is given. Also, strategies to overcome problems associated with variation in pK(a) values between different samples are addressed. Due to the flexible implementation of the hard-model equilibrium constraint in the multivariate curve resolution-alternating least squares method, this approach is expected to be useful also for analysis of other complex mixed equilibrium-based chemical systems.

• Dijksterhuis, G.B., & Gower, J.C (1991/2)
  The interpretation of generalized Procrustes analysis and allied methods. Food Quality and Preference, 3, 67-87.
  Various issues surrounding the use and interpretation of Generalised Procrustes Analysis and related methods are discussed. Included are considerations that have to be made before starting an analysis, how to handle different dimensionalities of data, when to consider fitting scaling factors and when not to, and the distinction between the number of dimensions that are needed to give an adequate fit and the need for graphical representation. The distinction between signal and noise plays an important part in explaining how different methods are suitable for exploring different aspects of the data, rather than being viewed as competing models with the same objectives. Explanations are largely set in a geometric context; a common Analysis of Variance framework allows all methods to be considered in a unified way and suggests some new ways in which these kinds of data may be analysed. The whole is illustrated by example analyses.

• Dillon, W.R., Frederick, D.G., & Tangpanichdee, V. (1985).
  Decision issues in building perceptual product spaces with multi-attribute rating data. Journal of Consumer Research, 12, 47-63.
  This paper considers decisions that face consumer researchers as they implement a perceptual product space analysis based on multi-attribute rating data. Decisions that affect the structure of the derived perceptual product space solution can be grouped into six major categories relating to issues of (1) data input, (2) mode, (3) preprocessing transformation, (4) choice/preference modeling, (5) technique, and (6) solution. The major difficulties of each decision area are provided whenever possible.

• Disner, S.F. (2001).
  Evaluation of vowel normalization procedures. Journal of the Accoustical Society of America, 67, 253-261.
  Vowel normalization procedures are commonly evaluated on the basis of how effectively they separate the vowels of a single test data set into distinct groups corresponding to the phonetic categories of that language. A quantifiable method of evaluation is proposed here, based on how much of he overall variance is removed from the data. This evaluation method is applied to the vowels of six different Germanic languages which have been normalized according to four different procedures. It is shown that no one normalization procedure is the most effective for all languages. Furthermore, some of the most successful of these normalizations introduce procedural artifacts into the data, and as a result the relative quality of vowels across languages or dialects is altered. In such cases, it is shown that comparisons of the normalized vowels of one language with the (separately) normalized vowels of another language are not valid if the vowel systems are different. Some reasons for the appearance of the procedural artifacts are discussed.

  
             

• Do, T., & McIntyre, N. S. (1999a).
  Pressure effects on aluminium oxidation kinetics using X-ray photoelectron spectroscopy and parallel factor analysis. Surface Science, 440, 438-450.
  The effects of water vapour pressure on oxidation kinetics of aluminium have been studied using X-ray photoelectron spectroscopy (XPS) and three-way parallel factor analysis (PARAFAC). While the first technique is a powerful experimental tool for surface oxidation studies, the PARAFAC technique is a sophisticated analytical tool for analysing XPS data. The XPS Al(2p) and O(1s) core level have been used to follow the oxide film growth on clean surfaces at room temperature as a function of oxidation time (ranging from 1 to 60 min) and pressure of water vapour (ranging from 2.0 x 10(-6) to 6.5 x 10(-4) Pa). The growth of thin oxide films on aluminium surfaces has been found to follow the Cabrera-Mott inverse logarithmic law in all pressure ranges studied. The pressure effects have shown that the defect formation reaction at the oxide film/gas interface is the rate determining process in the aluminium oxidation. The pressure dependence of oxidation kinetics can be explained on the basis of metal vacancies in the defect structure of thin aluminium oxide films.

• Do, T., & McIntyre, N. S. (1999b).
  Application of parallel factor analysis and X-ray photoelectron spectroscopy to the initial stages in oxidation of aluminium. Surface Science, 435, 136-141.
  The three-way parallel factor analysis (PARAFAC) has been used to decompose a set of XPS (X-ray photoelectron spectroscopy) spectra which result during the oxidation of aluminium surfaces by water vapour. Al(2p) and O(1s) core-level photoelectron lines have been used to follow oxide film growth on clean aluminium surfaces as a function of exposure time and pressure of water vapour. The PARAFAC solution provides new information on elemental processes in the very initial stages of oxidation kinetics, showing new components in the XPS spectrum, as well as their evolution through the range of time and pressure variables. Reaction of H2O vapour with aluminium results in attenuation of the metallic peak, binding energy (BE) at 72.87 +/- 0.05 eV, and increase of the oxidic peak, BE at 75.80 +/- 0.05 eV. An additional factor is identified, which suggests the formation of an interface metal hydride, with BE at 72.4(4) eV, as well as a concomitant oxide peak at 75.4(3) eV, At pressures above 1.3 x 10(-5) Pa this factor is diminished; this is presumably due to the increase in recombination of atomic hydrogen.

• Do, T., McIntyre, N.S., Harshman, R.A., Lundy, M.E., & Splinter, S.J. (1999c).
  Application of parallel factor analysis and x-ray photoelectron spectroscopy to the initial stages in oxidation of aluminium - I. The Al 2p photoelectron line. Surface and Interface Analysis, 27, 618-628.
  Three-way parallel factor analysis (PARAFAC) has been used to decompose a set of x-ray photoelectron spectroscopy (XPS) spectra that result during the oxidation of aluminium surfaces. The Al 2p core-level photoelectron lines have been used to follow oxide film growth on clean aluminium surfaces as a function of exposure time and pressure of water vapour. In this paper, a fine peak structure of the XPS Al 2p spectrum has been extracted using PARAFAC. The PARAFAC solution provides new information on elemental processes in the very initial stages of oxidation kinetics, showing new components in the XPS spectrum as well as their evolution through a range of time and pressure variables.

• Donchin, E., Gerbrandt, L.A., Leifer, L. & Tucker, L. (1972).
  Is the contingent negative variation contingent on a motor response? Psychophysiology, 9, 178-188.
  T3 on evoked potential data (7 subjects, 4 conditions, 125 time segments). Results are only partially presented.

  

• Dong, D., & McAvoy, T.J. (1996).
  Batch tracking via nonlinear principal component analysis. AIChE Journal, 42, 2199-2208.
  Batch processes are very important to the chemical and manufacturing industries. Techniques for monitoring these batch processes to ensure their safe operation and to produce consistently high-quality products are needed. Nomikos and MacGregor (1994) presented a multiway principal component analysis (MPCA) approach for monitoring batch processes, and test results show that the method is simple, powerful, and effective. MPCA, however, is a linear method, and most batch processes are nonlinear. Although data treatment techniques can remove some nonlinearity from the data, nonlinearity is still a problem when using mpca for monitoring. In this article a nonlinear principal component analysis (NLPCA) method (Dong and McAvoy, 1993) is used for batch process monitoring. Results show that this method is excellent for this problem. Another interesting extension of this approach involves multistage batch process monitoring which is illustrated through a detailed simulation study.

  
       

• Drava, G., Leardi, R., Portesani, A., & Sales, E. (1996).
  Application of chemometrics to the production of friction materials: Analysis of previous data and search of new formulations. Chemometrics and Intelligent Laboratory Systems, 32, 245-255.
  A friction material is a composite containing up to 18 different components which can be chosen among a large number of possible raw materials having different characteristics, like graphite, sulphides, metals, fibers, rubbers, resins and fillers. The requested form is obtained by moulding at controlled pressure and temperature. In order to prepare new formulations having good performances, the problem is to choose the best raw materials and to mix them in the optimal proportions. Since the quality of a formulation is not expressed by a single value, but several responses have to be taken into account at the same time (friction coefficient, comfort, wear, etc.), the analysis of the data obtained from different formulations is quite difficult. In this study an approach to the analysis of this kind of data is presented, in order to evaluate different products on the basis of a small number of 'quality indicators'. The techniques of experimental design are successfully applied in order to investigate the effect of process variables on the performances of the product and to perform a screening of the raw materials for new optimal formulations.

       

• Dueñas Cueva, J.M., Rossi, A.V., & Poppi, R.J. (2001).
  Modeling kinetic spectrophotometric data of aminophenol isomers by PARAFAC2. Chemometrics and Intelligent Laboratory Systems, 55, 125-132.
  This paper describes an applicaion of PARAFAC2 in the modeling of kinetic-spectophotometric data. The data were obtained by spectrophotometric monitoring of the reaction between aminophenol isomers and NaNO₂ in acid solution. The data set of various samples produces a three-way data array. This reaction has the property that isomers in different proportions produce different kinetic profiles. Due to this property, PARAFAC2 is suitable to model the system because it does not assume parallel proportional kinetic profiles. The results with PARAFAC2 were satisfactory, it being possible to recover the spectral and kinetic profiles, as well as the initial isomer concentrations with good accuracy.

• Dunn, T.R., & Harshman, R.A. (1982).
  A multidimensional scaling model for the size-weight illusion. Psychometrika, 47, 25-45.
  The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker's Three-mode Multidimensional Scaling (TMMDS) model or Carroll's Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. This article shows how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. The data were also analysed using INDSCAL, however, INDSCAL cannot represent individual differences in the strength of the illusion.

• Dunn, W.J., Hopfinger, A.J., Catana, C., & Duraiswami, C. (1996).
  Solution of the conformation and alignment tensors for the binding of trimethoprim and its analogs to dihydrofolate reductase: 3D-quantitative structure-activity relationship study using molecular shape analysis, 3-way partial least squares regression, and 3-way factor analysis. Journal of Medicinal Chemistry, 39, 4825-4832.
  Molecular recognition is the basis of rational drug design. However, the process by which a ligand recognizes and binds its receptor is complex. A general treatment for solving for the receptor-bound geometry (conformation and alignment) of a series of ligands is presented. A 20 (inhibitor structors) by 10 (compounds or ligands) by 4 (alignments) data array (included in the paper) with molecular shapes as measured by common overlap steric volume was available for analysis, as well as an inhibitor concentration as criterion variable. A PLS 3-way decomposition was applied to the complete data (see also Lohmöller & Wold, 1980) and a three-mode PCA (Tucker, 1966) to the three-way array using software developed by the authors. Random permutations of the criterion variable were used to validate the PLS solution.

  

• Durell, S.R., Lee, C.-H., Ross, R.T., & Gross, E.L. (1990).
  Factor analysis of the near-ultraviolet absorption spectrum of plastocyanin using bilinear, trilinear, and quadrilinear models. Archives of Biochemistry and Biophysics, 278, 148-160.
  Factor analysis was used to resolve the spectral component in the near-uv absorption spectrum of plastocyanin. The data set was absorption as a function of four variables: wavelength, species of plastocyanin, oxidation state of the copper center, and environmental pH. The data were fit with the traditional bilinear model as well as with trilinear and quadrilinear models. Trilinear and quadrilinear models have the advantage that they uniquely define the components, avoiding the indeterminacy of bilinear models. Bilinear analysis using the absorption spectra of tyrosine and copper metallothionein as targets resulted in a two-component solution which was nearly identical to that obtained using trilinear and quadrilinear models, for which no targets are required. The two-component models separate the absorption into tyrosine and copper center components. The absorption of tyrosine is found to be pH dependent in reduced plastocyanin, and the absorption magnitude of the reduced copper center is the same in the four different plastocyanin species. Further resolution is provided by a three-component quadrilinear model. The results indicate that there are at least two different electronic transitions which cause the absorption of the reduced copper center and that one of them couples to a tyrosine residue which is pH dependent. Correlation of the results with previous studies indicates that it is Tyr 83 which is the perturbed residue. The separation of the absorption of the copper center and Tyr 83 provides spectroscopic probes for the conformations of the north pole and east face reaction sites on the plastocyanin protein.

• D'Urso, P. (2004).
  Fuzzy C-means clustering models for multivariate time-varying data: Different approaches. International Journal of Uncertainty Fuzziness and Knowledgebases Systems, 12, 287-326.
  The classification of multivariate time-varying data finds application in several fields, such as economics, finance, marketing research, psychometrics, bioinformatics, medicine, signal processing, pattern recognition, etc. In this paper, by considering an exploratory formalization, we propose different unsupervised clustering models for multivariate data time arrays (objects x quantitative variables x times). These models can be classified in two different approaches: the cross sectional and the longitudinal approach. In the first case, after the objects, observed at each time, have been classified, comparison among the classifications made in different time instants will be done. In the second approach, we cluster the time trajectories of the objects; then, we obtain only one classification by comparing the instantaneous and evolutive features of the trajectories of the objects. In particular, in this work, the second approach is analyzed in detail, with reference to the so-called single and double step procedures. Geometric, correlative, instantaneous, evolutive and trend characteristics of the multivariate time arrays are taken into account in the different proposed clustering models. Furthermore, the fuzzy approach, that is particularly suitable in the dynamic classification, problem, has been considered. Extensions of a cluster-validity criterion for the proposed fuzzy dynamic clustering models are also suggested. A socio-economic example concludes the paper.

• D'Urso, P. & Vichi, M. (1998).
  Dissimilarities between trajectories of a three-way longitudinal data set. In A. Rizzi, M. Vichi, & H.-H. Bock (Eds.) Advances in data science and classification (pp. 585-592). Berlin: Springer.
  Methods are presented to evaluate dissimilarities between trajectories in a three-way longitudinal data (sets of multiple time series). The dissimilarity between trajectories is defined as a conic combination of the dissimilarities between trends, velocities and acccelerations of pairs of trajectories. The coefficients of the linear combinations are estimated maximising its variance. As an example trajectories of Italian regions are classiified by their employment dynamics.

• D'Urso, P., & Giordani, P. (2004).
  A least squares approach to principal component analysis for interval valued data. Chemometrics and Intelligent Laboratory Systems, 70, 179-192.
  Principal Component Analysis (PCA) is a well-known technique, the aim of which is to synthesize huge amounts of numerical data by means of a low number of unobserved variables, called components. In this paper, an extension of PCA to deal with interval valued data is proposed. The method, called Midpoint Radius Principal Component Analysis (MR-PCA), recovers the underlying structure of interval valued data by using both the midpoints (or centers) and the radii (a measure of the interval width) information. In order to analyze how MR-PCA works, the results of a simulation study and two applications on chemical data are proposed.

         

• Dyson, R., Maeder, M., Neuhold, Y. M. & Puxty, G. (2003).
  Analyses of three-way data from equilibrium and kinetic investigations Analytica Chimica Acta, 490, 99-108. In kinetic or equilibrium investigations it is common to measure two-way multiwavelength data, e.g. absorption spectra as a function of time or reagent addition. Often it is advantageous to acquire experimental data at various initial conditions or even on different instruments. A collection of these measurements can be arranged in three-dimensional arrays, which can be analysed as a whole under the assumption of a superimposed function, e.g. a kinetic model, and/or common properties of the subsets, such as molar absorptivity. As we show on selected formation equilibria (Zn2+/phen) and kinetic studies (Cu2+/cyclam) from our own research, an appropriate combination of multivariate data can lead to an improved analysis of the investigated systems. Crown Copyright (C) 2003 Published by Elsevier B.V. All rights reserved.

   

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