Three-Mode Abstracts, Part T

INDEX

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• Takane, Y., Young, F.W., & De Leeuw, J. (1977).
  Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features. Psychometrika, 42, 7-67.
  A new procedure is discussed which fits either the weighted or simple Euclidean model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used indiviual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants. The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems. The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.
  
  
• Takane, Y., Kiers, H. A. L., & De Leeuw, J. (1995).
  Component analysis with different sets of constrainst on different dimensions. Psychometrika, 60, 259-280.
  Many of the ''classical'' multivariate data analysis and multidimensional scaling techniques call far approximations by lower dimensional configurations. A model is proposed, in which different sets of linear constraints are imposed on different dimensions in component analysis and ''classical'' multidimensional scaling frameworks. A simple, efficient, and monotonically convergent algorithm is presented for fitting the model to the data by least squares. The basic algorithm is extended to cover across-dimension constraints imposed in addition to the dimensionwise constraints, and to the case of a symmetric data matrix. Examples are given to demonstrate the use of the method.
  
  
• Takane, Y.(2004).
  Matrices with special reference to applications in psychometrics. Linear Algebra and its Applications, 388, 341-361.
  Multidimensional scaling, item response theory, and factor analysis may be considered three major contributions of psychometricians to statistics. Matrix theory played an important role in early developments of these techniques. Unfortunately, nonlinear models are currently very prevalent in these areas. Still, one can identify several areas of psychometrics where matrix algebra plays a prominent role. They include analysis of asymmetric square tables, multiway data analysis, reduced-rank regression analysis, and multiple-set (T-set) canonical correlation analysis among others. In this article we review some of the important matrix results in these areas and suggest future studies.
  
  
• Takeuchi, H., Kroonenberg, P.M., Taya, H., & Miyano, H. (1986).
  An analysis of Japanese language on thermal sensation. Mathematical Linguistics, 15, 201-209.
  An analysis was conducted by the semantic differential method to clarify the relation of the words which express thermal sensation. Fifty subjects were required to rate twenty-two stimulus words by twenty-six semantic differential scales. The data were analyzed by three mode principal component analysis. And the meaning structure on thermal sensation was obtained.
  
  
• Tan, Y.X., Jiang, J.H., Wu, H.L., Cui, H. & Yu, R.Q. (2000).
  Resolution of kinetic system of simultaneous degradations of chlorophyll a and b by PARAFAC. Analytica Chimica Acta, 412, 195-202.
  A three-way resolution method based on PARAFAC analysis of fluorescence excitation/emission matrices (EEMs) is presented to study the black kinetic system of simultaneous degradations of chlorophyll a and b extracted with other interferents from fresh spinach. The excitation and emission spectral profiles as well as the kinetic concentration profiles of chlorophyll a, b and their degradation products, pheophytin a, b were resolved for this multi-component kinetic system. The degradation of chlorophyll to pheophytin under the optimized experimental conditions is confirmed to follow the first-order reaction model. The most obvious advantage of this method is that the complex black multi-component kinetic system can be resolved and studied in the presence of unknown interferents by applying PARAFAC, which is capable of resolving the three-way data array provided by EEMs and giving a unique solution to the analytical problem.
  
  
• Tanaka, K., & Shinotake, T. (1993).
  Analysis of clinical cases adopting three-mode factor analysis. Japanese Journal of Psychology, 64, 284-288.
  Studied sequentially the evaluations of the self and others in psychotherapy using the rating grid method (a repertory grid technique). Ss were a 25-yr-old female with somatization disorder and a 30-yr-old male with conversion disorder. The evaluations of each S were submitted to a three-mode factor analysis.
  
• Tates, A. A., Louwerse, D. J., Smilde, A. K., Koot, G. L. M., & Berndt, H. (1999).
  Monitoring a PVC batch process with multivariate statistical process control charts. Industrial & Engineering Chemistry Research, 38, 4769-4776.
  Multivariate statistical process control charts (MSPC charts) are developed for the industrial batch production process of poly(vinyl chloride) (PVC). With these MSPC charts different types of abnormal batch behavior were detected on-line. With batch contribution plots, the probable causes of these abnormalities were located. Examples are given for two different types of abnormalities, i.e., a bias in the batch loading and a control valve problem during polymerization. These examples show that MSPC charts in combination with batch contribution plots are a simple and powerful method for fault detection and identification of their cause in the operation and control of batch processes.
  
  
• Tassinari, G., & Vichi, M. (1994).
  La dinamica economica dei Paesi avanzati negli anni Ottanta: Riflessioni sulle traiettorie risultanti dall'analisi di matrici a tre vie. (Economic Dynamics of Advanced Countries in the Eighties: A Three-Way Matrices Analysis. With English summary.) Giornale-degli-Economisti-e-Annali-di-Economia, 53, 101-133.
  The paper deals with the comparison of the levels of economic performances among G7 countries in the 1980-86 period and with the analysis of their evolution. A three-way matrices analysis method (known as factorial matrices analysis, FAMA) is used, where the three dimensions are, respectively, countries, times and economic variables. The results show that growth rates of the same order might be matched by lacks of balance on the financial side, so that a synthesis of many economic variables is to be used to represent the economic performance of a country. The use of FAMA allows to identify, in every year, the changes in the variables that have caused the changes in the countries positions.
  
  
• Tatsunami, K., Kuwabara, R., Yago, N., Mimaya, J., Yamada, K., Sato, M. & Sato, Y. (1998).
  Application of three-way data clustering to analysis of lymphocyte subset numbers in Japanese hemophiliacs infected with HIV-1. In A. Rizzi, M. Vichi & H.-H. Bock (Eds.), Advances in data science and classification (pp. 627-632). Berlin: Springer
  Three-way clustering developed by Sato and Sato (1994) was used to analyse a number of CD4+ and CD8+ cells from 131 Japanese hemophiliacs infected with HIV-1 resulting in four clusters. In particular, times-series data of the CD4+ and CD8+ cell counts were analysed.
  
  
• Tauler, R. (1995).
  Multivariate curve resolution applied to second order data. Chemometrics and Intelligent Laboratory Systems, 30, 133-146.
  Application of multivariate curve resolution to second order data from hyphenated liquid chromatography with spectrometric diode array detection is shown. Chromatographic analysis of samples giving unresolved mixtures produces different data structures depending on the reproducibility of the elution process: (a) second order data where elution peaks of the same component in the different chromatographic runs have the same shape and appear at exactly the same elution times (synchronized); (b) second order data where elution peaks of the same component in the different chromatographic runs appear at different elution times (non-synchronized) although they are still of the same shape; and (c) second order data where elution peaks of the same component in the different chromatographic runs have different shapes and appear at different elution times. Multivariate curve resolution is easily adapted to analyze all these situations taking advantage in every case of the particular data structure. Multivariate curve resolution is also easily adapted to those situations where second order data has not a complete trilinear structure.
  
  
• Tauler, R. (2001).
  Calculation of maximum and minimum band boundaries of feasible solutions for species profiles obtained by multivariate curve resolution. Journal of Chemometrics, 15, 627-646.
  A method for the calculation of maximum and minimum band boundaries of feasible solutions corresponding to the species profiles estimated by multivariate curve resolution is presented. The method is based on a non-linear constrained optimization of an objective function defined by the ratio of the signal contribution of a particular species to the whole measured signal. Implementation of constraints such as normalization, closure, nonnegativity, unimodality and local rank/selectivity during the non-linear optimization and their effect on the calculation of the feasible solutions are studied in detail. Calculation of the band boundaries is shown for different simulated and real two-way data examples of increasing complexity.
  
  
• Tauler, R., & Barceló, D. (1993).
  Multivariate curve resolution applied to liquid chromatography-diode array detection. TrAC: Trends in Analytical Chemistry, 12,, 319-327.
  Multivariate curve resolution methods can be used for the quantitation of the overlapping components in a chromatographic peak. Initial qualitative solutions obtained by self-modelling curve resolution methods, such as evolving factor analysis, can be further optimized by simultaneous analysis of multiple chromatographic runs with alternating least-squares regression. Quantitation is achieved by constraining the pure unit spectra and elution profiles of the common analytes in the different chromatographic runs to be equal. The relationship between the proposed method and other higher-order calibration and resolution methods is examined.
  
  
• Tauler, R., & Casassas, E. (1992a).
  Application of factor-analysis to speciation in multiequilibria systems. Analusis, 20, 255-268.**
  Multiequilibria systems in solution have traditionally been investigated by least-squares curve-fitting methods. These approaches are difficult to apply to complicated systems because a chemical model must be defined based on the constituent species and their constants of formation. Model-free factor analysis is mainly seen as an independent method for investigating the number of species present in the system. Further developments such as target factor analysis, rank annihilation, evolving factor analysis and tensorial resolution, all of which can elucidate the nature and concentrations of the species involved, are still, however, very little used, in spite of the their clear advantages for understanding multiequilibria systems. A subject on which many controversies exist in the literature. Here we summarize results of the application of our new SPFAC procedure (based on recent advances in factor analysis techniques) to the study of multiequilibria systems in solution. SPFAC is a new and powerful method for the study of ionic multiequilibria systems; when macromolecules are present it offers an independent approach which avoids the rather cumbersome difficulties intrinsic to the multifactor contributions in such systems.
  
  
• Tauler, R., & Casassas, E. (1992b).
  Spectroscopic resolution of macromolecular complexes using factor-analysis: CU (II)- polyethyleneimine system. Chemometrics and Intelligent Laboratory Systems, 14, 305-317.
  The complexes of Cu(II) ion with polyethyleneimine formed during a VIS spectroscopic titration of solutions containing different amounts of Cu(II) and polyethyleneimine at various pH values have been studied using a new self-resolving approach based on different factor analysis techniques, including evolving factor analysis, error in factor analysis, cross-validation, target factor analysis, and rank annihilation. Three different macromolecular complex species between Cu(II) and polyethyleneimine were detected in the system, and their concentration profiles and individual spectra were estimated without any previous knowledge of the underlying chemical model (set of stoichiometric coefficients and set of stability constants). The method developed is proposed for use in the deduction of the metal complexing properties of macromolecular systems.
  
  
• Tauler, R., Kowalski, B., & Fleming, S. (1993).
  Multivariate curve resolution applied to spectral data from multiple runs of an industrial process. Analytical Chemistry, 65, 2040-2047.
  A method for extracting information from spectroscopic data gathered during process monitoring is described and applied to an industrial problem. The method allows the estimation of the changes in the concentrations of the components in the process as well as their pure spectroscopic responses. Three key aspects of the new method are as follows: (1) the use of evolving factor analysis to have an initial estimation of how the concentrations of the constituents change during the process; (2) the implementation of an alternating and constrained least-squares method to optimize both the spectra and the concentrations of the components in the process; (3) the development of a new approach for the simultaneous analysis of various runs of the same process to estimate the ratio of concentrations between the common components in the different runs.
  
  
• Tauler, R., Lacorte, S., Guillamon, M., Cespedes, R., Viana, P., & Barcelo, D.(2004).
  Chemometric modeling of main contamination sources in surface waters of Portugal. Environmental Toxicology and Chemistry, 23, 565-575.
  Various chemometric data analysis methods, such as principal components analysis, multivariate curve resolution-alternating least squares, parallel factor analysis, and Tucker3, are proposed and compared for the resolution and modeling of main contamination sources in a large environmental data array obtained in an exhaustive environmental monitoring program that examined the quality of surface waters of Portugal. The study covered the analysis of 19 priority semivolatile organic compounds (SVOCs) frequently found in a total number of 644 surface water samples, including 46 different sites from throughout Portugal and corresponding to a period of 14 months, from April 1999 to May 2000. Main contamination sources of the analyzed SVOCs were identified and interpreted according to their chemical composition and according to their resolved geographical and temporal distribution profiles.
  
  
• Tauler R., Marqués, I. & Casassas, E. (1998).
  Multivariate curve resolution applied to three-way trilinear data: Study of a spectrofluorimetric acid-base titration of salicylic acid at three excitation wavelengths. Journal of Chemometrics, 12, 55-76.
  Application of multivariate curve resolution (MCR) is shown for three-way data obtained in acid-base titrations of salicylic acid monitored using emission spectrofluorimetry at three different excitation wavelengths. Rank analysis of the augmented column- and row-wise matrices showed that the experimental fluorescent data have a trilinear structure. MCR allows unambiguous recovery of the species profiles in both orders. Results are compared with those obtained using the trilinear decomposition and generalized rank annihilation methods.
  
  
• Tauler, R., Smilde, A.K., Henshaw, J.M., Burgess, L.W., & Kowalski, B.R. (1994).
  Multicomponent determination of chlorinated hydrocarbons using a reaction-based chemical sensor. 2. Chemical speciation using multivariate curve resolution. Analytical Chemistry, 66, 3337-3344.
  A new multivariate curve resolution method that can extract analytical information from UV/visible spectroscopic data collected from a reaction-based chemical sensor is proposed. The method is demonstrated with the determination of mixtures of chlorinated hydrocarbons by estimating the kinetic and spectral profiles of the chemical species formed in the Fujiwara reaction. The three key aspects of the proposed method are (1) the initial estimation of the kinetic concentration profiles from evolving factor analysis; (2) the implementation of an alternating and constrained least squares method to optimize the determination of both the spectral and concentration profiles of the species present in the reaction, and (3) the development of a quantitative approach based on the simultaneous analysis of standards and unknowns for the determination of the initial concentration of the analytes in the mixtures.
  
  
• Tauler, R., Smilde, A., & Kowalski, B. (1995).
  Selectivity, local rank, three-way data analysis and ambiguity in multivariate curve resolution. Journal of Chemometrics, 9, 31-58.
  A new multivariate curve resolution method is presented and tested with data of various levels of complexity. Rotational and intensity ambiguities and the effect of selectivity on resolution are the focus. Analysis of simulated data provides the general guidelines concerning the conditions for uniqueness of a solution for a given problem. Multivariate curve resolution is extended to the analysis of three-way data matrices. The particular case of three-way data where only one of the orders is common between slices is studied in some detail.
  
  
• Tauler, R., Gargallo, R., Vives., M., & Izquierdo-Ridorsa, A. (1999).
  Resolution of temperature dependent conformational multiequilibria processes. Chemometrics and Intelligent Lagoratory Systems, 46, 275-+.
  Multivariate Curve Resolution (MCR) is applied to the study of temperature dependent conformational multiequilibria evolving processes. Experimental data sets are obtained by UV spectral monitoring of the melting behavior of the heteropolynucleotide poly(adenylic) acid-poly(uridylic) acid (poly(A)-poly(U)) and of the hompolynucleotides poly(adenylic) (poly(A)) acid and poly(urydilic) acid (poly(U)), i.e., recording UV spectra at different temperatures during the melting process of these polynucleotides, Separate study of every individual melting experiment by MCR did not give a satisfactory resolution of the heteropolynucleotide poly(A)-poly(U) melting process because of unresolved rotational ambiguities and rank deficiency problems. Conversely, the simultaneous MCR analysis of the melting process of poly(A)-poly(U) heteropolynucleotide together with the separate melting processes of the poly(A) and poly(U) homopolynucleotides, allowed the resolution of the species profiles and the elimination of the rank deficiency problems present in the individual analysis of the melting behavior of poly(A)-poly(U).
  
  
         
• Ten Berge, J.M.F. (1972).
  Difficulty factors, distribution effects, and the least squares simplex data matrix solution. Educational and Psychological Measurement, 32, 911-920.
  Horst (1965) raises Ferguson's dilemma again. He introduces a method for eliminating distribution factors. The method fails to deal adequately with these factors. A distinction is made between factors resulting from the effect of distribution shape discrepancy on product-moment correlation, and a factor referring to an attribute like "capacity for solving difficult problems." Some well-known alternative solutions for the problem of distribution factors are mentioned.
  
  
• Ten Berge, J.M.F. (1977).
  Orthogonal Procrustes rotation for two or more matrices. Psychometrika, 42, 267-276.
  Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.
  
  
• Ten Berge, J.M.F. (1983).
  A generalization of Kristof's Theorem on the trace of certain matrix products. Psychometrika, 48, 519-523.
  Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form X₁L₁X₂L₂...X_nL_n where the matrices L_i are diagonal and fixed and the X_i vary unrestri Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form X₁L₁X₂L₂...X_nL_n where the matrices L_i are diagonal and fixed and the X_i vary unrestrictedly and independently over the set of orthogonality constraints. The present paper contains a generalization of Kristof's theorem to the case where the X_i are merely required to be submatrices of orthonormal

  

• Ten Berge, J.M.F. (1986a).
  Rotatie naar perfecte congruentie en de multipele groep methode. Nederlands Tijdschrift voor de Psychologie en haar grensgebieden, 41, 218-225.
  This article contains an analysis of component comparisons across populations, using congruence. For any given component in a first population, a parallel component in any second population can be defined which has, up to a constant of proportionality, the same variable-component correlations or the same component scores coefficients, provided that the same variables have been used in the two populations. For this reason, congruence studies should not be concerned with amounts of congruence obtained, but should focus on the behavior of perfectly congruent components in terms of amounts of variance explained. From this perspective, congruence rotation applied to component weights appears to be closely related to Multiple Group Analysis. The present article is a non- technical summary of results published elsewhere in greater detail, and additionally, contains a description of a relevant computer programme.
  
  
• Ten Berge, J.M.F. (1986b).
  Rotation to perfect congruence and the cross-validation of component weights across populations. Multivariate Behavioral Research, 21, 41-64.
  This paper deals with srearegies of congruence studies, aimed at evaluating recoverableness of a given set of components from a first population in a second population where the same variables have been used. Five decisions inherent to congruence studies are analysed in detail. Confirmatory evidence with respect to recoverableness can be obtained from an independent component analysis for the second population, parallel to that of the first population. Disconfirmatory evidence requires oblique rotation to perfect congruence, which can always be attained. Rotation to perfect congruence is advocated as a new strategy, in which amounts of variance explained are of major concern. The perfect congruence strategy can be applied to variable-component correlations and to weights. The latter approach is to be preferred for two reasons. First, rotating weights to perfect congruence can be easily understood as a cross-validation method, closely related to the well-known multiple group method. Second, this approach appears to give more satisfactory results in practical applications than are obtained from rotating variable-component correlations to perfect congruence.
  
  
• Ten Berge, J.M.F. (1986c)
  Some relationships between descriptive comparisons of components from different studies. Multivariate Behavioral Research, 21, 29-40.
  This paper considers descriptive methods of comparing components from different studies, based on correlations between columns of component scores matrices, on congruence coefficients between columns of pattern, structure or component scores coefficient matrices, or on coefficient invariance. Contrary to common belief, it is shown that coefficients of invariance are unrelated to correlations between component scores. On the other hand, having a perfect coefficient of invariance is shown to be equivalent to having a perfect congruence between corresponding columns of the component scores coefficient matrices. A similar but weaker relationship between the latter congruence and congruence between columns of pattern matrices is demonstrated.
  
  
• Ten Berge, J.M.F. (1989).
  Convergence of PARAFAC preprocessing procedures and the Deming-Stephan method of iterative proportional fitting. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 53-63). Amsterdam: Elsevier.
  Before three-way arrays can be analyzed by PARAFAC, the arrays typically have to be preprocessed by removing means and/or standardizing mean squares in certain directions. This paper contains a taxonomy of the main preprocessing problems for PARAFAC. The Deming-Stephan method of iterative proportional fitting is shown to converge, except for a specific class of matrices. The PARAFAC preprocessing method of iterative rescaling within two modes is expressed in a form equivalent to the Deming-Stephan method. Accordingly, the convergence properties obtained for the Deming-Stephan method are shown to hold for iterative rescaling within two modes as well.
  
  
• Ten Berge, J.M.F. (1991a).
  A general solution for a class of weakly constrained linear regression problems. Psychometrika, 56, 601-609.
  This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.
  
  
• Ten Berge, J.M.F. (1991b).
  Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays. Psychometrika, 56, 631-636.
  A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. Kruskal pointed out that a 2 × 2 × 2 array has rank three or less, and that the subsets of those 2 × 2 × 2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2 × n × n arrays. Incidentally, it is shown that two n × n matrices can be diagonalized simultaneously with positive probability.
  
  
• Ten Berge, J. M. F. (2000).
  The typical rank of tall three-way arrays. Psychometrika, 65, 525-532.
  The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank of I × J × K arrays with K = 2 to a particular class of arrays with K ? 2. It is shown that the typical rank is I when the array is tall in the sense that JK - J < I < JK. In addition, typical rank results are given for the case where I equals JK - J .
  
  
• Ten Berge, J. M. F. (2004a).
  Partial uniqueness in CANDECOMP/PARAFAC. Journal of Chemometrics, 18, 12-16.
  A key property of CANDECOMP/PARAFAC is the essential uniqueness it displays under certain conditions. It has been known for a long time that, when these conditions are not met, partial uniqueness may remain. Whereas considerable progress has been made in the study of conditions for uniqueness, the study of partial uniqueness has lagged behind. The only well known cases are those of overfactoring, when more components are extracted than are required for perfect fit, and those cases where the data do not have enough system variation, resulting in proportional components for one or more modes. The present paper deals with partial uniqueness in cases where the smallest number of components is extracted that yield perfect fit. For the case of K x K x 2 arrays of rank K, randomly sampled from a continuous distribution, it is shown that partial uniqueness, with some components unique and others differing between solutions, arises with probability zero. Also a closed-form CANDECOMP/PARAFAC solution is derived for 5 x 3 x 3 arrays when these happen to have rank 5. In such cases, any two different solutions share four of the five components. This phenomenon will be traced back to a sixth degree polynomial having six real roots, any five of which can be picked to construct a solution.
  
  
• Ten Berge, J. M. F. (2004b).
  Simplicity and typical rank of three-way arrays, with applications to Tucker-3 analysis with simple cores. Journal of Chemometrics, 18, 17-21.
  In chemometric applications of Tucker three-way principal component analysis, core arrays are often constrained to have a large majority of zero elements. This gives rise to questions of non-triviality (are the constraints active, or can any core of a given format be transformed to satisfy the constraints?) and uniqueness (can we transform the components in one or more directions without losing the given pattern of zero elements in the core?). Rather than deciding such questions on an ad hoc basis, general principles are to be preferred. This paper gives an overview of simplicity transformations on the one hand, and typical rank results on the other, which are suitable to determine whether or not certain constrained cores are trivial.
  
  
• Ten Berge, J.M.F., Bekker, P.A., & Kiers, H.A.L. (1994).
  Some clarifications of the TUCKALS2 algorithm applied to the IDIOSCAL problem. Psychometrika, 59, 193-201.
  Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/ PARAFAC to fit the INDSCAL model.
  
  
• Ten Berge, J.M.F., De Leeuw, J., & Kroonenberg, P.M. (1987).
  Some additional results on principal components analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 52, 183-191.
  Kroonenberg and De Leeuw (1980) have developed an alternating least- squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. This paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/Harshman CANDECOMP/PARAFAC model.
  
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1989a).
  Convergence properties of an iterative procedure of ipsatizing and standardizing a data matrix, with applications to Parafac/Candecomp preprocessing. Psychometrika, 54, 231-236.
  Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the iterations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in PARAFAC/CANDECOMP.
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1989b).
  Fitting the off-diagonal DEDICOM model in the least-squares sense by a generalization of the Harman and Jones minres procedure of factor analysis. Psychometrika, 54, 333-337.
  Harshman's DEDICOM model provides a framework for analyzing square but asymmetric matrices of directional relationships among n objects or persons in terms of a small number of components. One version of DEDICOM ignores the diagonal entries of the matrices. A straightforward computational solution for this model is offered in the present paper. The solution can be interpreted as a generalized Minres procedure suitable for handling asymmetric matrices.
  
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1990).
  Simultane componenten-analyse voor twee of meer groepen personen. Nederlands Tijdschrift voor de Psychologie, 45, 221-226.
  Principal Components Analysis (PCA) is a familiar technique for summarizing a matrix of standard scores from persons on a set of variables. Occasionally, researchers have to deal with scores from several groups of persons on the same variables. In this paper a generalization of PCA, that provides common components for groups of persons, is described and illustrated. This generalization was first suggested by Millsap and Meredith, and was treated in an alternating least squares framework.
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1991a).
  A numerical approach to the approximate and the exact minimum rank of a covariance matrix. Psychometrika, 56, 309-315.
  A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.
  
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1991b).
  Some clarification of the CANDECOMP algorithm applied to INDSCAL. Psychometrika, 56, 317-326.
  Carroll and Chang have claimed that CANDECOMP applied to symmetric matrices yields equivalent coordinate matrices, as needed for INDSCAL. Although this claim had appeared to be valid for all practical purposes, it has gone without a rigorous mathematical footing. The purpose of the present paper is to clarify CANDECOMP in this respect. It is shown that equivalent coordinate matrices are not granted at global minima when the symmetric matrices are not Gramian, or when these matrices are Gramian but the solution not globally optimal.
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1996a).
  Optimality criteria for principal component analysis and generalizations. British Journal of Mathematical and Statistical Psychology, 49, 335-345.
  Principal components analysis can be derived from various criteria. Because these give essentially the same results, the question of which criterion should be used has not received much attention. Nevertheless, it can be argued that the approach of Pearson and Eckart & Young, based on variance explained by components, is more elegant and flexible than the (more popular) approach of Hotelling, which is concerned with variance that components have, rather than explain. When two or more correlation or covariance matrices, based on the same variables, are to be analyzed in generalized component analysis, the question of which criterion is used becomes of utmost importance. A taxonomy of generalized principal component methods is given. It appears than generalized component analysis based on the Hotelling criterion coincides with one particular generalization based on the criterion of Pearson and Eckart & Young.
  
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1996b).
  Some uniqueness results for PARAFAC2. Psychometrika, 61, 123-132.
  Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2.
  
• Ten Berge, J.M.F., & Kiers, H.A.L. (1999).
  Simplicity of core arrays in three-way principal component analysis and the typical rank of p × q × 2 arrays. Linear Algebra and its Applications, 294, 169-179.
  Interpreting the solution of a Principal Component Analysis of a three-way array is greatly simplified when the core array has a large number of zero elements. The possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand, and by mathematical analysis on the other. In the present paper, it is shown that a p × q × 2 array, with p greater than q greater than or equal to 2, can almost surely be transformed to have all but 2q elements zero. It is also shown that arrays of that form have three-way rank p at most. This has direct implications for the typical rank of p × q × 2 arrays, also when p = q. When p is greater than or equal to 2q, the typical rank is 2q; when q is less than p is less than 2q it is p, and when p=q, the rank is typically (almost surely) p or p+1. These typical rank results pertain to the decomposition of real valued three-way arrays in terms of real valued rank one arrays, and do not apply in the complex setting, where the typical rank of p × q × 2 arrays is also min[p,2q] when p is less than q, but it is p when p=q.
  
  
• Ten Berge, J.M.F., Kiers, H.A.L., & Commandeur, J.J.F. (1993).
  Orthogonal procrustes rotation for matrices with missing values. British Journal of Mathematical and Statistical Psychology, 46, 119-134.
  A method is offered for orthogonal Procrustes rotation of two or more matrices with missing values, and an extension to Generalized Procrustes Analysis is given. The method based on the idea that missing values, in arbitrary places, can be replaced by optimal values according nto the least squares criterion. Commandeur has offered a method of orthogonal Procrustes rotation and Generalized Procrustes Anlysis for the case where entire rows of the data matrices are missing etc.
  
  
• Ten Berge, J.M.F., Kiers, H.A.L., & De Leeuw, J. (1988).
  Explicit candecomp/parafac solutions for a contrived 2x2x2 array of rank three. Psychometrika, 53, 579-584.
  Kruskal, Harshman and Lundy have contrived a special 2 x 2 x 2 array to examine formal properties of degenerate CANDECOMP/PARAFAC solutions. It is shown that for this array the CANDECOMP/PARAFAC loss has an infimum of 1. In addition, the array will be used to challenge the tradition of fitting INDSCAL and related models by means of the CANDECOMP/PARAFAC process.
  
• Ten Berge, J.M.F., Kiers, H.A.L., & Krijnen, W.P. (1993).
  Computational solutions for the problem of negative saliences and nonsymmetry in INDSCAL. Journal of Classification, 10, 115-124.
  Carroll and Chang have derived the symmetric CANDECOMP model from the INDSCAL model, to fit symmetric matrices of approximate scalar products in the least squares sense. Typically, the CANDECOMP algorithm is used to estimate the parameters. In the present paper it is shown that negative weights may occur with CANDECOMP. This phenomenon can be surpressed by updating the weights by the Nonnegative Least Squares Algorithm. A potential drawback of the resulting procedure is that it may produce two different versions of the stimulus space matrix. To obviate this possibility, a symmetry preserving algorithm is offered, which can be monitored to produce non-negative weights as well.
  
  
• Ten Berge, J.M.F., Kiers, H.A.L., Murakami, T., & Van der Heijden, R. (2000).
  Transforming three-way arrays to multiple orthonormality. Journal of Chemometrics, 14, 275-284.
  This paper is concerned with the question to what extent the concept of rowwise or columnwise orthonormality can he generalized to three-way arrays. Whereas transforming a three-way array to multiple orthogonality is immediate, transforming it to multiple orthonormality is far from straightforward. The present paper offers an iterative algorithm for such transformations, and gives a proof of monotonical convergence when only two modes are orthonormalized. Also, it is shown that a variety of three-way arrays do not permit double orthonormalization. This is due to the order of the arrays, and holds regardless of the particular elements of the array. Studying three-way orthonormality has proven useful in exploring the possibilities for simplifying the core, to guide the search for equivalent direct transformations to simplicity; see Murakami et al. (1998) as an example. Also, it appears in various contexts of the mathematical study of three-way analysis.
  
  
• Ten Berge, J.M.F., Kiers, H.A.L., & Van der Stel, V. (1992).
  Simultaneous components analysis. Statistica Applicata, 4, 377-392.
  Simultaneous Components Analysis is a generalization of Principal Components Analysis for the situation where the same variables have been measured in two or more populations. A common matrix of component weights is used to ensure that the components are the same linear combinations of the same variables in each population. This common matrix of weights is determined such that the explained variance, summed over populations, is a maximum. The present paper deals with certain unknown properties of simultaneous components. The main result is that one always finds simultaneous components that explain a considerable amount of variance. Furthermore, it will be shown that neither using small samples, nor replacing optimal weights by simple weights, need to detract seriously from the usefulness of simultaneous components.
  
• Ten Berge, J.M.F., Knol, D.L., & Kiers, H.A.L. (1988).
  A treatment of the orthomax rotation family in terms of diagonalization, and a re-examination of a singular value approach to varimax rotation. Computational Statistics Quarterly, 3, 207-217.
  The two most familiar members of the Orthomax rotation family are Varimax and Quartimax rotation. It is well-known that Varimax rotation can be interpreted as diagonalizing a set of symmetric matrices simultaneously in the least-squares sense. This paper contains a generalization of this result to the entire family of Orthomax rotations. It follows that Varimax and Quartimax rotations can be carried out by applying the same rotation angle formulas to differently derived symmetric matrices. The Horst-Sherin method of Varimax rotation is based on iterative singular value decompositions. This method is re-examined as a diagonalization method. It is shown that the method converges monotonely if the symmetric matrices to be diagonalized are positive or negative semi-definite, and that this condition can always be implemented by properly adjusting the diagonal entries of the symmetric matrices. The SVD-approach can also be generalized to the case of column-wise orthonormal 'rotation' matrices, thus providing a solution for the orthogonally constrained case of INDSCAL.

  
  

• Ten Berge, J.M.F., & Sidiropoulos, N. D.(2002a).
  On uniqueness in CANDECOMP/PARAFAC. Psychometrika, 67, 399-409.
  One of the basic issues in the analysis of three-way arrays by CANDECOMP/ PARAFAC (CP) has been the question of uniqueness of the decompositon. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of the k-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP ssolution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It wil be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.

  
  

• Ten Berge, J. M. F. & Smilde, A. K. (2002b)
  Non-triviality and identification of a constrained Tucker3 analysis Journal of Chemometrics, 16, 609-612.
  Properties of estimated parameters of models of chemical systems are important. This paper focuses on two properties of such estimated parameters: triviality and uniqueness. If a chemical system is analyzed using a Tucker3 model, then the resulting core can often be rotated to a simple structure containing zeros. This means that it is possible that a prespecified pattern of zero and non-zero elements of the core, as used in e.g. constrained Tucker3 models, is not an active constraint, that is, the zeros can be obtained trivially for free. Once a non-trivial pattern of zeros in the core is specified, the question arises whether this pattern is sufficient for obtaining unique loadings. Both issues are discussed in this paper and it is shown that the model used by Gurden et al. (J. Chemometrics 2001; 15: 101-121) does essentially involve a non-trivial core and implies rotationally unique parameter estimates. Copyright (C) 2002 John Wiley Sons, Ltd.

  
  

• Ten Berge, J. M. F., & Sidiropoulos, N. D. (2002).
  On uniqueness in CANDECOMP/PARAFAC. Psychometrika, 67, 399-409.
  One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of the k-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank I or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher.

• Ten Berge, J. M. F., Sidiropoulos, N. D., & Rocci, R. (2004).
  Typical rank and indscal dimensionality for symmetric three-way arrays of order Ix2x2 or Ix3x3. Linear Algebra and its Applications, 388, 363-377.
  A peculiar property of three-way arrays is that the rank they typically have does not necessarily coincide with the maximum possible rank, given their order. Typical tensorial rank has much been studied over algebraically closed fields. However, very few results have been found pertaining to the typical rank of three-way arrays over the real field. These results refer to arrays sampled randomly from continuous distributions. Arrays that consist of symmetric slices do not fit into this sampling scheme. The present paper offers typical rank results (over the real field) for arrays, containing symmetric slices of order 2 x 2 and 3 x 3. Symmetric arrays often appear to have lower typical ranks than their asymmetric counterparts. This paper also examines whether or not the rank of a symmetric array coincides with the smallest number of dimensions that allow a perfect fit of INDSCAL. For all cases considered, this is indeed true. Thus, a full INDSCAL solution may require fewer dimensions for a symmetric array than a full CP decomposition applied to an asymmetric array of the same size. The reverse situation does not seem to arise. Next, we examine in which cases CP solutions inevitably are INDSCAL solutions. Finally, the rank-reducing impact of double standardizing the slices is discussed.

• Teppola, P., Mujunen, S. P., & Minkkinen, P. (1999).
  Addaptive Fuzzy C.Means clustering in process monitoring. Chemometrics and Intelligent Laboratory Systems, 45, 23-38.
  Quite often, quality control models fail because, e.g., the mean values are changing continuously. These kinds of changes, e.g., process drifts due to seasonal fluctuations, are common in an activated sludge waste-water treatment plant in Finland. Different Fuzzy C-Means (FCM) clustering algorithms were tested in order to cope with these kinds of seasonal effects. Firstly, a Principal Component Analysis (PCA) model was constructed in order to visualize the data set and reduce the dimensionality of the problem. Then, score values of the PCA were used in the FCM. The cluster centers represented the different process conditions (winter and summer seasons). Different algorithms were used to update the cluster centers or to give them some flexibility. The testing of different FCM algorithms was carried out by using a separate test set. The adaptive and the flexible FCM algorithms were compared to the basic non-adaptive FCM. For both cases, modifications are proposed and a simple strategy for updating the cluster centers is given.

• Teppola, P., & Minkkinen, P. (2000).
  Wavelet-PLS regression models for both exploratory data analysis and process monitoring. Journal of Chemometrics, 14, 383-399.
  Two novel approaches an presented which take into account the collinearity among variables and the different phenomena occurring at different scales. This is achieved by combining partial least squares (PLS) and multiresolution analysis (MRA). In this work the two novel approaches are interconnected. First, a standard exploratory PLS model is scrutinized with MRA. In this way, different events at different scales and latent variables are recognized. In this case, especially periodic seasonal fluctuations and long-term drifting introduce problems. These low-frequency variations mask and interfere with the detection of small and moderate-level transient phenomena. As a result, the confidence limits become too wide. This relatively common problem caused by autocorrelated measurements can be avoided by detrending. In practice, this is realized by using fixed-size moving windows and by detrending these windows Based on the MRA of the standard model, the second PLS model for process monitoring is constructed based on the filtered measurements. This filtering is done by removing the low-frequency scales representing low-frequency components, such as seasonal fluctuations and other long-term variations, prior to standard PLS modeling. For these particular data the results are shown to be superior compared to a conventional PLS model based on the non-filtered measurements. Often, model updating is necessary owing to non-stationary characteristics of the process and variables. As a big advantage, this new approach seems to remove any further need for model updating, at least in this particular case. This is because the presented approach removes low-frequency fluctuations and results in a more stationary filtered data set that is more suitable for monitoring.

• Terbeek, D., & Harshman, R. (1971).
  Cross-language differences in the perception of natural vowel sounds. UCLA Working Papers in Phonetics, 19, 26-38.
  One of the first applications of Harshman's Parafac algorithm and program. Twelve by twelve stimulus similarity matrices were presented to 5 German, 6 Thai and 6 English speakers. Separate Parafac analyses were used for each language group. The presence or absence of uniqueness was used to determine the dimensionality.
  
  
• Teufel, S. (1969).
  TUCK, Tuckers Modell einer drei-dimensionalen Faktorenanalyse. Ein FORTRAN IV-Programm. In F. Gebhardt, Statistische Programme des DRZ. Teil B: Einzelbeschreibungen. Programm Information PI-33 des Deutschen Rechenzentrum, Darmstadt.
  
       
• Thake, A. J., McLellan, P.J. & Forbes, J.F. (1999).
  Controller approximation using semidefinite programming. Industrial & Engineering Chemistry Research, 38, 2699-2708.
  Controller design, and the resulting control system performance, is frequently limited by the capabilities of the available control system hardware. When the control system hardware provides only a small set of fixed controller structures (e.g., proportional-integral-derivative, PID) or limited programming capabilities, it may be desirable to use the available functional elements in a manner such that the implemented fixed structure controller matches the performance of some reference advanced controller as closely as possible. This paper presents a systematic method for approximating the performance of an advanced controller with simpler fixed structure controllers using semidefinite programming. Tuning of the simple, fixed structure controller is obtained by matching a specific desired behavior of the advanced controller. The proposed method is illustrated using two case studies: the approximation of a linear quadratic regulator for a distillation column by a multiloop PID controller and cross-direction profile control for a paper machine.
  
• Thomas, K., & Otway, H.J. (1980).
  Public perceptions of energy system risks: Some policy implications. In T. O'Riordan & K. Turner, Progress in resource management and environmental planning, Vol. 2 (pp. 109-131). New York: Wiley.
  A questionnaire of 39 beliefs with respect to 5 energy sources was administered to 234 Austrian subjects via a Tucker 3 analysis. A solution of three source components, three subject components (the first two of which were considered related to response style), and five belief components was chosen for interpretation. For details reference is made to an unpublished report Thomas et al. (1980) [not in bibliography].
  
  
• Thomas, P.R., & Bain, J.D. (1982).
  Consistency in learning strategies. Higher Education, 11, 249-259.
  This article examines the reported use of surface and deep level learning strategies by first-year student teachers at an Australian College of Advanced Education. Students responded to a brief questionnaire measuring the learning strategies they adopted in different assessment situations. The article describes the development of this questionnaire, its factorial structure, and the predictive validity of its factors. High livels of achievement, on both objective tests and essay assignments, were found to be associated with the reported use of deep strategies. Three-mode factor analyses revealed high levels of consistency in the strategies reported for various learning contexts, implying that these were stylistic behaviours rather than strategic approaches to learning which were situation specific. The notion of consistency in learning strategies was considered in light of recent literature suggesting a greater extent of cognitive flexibility.
  
  
• Thomas, P.R., & Bain, J.D. (1984).
  Contextual dependence of learning approaches: The effects of assessments. Human Learning, 3, 227-240.
  A conceptual framework is presented in which learning activities are described in terms of cognitive operations applied to various features of study material. This framework was used to generate a learning activities questionnaire which college students completed after each of four assessments in their course. Three-mode factor analyses of the intercorrelations among learning activities identified transformational, reproductive, and skimming approaches to learning. Closed and open-ended assessment contexts were also identified from relationships among course assessments. Finally, three-mode procedures were used to establish the correlational interactions of the approach and context factors. In general, the two main approaches (transformational and reproductive) varied directly in each of the assessment contexts, contrary to the view (e.g. Marton) that they are mutually exclusive.
  
  
• Thygesen, L. G., Rinnan, R., Barsberg, S., & Moller, J. K. S. (2004).
  Stabilizing the PARAFAC decomposition of fluorescence spectra by insertion of zeros outside the data area. Chemometrics and Intelligent Laboratory Systems, 71, 97-106.
  The use of fluorescence spectroscopy for recording multiple excitation and corresponding emission wavelengths and the subsequent technique of analyzing the resulting fluorescence landscapes is a rather new method as opposed to the use of just a single excitation wavelength. In a fluorescence landscape, several light-scatter effects are usually present, and often the part of the landscape containing information on the chemical and/or physical characteristics of the sample is surrounded by two Rayleigh scatter lines. When such landscapes are decomposed using parallel factor analysis (PARAFAC) , the scatter effects may have detrimental effects on the resolved spectra, especially if the peaks from the analytes lie close to or on the Rayleigh scatter lines. Normally, all values close to and outside the Rayleigh scatter lines are set to missing values before decomposing the fluorescence landscapes by PARAFAC. In this paper, we introduce a novel pretreatment method applicable for two-dimensional fluorescence landscapes, where instead of inserting only missing values a mixture of zeros and missing values are inserted close to and outside the Rayleigh scatter lines. It is shown that, by the use of this technique, a physically and chemically meaningful decomposition is obtained, and furthermore the modeling converges faster. Constraining the PARAFAC solution to positive values in all modes gave results similar to those obtained for the unconstrained model, except that the loadings where less smooth and the number of iterations before convergence was smaller.
   
• Timmerman, M. E. (2001).
  Component analysis of multisubject multivariate longitudinal data. PhD. Dissertation, Department of Psychology, University of Groningen, Groningen, The Netherlands. Contents
  1. Introduction
  Longitudinal research; Sampling from the longitudinal axis; Two types of multisubject longitudinal data: longitudinal two-way data and multiple time series; Two types of multivariate multisubject longitudinal data
  2. Component models for longitudinal three-way data
  Introduction; Notational issues and some matrix algebraic properties; Component models for three-way data;Fitting the component models for three-way data; Transformational freedom within the component models for three-way data; Issues in the application of the component models to three-way data; Preprocessing three-way data; Choice of a specific model and the numbers of components; Interpretation of three-way component models applied to longitudinal three-way data.
  3. Occasion components as evaluations of latent curves: possibilities for constraints to the time mode
  4. The CP and Tucker3 models with smoothness constraints
  Introduction; The choice of a smoother; How to smooth in the Tucker3 model and CP model?; Comparing constrained with unconstrained CP and Tucker3 models; Construction of the data for the simulation study; Analyses of simulation data; Criteria of interest; Results of the simulation studies; Empirical example: Learning to read study (I); Discussion and conclusion.
  5. Structured latent curve component models for longitudinal three-way data
  Introduction; Structured latent curve two-way component models for growth data; The SLC two-way component model for data measured at equal time points; Fitting the SLC two-way component model to data with equal measurements; The SLC two-way component model for growth data measured at unequal time points; The SLC Tucker3 model for longitudinal three-way data; Fitting the SLC Tucker3 model to data; Transformational freedom and interpretation in the SLC Tucker3 model; Empirical example: Learning to read study (II); Discussion and conclusion.
  6. Simultaneous Component Models of Multisubject Multivariate Time Series
  Introduction; Four models for simultaneous component analysis; Preprocessing of raw data before fitting the SCA model to data; SCA with invariant Pattern (SCA-P); Constrained versions of SCA-P; SCA with PARAFAC2 constraints (SCA-PF2); SCA with INDSCAL constraints (SCA-IND); SCA with Equal average Cross-Products constraints (SCA-ECP); Transformational freedom in the SCA-P, SCA-PF2, SCA-IND and SCA-ECP models; Model selection; Fitting the four SCA models to the data; Fitting the SCA-PF2 model; Fitting the SCA-IND model to data; Fitting the SCA-ECP model to data; Starting values of the parameters; Empirical examples of simultaneous component analyses; Empirical example 1: Mood in individuals with Parkinson's disease; Empirical example 2: The Big Five as states; Discussion and conclusion.
  7. Lagged Simultaneous Component Models of Multisubject Multivariate Time Series
  Introduction; The dynamic factor model; Lagged SCA models; LSCA-P; LSCA-PF2, LSCA-IND and LSCA-ECP; Fitting the LSCA models to data; Transformational freedom in the LSCA models; Testing the LSCA-P and LSCA-IND algorithms; Empirical example: Mood in individuals with Parkinson's disease; Discussion and conclusion.
  8. Conclusion
  Summary; Discussion and future work
  
  
• Timmerman, M.E., & Kiers, H.A.L. (2000).
  Three-mode principal components analysis: Choosing the numbers of components and sensitivity to local optima. British Journal of Mathematical and Statistical Psychology, 53, 1-16.
  A method that indicates the numbers of components to use in fitting the three- mode principal components analysis (3MPCA) model is proposed. This method, called DIFFIT, aims to find an optimal balance between the fit of solutions for the 3MPCA model and the numbers of components. The achievement of DIFFIT is compared with that of two other methods, both based on two-way PCAs, by means of a simulation study. It was found that DIFFIT performed considerably better than the other methods in indicating the numbers of components. The 3MPCA model can be estimated by the TUCKALS3 algorithm, which is an alternating least squared algorithm. In a study of how sensitive TUCKALS3 is at hitting local optima, it was found that, if the numbers of components are specified correctly, TUCKALS3 never hits a local optimum. The occurrence of local optima increased as the difference between the numbers of underlying components and the numbers of components as estimated by TUCKALS3 increased. Rationally initiated TUCKALS3 runs hit local optima less often than randomly initiated runs.
  
  
• Timmerman, M.E., & Kiers, H.A.L. (2002).
  Three-way component analysis with smoothness constraints. Computational Statistics & Data Analysis, 40, 447-470.
  Tucker3 Analysis and CANDECOMP/PARAFAC (CP) are closely related methods for three-way component analysis. Imposing constraints on the Tucker3 or CP solutions can be useful to improve estimation of the model parameters. In the present paper, a method is proposed for applying smoothness constraints on Tucker3 or CP solutions, which is particularly useful in analysing functional three-way data. The usefulness of smoothness constraints on Tucker3 and CP solutions is examined by means of a simulation experiment. Generally, the results of the experiments indicate better estimations of the model parameters. An empirical example illustrates the use of smoothness constraints. The constrained model is more stable and easier to interpret than the unconstrained model. (C) 2002 Elsevier Science B.V. All rights reserved.
  
  
• Timmerman, M. E., & Kiers, H. A. L. (2003).
  Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences. Psychometrika, 68, 105-121.
  A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.
             
• Togkalidou, T., Tung, H. H., Sun, Y. K., Andrews, A., & Braatz, RD(2005).
  Solution concentration prediction for pharmaceutical crystallization processes using robust chemometrics and ATR FTIR spectroscopy. Organic Process Research & Development, 6, 317-322.
  In the pharmaceutical industry, a vast number of compounds are produced by solution crystallization, making the design and development of such processes of critical importance. The kinetics of crystal growth and nucleation, the fundamental mechanisms of a solution crystallization process, are strongly dependent on supersaturation (the difference between solution concentration and the saturation concentration). The present study uses attenuated total reflection (ATR) Fourier transform infrared (FTIR) spectroscopy, coupled with robust chemometric techniques, for the on-line measurement of solution concentration of two pharmaceutical compounds in multicomponent systems in the presence of impurities and over a wide range of temperature. To our best knowledge, this is the first time that ATR FTIR spectroscopy has been applied to a multicomponent pharmaceutical system. The resulting models show high accuracy, in predicting the solution concentration and are applied successfully in measuring the solubility for the cases of cooling and antisolvent crystallization.
  
  
• Tomasi, G., & Bro, R. (2005).
  PARAFAC and missing values. Chemometrics and Intelligent Laboratory Systems, 75, 163-180.
  Missing values are a common occurrence in chemometrics data, and different approaches have been proposed to deal with them. In this work, two different concepts based on two algorithms are compared in their efficiency in dealing with incomplete data when fitting the PARAFAC model: single imputation (SI) combined with a standard PARAFAC-alternating least squares (ALS) algorithm, and fitting the model only to the existing elements using a computationally more expensive method (Levenberg–Marquadt) appropriately modified and optimised. The performance of these two algorithms and the effect of the incompleteness of the data on the final model have been evaluated on the basis of a Monte Carlo study and real data sets with different amounts and patterns of missing values (randomly missing values, randomly missing spectra/vectors, and systematically missing spectra/vectors). The evaluation is based on the quality of the solution as well as on computational aspects (time requirement and number of iterations). The results show that a PARAFAC model can be correctly determined even when a large fraction of the data is missing (up to 70%), and that the pattern matters more than the fraction of missing values. Computationally, the Levenberg–Marquadt-based approach appeared superior for the pattern of missing values typical of fluorescence measurements when the fraction of missing elements exceeded 30%.
       
• Treat, T., McRall, R. M., Viken, R. J., Nosofsky, R. M., MacKay, D. B., & Kruschke, J. K. (2002).
  Assessing clinically relevant perceptual organization with multidimensional scaling techniques. Psychological Assessment, 14, 239-252.
  Multidimensional scaling (MDS) techniques provide a promising measurement strategy for characteriz-ing individual differences in cognitive processing, which many clinical theories associate with the development, maintenance, and treatment of psychopathology. The authors describe the use of deter-ministic and probabilistic MDS techniques for investigating numerous aspects of perceptual organization, such as dimensional attention, perceptual correlation, within-attribute organization, and perceptual variability. Additionally, they discuss how formal quantitative models can be used, in conjunction with MDS-derived representations of individual differences in perceptual organization, to test theories about the role of cognitive processing in clinically relevant phenomena. They include applied examples from their work in the areas of eating disorders and sexual coercion.
  
  
• Trendafilov, N.T. (2002).
  GIPSCAL revisited. A projected gradient approach. Statistics and Computing, 12, 135-145.
  A model for analysis and visualization of asymmetric data-GIPSCAL-is reconsidered by means of the projected gradient approach. GIPSCAL problem is formulated as initial value problem for certain first order matrix ordinary differential equations. This results in a globally convergent algorithm for solving GIPSCAL. Additionally, first and second order optimality conditions for the solutions are established. A generalization of the GIPSCAL model for analyzing three-way arrays is also considered. Finally, results from simulation experiments are reported.
  
  
• Trendafilov, N.T. & Lippert, R.A.(2002).
  The multimode Procrustes problem. Linear Algebra and its Applications, 349, 245-264.
  In this paper, we consider a generalization of the well-known Procrustes problem relevant to principal component analysis of multidimensional data arrays. This multimode Procrustes problem is a complex constrained minimization problem which involves the simultaneous least-squares fitting of several matrices. We propose two solutions of the problem: the projected gradient approach which leads to solving ordinary differential equations on matrix manifolds, and differential-geometric approach for optimization on products of matrix manifolds. A numerical example concerning the three-mode Procrustes illustrates the developed algorithms. (C) 2002 Elsevier Science Inc. All rights reserved.
  
  
• Trevisan, M. G., & Poppi, R. J. (2003).
  Determination of doxorubicin in human plasma by excitation-emission matrix fluorescence and multi-way analysis. Analytica Chimica Acta, 493,69-81.
  Direct determination of doxorubicin (DXR), a cytotoxic anthracycline antibiotic, in human plasma was accomplished based on excitation-emission matrix (EEM) fluorescence measurements and multi-way chemometric methods based on parallel factor analysis (PARAFAC) and N-PLS. Several different procedures, such as residual analysis, core consistency diagnostic (CONCORDIA) and split-half analysis were employed to determine the correct number of factors in PARAFAC. These procedures converged to a choice of two factors, attributed to DXR and to the sum of two fluorescence species present in the plasma. Sample PARAFAC loadings were employed to build a regression model against concentration, resulting in a RMSECV of 0.060 mug ml(-1). N-PLS using two factors produced a RMSECV of 0.045 mug ml(-1). Figures of merit (FOM), such as sensitivity (SEN), selectivity (SEL) and limit of detection (LD) were determined for both PARAFAC and N-PLS.
  
  
• Triandis, H.C. (Ed.) (1972).
  The analysis of subjective culture. (pp. 47-51). New York, NY: Wiley-Interscience.
  Contains a summary of an unpublished study by Triandis et al. (1967) on interpersonal attitudes (p.49,50), and a completely non-technical and non-numerical detailed description of results of obtained with T3 (H.C. Triandis & V. Vassiliou, A comparative analysis of subjective culture, p.299-335).
  
• Triandis, H.C. (1977).
  Subjective culture and interpersonal relations across cultures. Annals of the New York Academy of Sciences, 285, 418-434.
  
  
• Triandis, H.C., Feldman, J.M., Weldon, D.E. & Harvey, W.M. (1975).
  Ecosystem distrust and the hard-to-employ. Journal of Applied Psychology, 60, 44-56.
  T3 was used as one of the analyses in a large scale project. It was employed to examine a concepts x judgments x subjects matrix. The precise dimensions are not given. The results are only partially presented, and the effectiveness of the method for the data is difficult to assess or check.
  
• Triandis, H.C., Malpass, R.S., & Feldman, J. (1976).
  Method and a sample of results. In H.C. Triandis (Ed.), Variations in black and white perceptions of the social environment. Urbana, IL: University of Illinois Press.
  In chapter 5 an overview is given of the methods of data collection and analysis which are the basis of the other chapters. As an example, a three-mode analysis of 20 behavioral item scales, 104 role pairs, 89 individuals based on cross-products matrices is included. Oblique factor rotations with counterrotation of core matrix.
  
  
• Triandis, H.C., Tucker, L.R., Koo, P. & Stewart, T. (1967).
  Three-mode factor analysis of the behavioral component of interpersonal attitudes. Technical Report No.50, Department of Psychology, University of Illinois, Urbana, Ill.
  250 subjects from Japan, India and USA responded to semantic and behavioral differential scales. The interpersonal attitudes of the subjects were assessed with respect to stimulus persons varying in sex, age, occupation, and religion. In part reported in H.C. Triandis (Ed.), 1972, p.49- 50.
  
  
• Trick, L., & Katz, A.N. (1986).
  The domain interaction approach to metaphor processing: Relating individual differences and metaphor characteristics. Metaphor and Symbolic Activity, 1, 185-213.
  Tourangeau and Sternberg (1981) proposed a sophisticated model to explain how people comprehend and appreciate metaphors of the form "Robert Redford is the peacock of actors". This study conceptually extended this work by incorporating methodological improvements that permit the examination of individual differences. The paper contains an appendix discussing PARAFAC analyses and a complete four-factor solution.
  
  
   
• Tsuychiya, T. (1996).
  Scaling methods for qualitative three-mode data by partitioning items into G groups. Japanese Journal of Educational Psychology, 44, 425-434.
  This paper proposes three scaling methods for qualitative three-mode three-may data. A real data set collected to investigate the impression of pictures by the semantic differential (SD) method is analyzed as numerical examples. The data consists of three modes; raters, SD items and pictures. All the models classify I items into G groups to construct uni- dimensional scale in each group by introducing fuzzy c-means criterion into homogeneity analysis. In the first model, each scale score is expressed as a function of a design matrix. The data is analyzed assuming that scale scores of all raters to the same picture are equal. Appropriate scales are, however, not constructed indicating that there are differences among the raters. In the second model, rank of scale scores is restricted to R(g) in order to explore the differences of raters. In the third model, raters are clustered into D groups to find which raters are different. The selection of R(g) or D is perfomed by means of increasing the parameter value until appropriate classification is obtained.
     
• Tu, X.M. (1995).
  Nonparametric-estimation of survival distributions with censored initiating time, and censored and truncated terminating time: application to transfusion data for acquired-immune-deficiency-syndrome. Applied statistics, 44, 3-16.
  The analysis of survival data with censored initiating time, and censored and truncated terminating time arises in same recent epidemiological studies. The transfusion-related acquired immune deficiency syndrome (AIDS) data of the centers for disease control (CDC) are a typical example. The initiating time in this case is the time of infection by the human immunodeficiency virus and is not observed for every patient either because of unrecorded transfusion times or multiple transfusions. The terminating time here is the onset of AIDS and is truncated, the result at being able to report within an observational period only a proportion of the infected cases which came down with AIDS in the time period. We consider nonparametric estimation of the survival as well as the initiating time distributions assuming that they are independent and non-informatively censored. We propose a simple algorithm to obtain the maximum likelihood estimates for the discrete formulation of the problem and apply it to estimating the aids latency and infection distributions for four age groups of transfusion-related aids from the CDC surveillance database.
  
  
• Tu, X.M., & Burdick, D.S. (1992).
  Resolution of trilinear mixtures: Application in spectroscopy. Statistica Sinica, 2, 577-593.
  Recent methodological development in three-way array analysis allows unique resolution of multicomponent linear mixtures in the absence of noise. When noise is present, an iterative linearization procedure based on a least squares formulation was proposed by appellof and davidson, but it requires good initial guesses which may be difficult to find in practical situations. In the paper, we describe an alternative procedure based on eigenanalysis, and we discuss the relationship between the two procedures. Even though the alternative procedure does not aim to minimize the squared residuals, it is however, non-iterative an may be used to find initial values for the least squares procedure. An example using phase-resolved fluorescence spectroscopy data seems to indicate that only minor improvement may be expected from using the least squares procedure, and, for practical purposes, the estimates from the eigenanalysis procedure may be close enough to dispense with the least squares procedure.
  
  

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• Tu, X.M., Meng, X.L., & Pagano, M. (1994).
  On the use of conditional maximization in chemometrics. Journal of Chemometrics, 8, 365-370.
  The purpose of this short communication is to illustrate the use of conditional maximization (CM) in chemometric applications. The CM algorithm is useful in reducing the computational complexity when a high-dimensional and complicated maximization problem arises from fitting chemometric models. It can also be efficiently combined with the expectation-maximization (EM) algorithm for handling incomplete data, a problem that sometimes arises when only a part of the intended data can be collected. Three models from fluorescence spectroscopy are used for illustration.
  
  
• Tucker, L.R.(1958).
  Determination of parameters of a functional relation by factor analysis.Psychometrika, 23, 19-23.
  Consideration is given to determination of parameters of a functional elation between two variables by the means of factor analysis techniques. If the function can be separated into a sum of products of functions of the individual parameters and corresponding functions of the independent variable, particular values of the functions of the parameters and of the functions of the independent variables might be found by factor analysis. Otherwise approximate solutions may be determined. These solutions may represent important results from experimental investigations.
  
  
• Tucker, L.R. (1963a).
  Implications of factor analysis of three-way matrices for measurement of change. In C.W. Harris (Ed.), Problems in measuring change. Madison: University of Wisconsin Press, (Pp.122-137).
  The first introduction in the literature of T3. The main principles are presented, and some computational problems are discussed. The missing details appear in later papers (Tucker, 1964, 1966; Levin, 1965), illustrated with an articial example.
  
• Tucker, L. R. (1963b).
  An individual differences model for multidimensional scaling.Psychometrika, 28, 333-367.
  A quantitative system is presented to permit the determination of separate multidimensional perceptual spaces for individuals having different viewpoints about stimulus interrelationships. The structure of individual differences in the perception of stimulus relationships is also determined to provide a framework for ascertaining the varieties of consistent individual viewpoints and their relutionships with other variables.
  
  
• Tucker, L.R. (1964).
  The extension of factor analysis to three- dimensional matrices. In H. Gullikson & N. Frederiksen (Eds.), Contributions to mathematical psychology. New York: Holt, Rinehart and Winston (pp. 110-119).
  A more detailed discussion of T3 than Tucker's (1963) paper, but the mathematics are still somewhat awkward. The transformational freedom in the model is also treated in more detail. The same artificial example is used as in 1963.
  
  
• Tucker, L.R. (1966a).
  Experiments in multimode factor analysis. In A. Anastasi (ed.), Testing Problems in Perspective (pp. 369- 379), Washington, DC: American Council on Education. Reprinted from: Proceedings of the 1964 Invitational Conference on Testing Problems. Princeton, N.J.: Educational Testing Service, 1965 (Pp. 46-57).
  Presents a very global description of T3. Illustrated by analyses presented in detail in Hoffman & Tucker (1964), Tucker (1967), and Levin (1963).
  
• Tucker, L.R. (1966b).
  Learning theory and multivariate experiment. Illustration by determination of generalized learning curves. In R.B. Cattell (Ed.), Handbook of multivariate experimental psychology (pp. 476-501). Chicago, IL: Rand McNally.
  This paper is a precursor of Tucker's work in three-mode component analysis. Several groups were analysed separately and the results were informally compared after rotation for learning curves.
  
  
• Tucker, L.R. (1966c).
  Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279-311.
  The basic paper on T3 which presents a detailed and mathema- tically coherent description of the model. Also a full discussion on the freedom of rotation is given. It contains an outline of the notation which is used by many later authors, and introduces some new terminology. Three computational methods are presented, all in the spirit of principal component analysis rather than factor analysis. Method I is a straightforward application of the basic formulas. Method II and III are applicable to the analysis of data with one (very) large mode. Method III is particular appropriate for 'multitrait-multimethod' type matrices. An extension of the general method is described in which allowance is made for unique variances for each combination variable. This model uses a version of Method III for the analysis. A fictitious body of data is used to illustrate several points.
  
  
• Tucker, L.R. (1967).
  Three-mode factor analysis of Parker-Fleishman complex tracking behavior data. Multivariate Behavioral Research, 2, 139-151.
  T3 was used to analyse data from a simulation of tracking during airborne radar intercept missions (4 accuracy measures, 10 stages of practices and 203 individuals). More in particular the available multimeasures-multistage method matrix of correlations with estimated communalities was used with Tucker's method III (1966). Extensive information on, and interpretation of the solution.
  
• Tucker, L.R. (1972a).
  Relations between multidimensional scaling and three-mode factor analysis. Psychometrika, 37, 3-27(a).
  The presented multidimensional scaling variant of T3, which later came to be called 'three-mode scaling', assumes that the first two modes are equal, and that the input consists of a scalar product matrix for each of subjects. This results in an object space, a person space and a core matrix, of which the frontal planes contain the subject components' weights for the components of the common object space as well as their angles between those dimensions. Also the extended core matrix is in- troduced as an aid for interpretation. Some special transfor- mations of the core matrix are suggested. The technique is illustrated with data from an adjective similarity study (all 66 pairs from 12 adjectives were judged as to their similarity by 87 students). The adjectives were designed to form a circle, which indeed was found, but not properly recognized. The interpretation of the core matrix and person space was made by using 'conceptual' individuals.
  
  
• Tucker, L.R. (1972b).
  Use of three-mode factor analysis in MDS. Paper presented at the Workshop on Multidimensional Scaling, University of Illinois, 7-10 June.
  Summary presentation of theory in Tucker (1972a). Illustrations from Helm's color data and Wish' relations- between-nations data.
  
  
• Tucker, L.R. (1975).
  Three-mode factor analysis applied to multidimensional scaling. Paper presented to the U.S.-Japan Seminar on Theory, Methods, and Applications of Multidimensional Scaling and Re- lated Techniques, La Jolla, Calif., August 20-24.
  A concise overview of three-mode factor analysis, the T2 model, three-mode scaling. Illustrated with artificial data.
  
• Tucker, L.R. (1992).
  Remarks on the study of the variety of individuals. Multivariate Behavioral Research, 27, 635-647.
  Techniques for the study of the variety of individuals are discussed. A contrast is pointed out between the study of mean performances and the study of covariations. These are two components of the performance of individuals. For the study of the variety of individuals, instead of keeping these two components seperate, these two components should be combined into a single analysis. Examples of techniques which combine these two components are discussed.
  
  
• Tucker, L.R. & Messick, S. (1963).
  An individual differences model for multidimensional scaling. Psychometrika, 28, 333-367.
  The point-of-view approach to the analysis of individual differences for dissimilarity data (a fore-runner of T2/T3) is developed. The subjects are factored to construct 'idealized individuals' representing different points-of-view. The coordinates of the 'idealized individuals' in the subject space are used to construct estimated distances of all stimulus-pairs for each of the idealized individuals. Separate MDS analyses are performed to obtain separate stimulus spaces for each 'point-of-view'. Illustrated with political judgment data.
  
• Turecek, F., & Gu, M. (1995).
  Differentiation of olefin isomers by survivor-ion mass-spectrometry. Journal of mass spectrometry, 30, 144-152.**
  Survivor-ion mass spectrometry is a method that relies on the selective monitoring of non-dissociating ions that underwent collisional neutralization acid reionization. These charge-permutation processes are found to modulate the relative intensities of precursor ions, both molecular ions and fragments, formed by electron impact from C₆H₁₀, C₆H₁₂, C₆H₁₂O, and xylene isomers. The relative intensities of highly-unsaturated ions, such as C₅H₅+, C₄H₂+., C₃H₃+, and C₂H₂+., and enol ions are enhanced in the survivor-ion spectra in dependence on the neutralization gas. Isomer differentiation through survivor-ion spectra is achieved for conjugated hexadienes, 1,4-hexadiene and cyclohexene, but not for hexenols and xylenes.
  
           
• Tzeng, O.C.S. (1972).
  Differentiation of affective and denotative meaning systems in personality ratings via three-mode factor analysis. Unpublished doctoral thesis, University of Illinois, Urbana - Champaign. ( Dissertation Abstracts International, 1973, 34 (2-B), 864.)
  (See Tzeng, 1975, 1977a).
  
  
• Tzeng, O.C.S. (1975).
  Differentiation of affective and denotative meaning systems and their influence in personality ratings. Journal of Personality and Social Psychology, 32, 978-988.
  Fifty Belgian males scored 40 person concepts on 40 semantic scales to investigate person perception. Component analysis loadings from T3 were rotated by a complex scheme on so-called marker scales of the E.P.A. dimensions and on non-marker scales with the object to separate the affective from the denotative semantic space. In conjunction with the core matrix three subject components were interpreted.
  
• Tzeng, O.C.S. (1977a).
  Differentiation of affective and denotative semantic subspaces. Annals of the New York Academy of Sciences, 285, 476-500.
  In each of four countries 80-160 subjects scored 24-42 concepts on 40-60 semantic differential scales. The scales were separated in marker (affective) and non-marker (denotative scales); see Tzeng, 1975. T3 was applied for each country. Comparisons between the component matrices and the core matrices of the four countries are given.
  
  
• Tzeng, O.C.S. (1977).
  Individual differences in self-conception: multivariate approach. Perceptual and Motor Skills, 45, 1119-1124.
  Investigation of determinants of the affective and denotative semantic structures in the process of self-conception. T3 ap- plied to semantic differential ratings of 11 ego-related con- cepts on a representative set of 29 (?) scales from a homoge- neous group of 29 male college sophomores. The retained factors (after varimax) were correlated with other variables. Virtually no numerical details.
  
• Tzeng, O. C. S. (1994).
  Insufficiency of product-moment correlation for factor-analysis of bipolar measurements. Perceptual and Motor Skills, 79, 1647-1665.
  This study examined the problems in using product-moment correlations for the 2 general functions of factor analysis, identification (or verification) and use of psychological constructs embedded in bipolar ratings. For illustration, a set of fictitious semantic differential ratings, with known structural similarities and differences between concept profiles, were factored. The results were assessed against 12 criteria for evaluation derived from simultaneous considerations of issues in statistics, measurement, and psychosemantics. It was concluded that solutions from factor analysis of r coefficients do not identify or verify underlying attributional relationships among concepts implicit in subjects' cognitive processes and manifest in ratings. An alternative index, Beta Coefficient, was then used to illustrate its advantage in analysis of the hypothetical bipolar ratings. Implications of the present study are discussed.
  
  
• Tzeng, O. C., & Everett, A.V. (1985).
  A cross-cultural perspective of self-related conceptions in adolescence. International Journal of Psychology, 20, 329-348.
  Compared intercultural similarities and differences in conceptions of 5 social determinants involving adolescents from 30 language/culture communities. The ratings of 30 concepts from the Atlas that represent the 5 major problem areas of adolescence (work, social and sexual identities, economy, and a philosophy of life) were analyzed with the 3-mode multidimensional scaling with points-of-view solution technique developed by the 1st author and D. Landis.
  
  
• Tzeng, O.C.S. & Landis, D. (1978).
  Three-mode multidimensional scaling with points of view solutions. Multivariate Behavioral Research, 13, 181-213.
  A rather hybrid stringing together of multivariate procedures called "three-mode point of view analysis" (3M-POV). Including in sequence Tucker's (1972) analysis, hierarchical clustering, averaging over individuals in a cluster to obtain coefficients for an 'idealized individual', computing frontal core planes for these idealized individuals, subjecting these planes to an eigen-decomposition and some additional rotation(s). Illustrated with data from Osgood's cross-cultural research.
  
• Tzeng, O.C.S., & Osgood, C.E. (1976).
  Validity tests for componential analyses of conceptual domains: A cross-cultural study in methodology. Behavioral Science, 21, 69-85.
  An analytic prediction model for evaluation of the validities of intuited denotative features in componential analyses is outlined and illustrated for a set of concepts referring to time. The intuited components function as independent variables and the measured affective meaning of the time-related concepts function as dependent variables, both represented in Euclidean spaces. Predictions of attributions of affect from the denotative components proved to have very high predictive, construct and content validities for both the cross- cultural (mean) and the majority of the culture-indigenous interconcept squared distance matrices. Although this model for validity checking is limited to those components which are predictive of affect attribution, the external criterion, it represents a type of social system analysis in which objective culture as reflected in the denotative components for a conceptual domain is related to subjective culture, i.e., evaluative, potency and activity feelings, via the mediation of the cognitive systems of the individual human beings who make the semantic differential judgments. Similarities and differences in both objective (weights given to components) and subjective (variations in affect attribution) cultures can be quantitatively expressed.