Three-Mode Abstracts, Part C

INDEX

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• Cant, R.G. (1971).
  Changes in the location of manufacturing in New Zealand 1957-1968: An application of three-mode factor analysis. New Zealand Geographer, 27, 38-55.
  The data of 23 employment districts with 26 industries and in- dustrial indicators over 11 years were analysed with T3. A conceptual discussion of T3 is included as well. Although the component matrices were relatively clear, the 7x9x2 core matrix proved to be too complex for easy interpretation.

• Cao, Y. Z., Chen, H., Wu, H. L.& Yu, R. Q. (2003).
  An algorithm for three-way data analysis that alternatively minimizes coupled vector (COV) resolution error and PARAFAC error. Analytical Sciences 19, 753-756.
  A novel algorithm, alternatively minimizing coupled vector (COV) resolution error and PARAFAC error algorithm, is proposed in this paper. This algorithm can overcome the problem of slow convergence and is insensitive to the estimation of component number, such problems are unavoidable while using the traditional parallel factors analysis (PARAFAC) algorithm. In other words, this algorithm is capable of improving the computing speed and providing accurate resolutions provided that the number of factors used in the computation is no less than that of the actual underlying ones. The characteristic performances were demonstrated with a novel fluorescence data array.

• Cao, Y. Z., Chen, Z. P., Mo, C. Y., Wu, H. L.& Yu, R. Q. (2000).
  A PARAFAC algorithm using penalty diagonalization error(PDE) for three-way data array resolution. The Analyst, 125, 2303-2310.
  A modified parallel factors analysis (PARAFAC) algorithm with the penalty diagnolization error (PDE) was developed. This algorithm can overcome the slow convergence problem of the traditional PARAFAC method and is insensitive to the number of components, i.e., it is much faster than PARAFAC and insensitive to overestimation of the dimensionality of the model. The characteristic performance was demonstrated by treating simulated and real excitation–emission fluorescence data for samples of naphthalene, 1-naphthol and 2-naphthol with satisfactory results.

• Carlier, A. (1985b).
  Application de l'analyse factorielle des évolutions et de l'analyse intra-periodes. Statistiques et Analyse de Données, 10, 27-53.
  The methods considered in this paper are local principal components analysis using graphs defined on sets of individuals. Two different choices of graph lead respectively to factorial analysis of evolutions and intra-period analysis. Different choices of coding, weighting and metrics, which respect the property of distributional equivalence, are used for the data set.

• Carlier, A. (1986).
  Factor analysis of evolution and cluster methods on trajectories. In F. de Antoni, N. Lauro, & A. Rizzi (Eds.), Compstat 1986 (pp. 140-145). Heidelberg: Physica Verlag.
  This paper studies the evolution of objects indexed by i and associated with the lines of a two-way contingency table, observed at several times. At each time t, the j-th column of the slice {nijt} is associated with the j-th category of one or several qualitative variables. The adaption of the Factor Analysis of Evolutions (FAE) to such a cube data, produces factors, correlation circles and graphical displays of trajectories in factorial planes. Here, after a brief overview of FAE, a hierarchical agglomerative clustering (HAC) method is proposed, which is complementary to FAE. Thus the results of the HAC can be represented by mean of classes of trajectories in the factorial planes found independently using FAE. In this way, the difficulty of interpreting a large number of trajectories in the factorial planes is overcome by using the HAC method.

  

• Carlier, A. & Ewing, G. (1992).
  Longitudinal analysis of contingency tables with application to demographic data. Computational Statistics, 7, 329-353.
  This paper adresses the complexity of contingency table data concerned with dynamic population and describes a method of estimating the principal components of the multi-category transformation of geographical locations- or other kinds of units - and of defining their temporal trajectories in a reduced factor space. Locations with similar trajectories are then grouped with a view to discovering the major features of the trend and the kinds of areas sharing common transformations.

• Carlier, A., & Kroonenberg, P.M. (1996).
  Decompositions and biplots in three-way correspondence analysis. Psychometrika, 61, 355-373.
  In this paper correspondence analysis for three-way contingency tables is presented using three-way generalisations of the singular value decomposition. It is shown that in combination with Lancaster's (1951) additive decomposition of interactions in three-way tables, a detailed analysis is possible of the deviations from independence. Finally, biplots are shown to produce powerful graphical representations of the results from three-way correspondence analyses. An example from child development is used to illustrate the theoretical developments.

• Carlier, A., Lavit, C., Pagés, M., Pernin, M.-O., & Turlot, J.-C. (1989).
  A comparative review of methods which handle a set of indexed data tables. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 85-101). Amsterdam: Elsevier.
  This paper presents a comparative review of methods used to analyze data tables indexed fairly often by time. It has been worked out by the "changing data" study group of the GRECO CNRS 59 (Groupe de REcherche COordonnée "Analyse des Données et Informatique" - Centre National de la Recherche Scientifique). We are interested in longitudinal exploratory analysis of multivariate time series which are too short, or too heterogeneous to be analyzed by classical time series methods. This review is restricted to methods based on factorial techniques, or on different generalizations of them.

• Carroll, J.D. (1972).
  Individual differences and multidimensional scaling. In R.N. Shepard, A.K. Romney & S.B. Nerlove, Multidimensional scaling: Theory and applications in the behavioral sciences, 1: Theory (pp. 105-155). New York: Seminar Press.
  This paper will discuss individual differences in judgments of similarity which presumably reflect differing "perceptual structures," and in judgments of preference (or other "dominance" judgments) which reflect different ways of using those perceptual structures (whether they themselves are the same or different).

  

• Carroll, J.D. & Arabie, P. (1980).
  Multidimensional scaling. Annul Review of Psychology, 31, 607-649.
  This paper presents a taxonomy of the field of multidimensional scaling. Virtually all proposed scaling models are placed in the taxonomy, including T2 and T3 and related three-way models (cf. pp.630-636).

• Carroll, J.D., & Arabie, P. (1983).
  INDCLUS: an individual differences generalization of the ADCLUS model and the MAPCLUS algorithm. Psychometrika, 48, 157-169. [Also published in H.G. Law, C.W. Snyder Jr, J.A. Hattie, & R.P. McDonald (Eds.) (1984). Research methods for multimode data analysis (pp. 518-534). New York: Praeger.]
  We present a new model and associated algorithm, INDCLUS, that generalizes the Shepard-Arabie ADCLUS (ADditive CLUStering) model and the MAPCLUS algorithm, so as to represent in a clustering solution individual differences among subjects or other sources of data. Like MAPCLUS, the INDCLUS generalization utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach to impose discrete (0,1) constraints on parameters defining cluster membership. All subjects in an INDCLUS analysis are assumed to have a common set of clusters, which are differentially weighted by subjects in order to portray individual differences. As such, INDCLUS provides a (discrete) clustering counterpart to the Carroll-Chang INDSCAL model for (continuous) spatial representations. Finally, we consider possible generalizations of the INCLUS model and algorithm.

• Carroll, J.D., & Arabie, P. (1996).
  Multidimensional Scaling. In M.H. Birnbaum (Ed.), Handbook of perception and cognition, Vol. 3: Measurement, judgment and decision making (pp. 179-250). San Diego, CA: Academic Press, Inc.
  The authors present a wide-ranging overview of the field of multidimensional scaling with an extensive discussion of (1)spatial individual differences models such as INDSCAL and IDIOSCAL and their extensions, (2) cluster methods such as INDCLUS and (3) tree models for three-way data.

  

• Carroll, J.D. & Chang, J.J. (1970).
  Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition. Psychometrika, 35, 283- 319.
  A special, restricted variant of T2 - the weighted Euclidian distance model, called INDSCAL - is presented within the context of multidimensional scaling. The method to estimate the parameters proceeds via the minimization of a loss function. It is illustrated with the Miller & Nicely data and Wish' data on the perception of nations.

  

• Carroll, J. D., & Chang, J. J. (1972).
  IDIOSCAL: A generalization of INDSCAL allowing IDIOsyncratic reference system as well as an analytic approximation to INDSCAL. Paper presented at the Spring Meeting of the Classification Society of North America. Princeton NJ, March.
  
  
• Carroll, J.D., & Chaturvedi, A. (1995).
  A general approach to clustering and multidimensional scaling of two-way, three-way, or higher-way data. In R.D. Luce, M. D'Zmura, D.D. Hoffman, G.J. Iverson, & A.K. Romney (Eds.), Geometric representations of perceptual phenomena: Papers in honor of Tarow Indow on his 70th birthday (pp. 295-318). Mahwah, NJ: Erlbaum.
  CANDCLUS is presented as a multiway generalisation of the (S)INDCLUS model and estimation procedure, allowing binary parameters for any p of the N modes, and continuous parameters for the remaining N - p modes. A generalised separability property allows straightforward conditional OLS estimation for each mode modelled by binary parameters. Special cases include CANDECOMP, fit "one dimension at a time", and a model in which all N modes are modelled via overlapping cluster structures. Hybrid models, in which a given mode can be modelled by hybrid mixture of continuous (spatial) and discrete (clusterlike) dimensions can also be fit within this framework. A generalisation of CANDCLUS, called MUMCLUS (multimode clustering) is also formulated and discussed theoretically.

• Carroll, J. D., Clark, L. A., & DeSarbo, W. S. (1984).
  The representation of three-way proximity data by single and multiple tree structure models. Journal of Classification, 1, 25-74.
  Models for the representation of proximity data (similarities/dissimilarities) can be categorized into one of three groups of models: continuous spatial mo~lels, discrete nonspatial models, and hybrid models (which combine aspects of both spatial and discrete models). Multidimensional scaling models and associated methods, used for the spatial representation of such proximity data, have been devised to accommodate two, three, and higher-way arrays. At least one model/method for overlapping (but generally non-hierarchical) clustering called INDCLUS (Carroll and Arabic I983) has been devised for the ease of three-way arrays of proximity data. Tree-fitting methods, used for the discrete network representation of such proximity data, have only thus far been devised to handle two-way arrays. This paper develops a new methodology called INDTREES (for INdividual Differences in TREE Structures) for fitting various (discrete) tree structures to three-way proximity data. This individual differences generalization is one in which different individuals, for example, are assumed to base their judgments on the same family of trees, but are allowed to have different node heights and/or branch lengths. We initially present an introductory overview focussing on existing two-way models. The INDTREES model and algorithm are then described in detail. Monte Carlo results for the INDTREES fitting of four different three-way data sets are presented. In the application, a single ultrametric tree is fitted to three-way proximity data derived from intention-to-buy-data for various brands of over-the-counter pain relievers for relieving three common types of maladies. Finally, we briefly describe how the INDTREES procedure can be extended to accommodate hybrid modelling, as well as to handle other types of applications.

• Carroll, J.D., & De Soete, G. (1991).
  Toward a new paradigm for the study of multiattribute choice behavior. Spatial and discrete modeling of pairwise preferences. American Psychologist, 46, 342-351.
  Some recent probabilistic choice models for paired comparisons data are reviewed. First, the differences between various forms of stochastic transitivity are explained. It is then argued that in order to be realistic, a pairwise choice model should be a moderate utility model; that is, it should imply moderate stochastic transitivity, but not necessarily strong stochastic transitivity. Subsequently, three recent families of pairwise choice models are introduced that satisfy this requirement. These models are formulated mainly in an informal manner, stressing their rationale rather than their formal derivation. The models are illustrated from data about celebrities collected by Rumelhart and Greeno (1971). In the conclusion it is argued that the present models may very well form the basis for a new paradigm for studying multiattribute choice behavior.

• Carroll, J.D., De Soete, G., & Kamensky, V. (1992).
  A modified CANDECOMP algorithm for fitting the latent class model: Implementation and evaluation. Applied Stochastic Models and Data Analysis, 8, 303-309.
  In this paper an implementation is discussed of a modified CANDECOMP algorithm for fitting Lazarsfeld's latent class model. The CANDECOMP algorithm is modified such that the resulting parameter estimates are non-negative and 'best asymptotically normal'. In order to achieve this, the modified CANDECOMP algorithm minimizes a weighted least squares function instead of an unweighted least squares function as the traditional CANDECOMP algorithm does. It is found that, with appropriate weights, the modified CANDECOMP algorithm yields solutions that are nearly identical with those obtained by means of the maximum likelihood procedure. While the modified CANDECOMP algorithm tends to be computationally more intensive than the maximum likelihood method, it is very flexible in that it easily allows one to try out different weighting schemes.

  

• Carroll, J.D., De Soete, G., & Pruzansky, S. (1989).
  Fitting of the latent class model via iteratively reweighted least squares CANDECOMP with nonnegativity constraints. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 463-472).
  Amsterdam: Elsevier.
  A modified CANDECOMP algorithm is proposed for fitting Lazarsfeld's Latent Class model. The new algorithm remedies two problems that exist with using regular CANDECOMP for fitting the Latent Class model. The first problem results from the fact that the least squares estimates obtained with regular CANDECOMP are not BAN, while the second problem arises from the fact that the regular CANDECOMP algorithm does not ensure that the parameter estimates are nonnegative. The new algorithm uses a nonnegatively constrained iteratively reweighted least squares estimation procedure to overcome these shortcomings.

• Carroll, J.D., & Green, P.E. (1988).
  An INDSCAL-based approach to multiple correspondence analysis. Journal of Marketing Research, 25, 193-203.
  Current methods of multiple correspondence analysis (MCA) provide configurations that are expressed in terms of principal axes. These solutions are not invariant over rotations. The authors propose an approach to MCA that entails an INDSCAL analysis of normalized Burt matrices (as commonly obtained from MCA). The resulting configuration is uniquely oriented and dimension weights also are obtained for each contributory data set. The method is applied to survey data describing relationships among respondent demographic characteristics and recent car purchases.

• Carroll, J.D., & Green, P.E. (1997).
  Psychometric methods in marketing research: Part II, Multidimensional scaling. Journal of Marketing Research, 34, 193-204.
  In this paper the history and maturation of MDS in marketing is discussed, including models and applications of individual differences models, constrained MDS, stochastic MDS modeling, normatively based MDS models for optimal product design, and scaling models developed for analysis of scanner data. Included is an appraisal of the state of practice of MDS in marketing, some of the problems associated with the gap between models and applications, and some suggestions for increasing the practical utility of MDS in marketing research.

  

• Carroll, J.D., & Pruzansky, S. (1986).
  Discrete and hybrid models for proximity data. In W. Gaul & M. Schader (Eds.) Classification as a tool of research (pp. 47-59). Amsterdam: Elsevier.
  Discrete models such as tree structure or overlapping clustering models, typically associated with clustering methods, can also be viewed as mdoels of proximity data (e.g. human judgments of similarity). Such models and various generalizations (e.g. multiple tree structure models) are considered, as well as methods for fitting them. Further generalization to "hybrid" models, combining such discrete structures with the kind of continuous spatial structure most commonly associated with multidimensional scaling, factor component analysis, are also considered. Finally, extentions to handle nonsymmetric proximity data, as well as to three-way (individual differences) data are discussed.

• Carroll, J.D., Pruzansky, S., & Green, P.E. (1977).
  Estimation of the parameters of Lazarsfeld's latent class model by application of canonical decomposition (CANDECOMP) to multi-way contingency tables. (Technical Memorandum no. 77-1229-7). Murray Hill, NJ: Bell Laboratories.

  Lazarsfeld's latent class model assumes that an observed multi-way contingency table can be decomposed into a weighted sum of more basic contingency tables. Each of these basic tables exhibits statistical independence among the factors defining the table. CANDECOMP is a method for least squares fitting of a multilinear model, the CANDECOMP model, to general multi-way data tables. We show that the latent class model is a special case of the CANDECOMP model, so that the CANDECOMP procedure provides a method for least squares fitting of the latent class model. A program, called CANLAC, has been written for estimating the parameters of the latent class model using the CANDECOMP procedure. Applications of the program to both artificial and real data are described.

• Carroll, J.D., Pruzansky, S., & Kruskal, J.B. (1980).
  CANDELINC: A general approach to multidimensional analysis of many-way arrays with linear constraints on parameters. Psychometrika, 45, 3-24.
  Some general multilinear models are introduced, and procedures are described to treat data consisting of a many-way array. A very general theorem is proven which says among other things that a least squares solution for the linearly constrained multi-mode factor analysis model can be found by finding a least squares solution for the reduced model (e.g. those of Sands & Young, 1980; Kroonenberg & De Leeuw, 1980; Lohmöller and Wold, 1980. The main discussion is on various approaches to constrained multidimensional scaling.

• Carroll, J.D., & Winsberg, S. (1995).
  Fitting an extended INDSCAL model to three-way proximity data. Journal of Classification, 12, 57-71.
  The INDSCAL individual differences scaling model is extended by assuming dimensions specific to each stimulus or other object, as well as dimensions common to all stimuli or objects. An "alternating maximum likelihood" procedure is used to seek maximum likelihood estimates of all parameters of this EXSCAL (Extended INDSCAL) model, including parameters of monotone splines assumed in a "quasi-nonmetric" approach. The rationale for and numerical details of this approach are described and discussed, and the resulting EXSCAL method is illustrated on some data on perception of musical timbres.

• Carroll, J. D. & Wish, M. (1974).
  Models and methods for three-way multidimensional scaling . In D.H. Krantz, R.C. Atkinson, R.D. Luce, & P. Suppes (Eds.), Contemporary developments in mathematical psychology. Vol. 2 (pp. 57-105). San Francisco: W.H. Freeman.
  C. & W. present primarily the INDSCAL model, but also treat in some detail the differences and similarities of IDIOSCAL, PARAFAC2, three-mode scaling, and INDSCAL. Probably the clearest statement of the relationships between these models.

  

• Casassas, E., Tauler, R., & Marqués, I. (1994).
  Interactions of H+ and Cu(II) ions with poly(adenylic acid): study by factor-analysis. Macromolecules, 27, 1729-1737.**
  The interactions of H+ and Cu(II) ions with poly(adenylic acid) (polya) are investigated using different factor analysis techniques. From the changes in the ultraviolet and circular dichroism absorption spectra obtained in the course of spectroscopic acid-base titrations of poly(adenylic acid) solutions containing different amounts of Cu(II) ion, the unitary spectra, the concentrations, and the stabilities of the macromolecular species detected are estimated. Four spectroscopically different species are detected for poly(adenylic acid) in the pH interval 1.5-6.8. Three of them are different conformations of the protonated form of polya, and the other is the unprotonated form of polya. One new species is detected in the presence of Cu(II) ion.

• Casin, Ph. (1995).
  L'analyse discriminante de tableaux évolutifs. [Discriminant analysis of longitudinal tables]. Revue de Statistique Appliquée, 43, 73-91.
  ADT is a method of analysing 3-way data tables using CARROLL's criteria of generalized canonical analysis, under a constraint of orthogonalisation in each subspace. ADTE is the application of ADT at indexed tables.

• Casin, Ph. (2001).
  A generalization of principal component analysis to K sets of variables. Computational Statistics & Data Analysis, 35, 417-428.
  The aim of this paper is to introduce a new method, generalized principal component analysis (GPCA), which is a generalization of principal component analysis (PCA), to several data tables. GPCA is a method for both finding common dimensions in several sets of variables and giving a description of each set of variables: GPCA takes into account both the correlation structure within sets and relationships between sets. Two sorts of orthogonal basis are provided; the first basis is useful to represent each set of variables (as PCA does) and the second is useful to represent associations between sets of variables (as canonical correlation analysis does). An example using real data (evolution of five characteristics of car markets from 86 to 93 for 8 countries) illustrates the method.

• Cattell, R.B. (1966).
  The data box: Its ordering of total resources in terms of possible relational systems. In R.B. Cattell (Ed.), Handbook of multivariate experimental psychology (pp. 67-128). Chicago: Rand McNally.
  This is a fundamental paper on the classification of different types of data. It introduces the Data Box or Basic Data Relation matrix and discusses ways to analyse various slices (facets) of the Data Box, such as P- (Q-, S-, and T-) techniques. The article also treats scales and standardizations for various facets.

• Cattell, R.B. (1980).
  The separation and evaluation of personal and environmental contributions to behavior by the person-centered model (PCER). Multivariate Behavioral Research, 15, 371-402.
  The person-centered environmental representation (PCER) model aims to bring extra-personality determiners of behavior measures (stimulus, response form, situation, observer) as known values into the behavioral equation. Illustration begins with the simplest 3-dimensional data box case - person, test, situation - in which the response score is a product of the characteristic trait score of the individual and the characteristic interaction of that trait with that kind of performance. In extending this to a particular kind of situation it is recognized that a test-situation interaction term must, realistically, be added.
  The problem in expriments is to infer the magnitude of the contributors from the 3-sums in the cells of the empirically given data box frame. Three ways of attempting this are described and the third, judged most practical, is illustrated in a plasmode of data typical of experiment. It is shown that the discovery from the frame data of the factor structure originally put into the contributory test and situation matrices is reasonably good, and it is pointed out that since this can be achieved with only 10 situations, 12 tests and 14 people, the smoothing effect of more typical large samples makes the method practicable. If further research with real data confirms the speculation that the personality factors determining situation reaction are the same as those determining test reaction, a split loading model could be used permitting, as a by-product, assessment of relative variance contributions of differences among persons, stimuli, response forms, and observers.

  
         

• Cecil, T.L., & Rutan, S.C. (1991).
  Fluorescence detection in liquid-chromatography with an intensified diode-array detector. Journal of Chromatography, 556, 495-503.**
  The capabilities of full fluorescence spectral detection for the analysis of polyaromatic hydrocarbons after a liquid chromatographic separation were evaluated. The limits of detection and the linear dynamic range were determined for three test compounds, benzo[a]pyrene, perylene and anthracene. With the instrument used, detection limits in the micromolar concentration range (ca. 100 ng injected), and a linear dynamic range of at least three orders of magnitude were observed. In addition, the pair of isomers 1- and 2-aminoanthracene, which are incompletely resolved both spectroscopically and chromatographically, were studied. Finally, a commercially available mixture of polyaromatic hydrocarbons was characterized, and it was possible to resolve the chromatographically unresolved pair of isomers benzo[a]anthracene and chrysene. The data analysis methods used included kalman filter-based methods for adaptive subtraction of background responses, shift correction and linear regression analysis of overlapped responses. The results indicate that full fluorescence spectral detection in liquid chromatography should be an increasingly useful technique.

• Ceulemans, E. & Van Mechelen, I. (2003).
  Tucker3 hierarchical classes analysis. Psychometrika, 68,413-433. This paper presents a new model for binary three-way three-mode data, called Tucker3 hierarchical classes model (Tucker3-HICLAS). This new model generalizes Leenen, Van Mechelen, De Boeck, and Rosenberg's (1999) individual differences hierarchical classes model (INDCLAS). Like the INDCLAS model, the Tucker3-HICLAS model includes a hierarchical classification of the elements of each mode, and a linking structure among the three hierarchies. Unlike INDCLAS, Tucker3-HICLAS (a) does not restrict the hierarchical classifications of the three modes to have the same rank, and (b) allows for more complex linking structures among the three hierarchies. An algorithm to fit the Tucker3-HICLAS model is described and evaluated in an extensive simulation study. An application of the model to hostility data is discussed.

• Ceulemans & Van Mechelen, I. (2003).
  Uniqueness of N-way N-mode hierarchical classes models. Journal of Mathemaical Psychology47, 259-264.
  This paper presents two uniqueness theorems for the family of hierarchical classes models, a collection of order preserving Boolean decomposition models for binary N-way N-mode data. The theorems are compared with uniqueness results for the closely related family of N-way N-mode principal component models. It is concluded that the two-way two-mode PCA and N-way N-mode TuckerN models suffer more from a lack of identifiability than their hierarchical classes analogues, whereas the uniqueness conditions for N-way N-mode PARAFAC/CANDECOMP models are less restrictive than the ones derived for their N-way N-mode hierarchical classes counterparts. (C) 2003 Elsevier Science (USA). All rights reserved.

• Ceulemans, E., Van Mechelen, I., & Kuppens, P.(2004).
  Adapting the formal to the substantive: Constrained Tucker3-HICLAS. Journal of Classification, 21,19-50.
  Ceulemans, Van Mechelen, and Leenen (2003) recently presented the Tucker3 hierarchical classes model for three-way three-mode binary data. This model includes a hierarchical classification of the elements of each mode involved in the data, and a linking structure among the three hierarchies. In substantive applications of this model, one may wish to incorporate substantive knowledge in the data analysis. In this paper, we show how the latter may be achieved by imposing constraints on the model. As the number of possible Tucker3-HICLAS constraints is huge, we present a comprehensive taxonomy of a wide range of constrained Tucker3-HICLAS models. Furthermore, we investigate which features of the existing Tucker3-HICLAS algorithm have to be adapted for fitting the different types of constrained Tucker3-HICLAS models. Finally, the constrained Tucker3-HICLAS approach is illustrated with applications from psychiatric diagnosis and personality research.

       

• Chapman, S.C., Crossa, J., Basford, K.E., & Kroonenberg, P.M. (1997).
  Genotype by environment effects and selection for drought tolerance in tropical maize. II. Three-mode pattern analysis. Euphytica, 95,11-20.
  A selection program in three tropical maize populations aimed to improve tolerance of mid-season to late season drought environments while maintaining grain yield potentials. Three-mode (genotype x environments x attributes) pattern analysis (three-mode mixture method clustering and three-mode PCA) was used to evaluate ten evaluation trials. Four groups were identified and a 3x2x3 solution for the three-mode PCA with 70% explained variation were used and yielded highly interpretable results.

  

• Chapman, R.M., & McCrary, J.W. (1995).
  EP component identification and measurement by principal components analysis. Brain and Cognition, 27, 288-331.
  Principal component analysis is shown to have a number of desirable characteristics for systems of Evoked Potential measurement. In addition to discussing the usual two-mode model of PCA, an extension of PCA to a three-mode model is described that provides separate parameters for (1) waveforms over time, (2) coefficients for spatial distribution, and (3) scores telling the amount of each component in each EP. Choices to be made in applying PCA are considered.

• Chaturvedi, A., & Carroll, J.D. (1994).
  An alternating combinatorial optimization approach to fitting the INDCLUS and generalized INDCLUS models. Journal of Classification, 11, 155-170.
  This paper presents a general approach for fitting the ADCLUS, INDCLUS and potentially a special case of the GENNCLUS models. The proposed approach, based largely on a separability property observed for the least squares loss function being optimized, offers increased efficiency and other advantages over existing approaches like MAPCLUS for fitting the ADCLUS model, and the INDCLUS method for fitting the INDCLUS model. The new procedure (called "SINDCLUS") is applied to three sets of empirical data to demonstrate the effectiveness of the SINDCLUS methodology. Finally, some potentially usefull extensions are discussed.

• Chen, G., & McAvoy, T. J. (1998).
  Predictive on-line monitoring of continuous processes. Journal of Process Control, 8, 409-420.
  For safety and product quality, it is important to monitor process performance in real time. Since traditional analytical instruments are usually expensive to install, a process model can be used instead to monitor process behavior. In this paper, a monitoring approach using a multivariate statistical modeling technique, namely multi-way principal component analysis (MPCA), is studied. The method overcomes the assumption that the system is at steady state and it provides a real time monitoring approach for continuous processes. The monitoring approach using MPCA models can detect faults in advance of other monitoring approaches. Several issues which are important for the proposed approach, such as the model input structure, data pretreatment, and the length of the predictive horizon are discussed. A multi-block extension of the basic methodology is also treated and this extension is shown to facilitate fault isolation. The Tennessee Eastman process is used for demonstrating the power of the new monitoring approach.

• Chen, J. H., & Yen J. H. (2003).
  Three-way data analysis with time lagged window for on-line batch process monitoring. Korean Journal of Chemical Engineering, 20, 1000-1011.
  In this paper, on-line batch process monitoring is developed on the basis of the three-way data structure and the time-lagged window of process dynamic behavior. Two methods, DPARAFAC (dynamic parallel factor analysis) and DTri-PLS (dynamic trilinear partial least squares), are used here depending on the process variables only or on the process variables and quality indices, respectively. Although multivariate analysis using such PARAFAC (parallel factor analysis) and Tri-PLS (trilinear partial least squares) models has been reported elsewhere, they are not suited for practicing on-line batch monitoring owing to the constraints of their data structures. A simple modification of the data structure provides a framework wherein the moving window based model can be incorporated in the existing three-way data structure to enhance the detectability of the on-line batch monitoring. By a sequence of time Window of each batch, the proposed methodology is geared toward giving meaningful results that can be easily connected to the current measurements without the extra computation for the estimation of unmeasured process variables. The proposed method is supported by using two sets of benchmark fault detection problems. Comparisons with the existing two-way and three-way multiway statistical process control methods are also included.

• Chen, Z. P., Li, Y., & Yu, R. Q. (2001a).
  Pseudo alternating least squares algorithm for trilinear decomposition. Journal of Chemometrics, 15, 149-167.
  In chemistry, PARAFAC is one of the most widely used algorithms for trilinear decomposition. However, the problem of PARAFAC requiring an accurate estimation of the number of factors in the system under study limits its applications to some extent. This troublesome problem has been tackled by the pseudo alternating least squares (PALS) algorithm designed in this paper. PALS is a unique algorithm which tries to alternately optimize three different objective functions to obtain the solutions for the trilinear decomposition model. It has the outstanding feature of being resistant to the influence of N (the number of factors chosen in calculation), which has been proved mathematically under some mild conditions. Although the optimization procedure of PALS is different from that of PARAFAC, an alternating least squares scheme, and hinders a straightforward analysis of its convergence properties, studies on simulated as well as real data arrays reveal that PALS can often converge to satisfactory results within a reasonable computation time, even if excess factors are used in calculation.

• Chen, Z. P., Wu, H. L., Li, Y., & Yu, R. Q. (2000a).
  Novel constrained PARAFAC algorithm for second-order linear calibration. Analytica Chimica Acta, 423, 187-196.
  A novel constrained PARAFAC was designed to mitigate the influence of model deviations on the predictive accuracy in second-order calibration. It combines the two steps of decomposition and calibration of PARAFAC in second-order calibration into just one step by imposing the concentrations of the calibration samples on the model parameters in the third mode as constraints. Such a scheme not only simplifies the whole calibration procedure, but also renders some extra advantages over PARAFAC when model deviations such as Rayleigh scattering in fluorescence spectroscopy present in measurements. The results for fluorescent and HPLC data arrays have demonstrated that the relative deviations of the predicted concentrations could be significantly reduced when the constrained version of PARAFAC replaces PARAFAC to do second-order linear calibration.

• Chen, Z. P., Wu, H. L., Jiang, J. H., Li, Y. & Yu, R. Q. (200b0).
  A novel trilinear decomposition algorithm for second-order linear calibration. Chemometrics and Intelligent Laboratory Systems, 52, 75-86.
  A novel trilinear decomposition algorithm for second-order linear calibration called self-weighted alternating trilinear decomposition (SWATLD) has been designed in this paper. Experiments show SWATLD has the features of fast convergence and being insensitive to the excess factors used in calculation. Due to the unique optimizing scheme employed, SWATLD is much more efficient than the ordinary PARAFAC algorithm. In terms of the variance, the performance of SWATLD is very stable when the number of factors used in calculation varies (as long as it is no less than the actual number of factors). Such a feature will facilitate the analysis of three-way data arrays, since it is now unnecessary to spend a lot of time and effort to accurately determine the number of underlying factors in the system studied as does in PARAFAC. Furthermore, as far as the deviations of the results are concerned, experiments show SWATLD can supply acceptable results in most cases.

• Chen, Z. P., Wu, H. L., & Yu, R. Q. (2001b).
  On the self-weighted alternating trilinear decomposition algorithm - the property of being insensitive to excess factors used in calculation. Journal of Chemometrics, 15, 439-453.
  PARAFAC is well known as an iterative trilinear decomposition method. In practice, an accurate estimation of the number of underlying factors is required; otherwise it is difficult to guarantee the chemical meaning of the results obtained. The absence of a versatile chemometric method for factor estimation has often caused problems for analysts. With the advent of ATLD followed by SWATLD, the above relatively strict constraint can be relaxed. Experiments have shown that the profiles of the underlying factors can be extracted by ATLD (Wu ed al., J. Chemometrics 1998; 12: 1) and SWATLD (Chen et at, Chemometrics Intell. Lab. Syst. 2000; 52: 75) as long as the number of factors used in calculation is no less than the number of actual factors. In other words, an overestimation of the number of factors will not affect the obtainment of chemically meaningful results by SWATLD and ATLD. In this paper the authors provide some simple mathematical explanations of this valuable property of SWATLD. Along with these explanations, some other properties of SWATLD as well as some guidelines for designing new trilinear decomposition methods are also discussed.

• Chen, Z. P., Liu, Z., Cao, Y. Z., & Yu, R. Q. (2001c).
  Efficient way to estimate the optimum number of factors for trilinear decomposition. Analytica Chimica Acta, 444, 295-307.
  In trilinear decomposition, one first tries to estimate the number of underlying factors in the system studied, and then employs trilinear decomposition methods such as PARAFAC to obtain the desired characteristic profiles of the underlying factors and their relative contributions. Since the results of PARAFAC are heavily dependent on the estimation of the underlying factors, either overestimation or underestimation of the underlying factors will lead the results of PARAFAC to be erroneous. Most of the existing factor-determining methods are established on the basis of factor analysis. These procedures are originally designed for two-way data sets. Only after the three-way data array was unfolded into two-way data set, could then these factor-determining methods be used. It is obvious that the trilinear character of the data array is not utilized in the factor-determining procedure. With a view to cope with non-ideal experimental conditions, such as heavy collinearity and varying backgrounds, the present authors advocated incorporating the advantages of trilinear data array into the factor-determining procedure. Hence, a novel factor-determining method has been proposed specifically for trilinear decomposition. Experiments have demonstrated that the proposed method has the features of easy implementation and excellent performance even when heavy collinearity and varying backgrounds are present.

• Chen, Z. P., & Yu, R. Q. (2003).
  Mitigating model deficiency in three-way data analysis by the combination of background constraining and iterative correcting techniques. Analitica Chimica Acta, 478, 171-180. PARAFAC is a popular model for trilinear data analysis in analytical chemistry. The prerequisite for the successful application of PARAFAC in analytical chemistry is that the three-way data array should follow a trilinear model, which is always violated by the presence of deviations such as Rayleigh scattering in fluorescence spectroscopy. In order to mitigate the influence of model deviations, background constraining and iterative correcting techniques are advocated in this contribution. The method established on these two techniques can nearly eliminate the effect of model deviation on the chemical loading parameters estimated. Compared with other methods for mitigating model deviations, the proposed method requires no prior knowledge about the chemical loading parameters. It is also unnecessary to assign weights to data entities as the weighted PARAFAC of Anderson does. Its implementation is comparable to PARAFAC-ALS and can be programmed to be completely automatic. Its performance has been demonstrated by fluorescent and chromatographic experiments.

• Chen, Z. N., Zhu, E. Y., & Chen, N. Y. (1993).
  Application of orthogonal expansion to mapping and modeling. Journal of Chemometrics, 7, 243-253.
  In this paper we discuss the orthogonal expansion of data matrices and its application to mapping and modelling. Two new methods, modified optimal discriminant plane (MODP) for mapping and order Gram-Schmidt orthogonalization (OGSO) for modelling, are proposed and examples are given.

• Chessel, D., & Mercier, P. (1993).
  Couplage de triplets statistiques et liaisons espèces-environnement. In J. D. Lebreton & B. Asselain (Eds.), Biométrie et Environnement (pp.15-44). Masson, Paris. We present a review of linear statistical methods used for the analysis of a couple of data tables consisting in a vegetional or a faunistical data set and an environmental data set. According to the linear approach (exploratory data analysis), we differentiate nonsymmetrical and symmetrical methods. In the nonsymmetrical case, the analysis of the two data sets is direct in the sense that the variables of one set are recognized as predictors and thos of the second as responses. The multivariate methods using instrumental variables (such as Canonical Correspondence Analysis of Ter Braak, 1986) give a convenient solution to this approach. In the symmetric case, each data set is taken into account on an equal footing. This paper demonstrates that the interbattery analysis of Tucker (1985), the correspondence analysis of an ecological profile data table of Romance (1972c) and the canonical analysis on categorical variables according to Cazes (1980) can be presented in a general theorical framework of connected analysis of a couple of data sets.

• Chiarello, E., & Chassery, J.M. (1997).
  Typology of spatial structures of images having the same color set. Journal of Mathematical Imaging and Vision, 7, 359-374.
  A previous study proposed a new model for generating random spatial patterns for modelling the dispersion of different colors in an image (Chiarello et al., (1996)). These simulations represented spatial structures in the sense of landscape ecology and they had to be compared to a real image. Thus, the aim of the present study was to measure the discrepancy between a set of simulations of multicolored mosaics and an observed pattern in order to build a Monte Carlo test. The multicolored mosaics were considered as random closed sets and described with the hitting function for all pairs of colors. This description provided large three-dimensional data tables (distances × color pairs × images) that were analyzed with the help of multiway data analyses. The partial triadic analysis was used. It provided a synthesis of the hitting function since the intrastructure enabled a typology of the distances for each image: The factorial coordinates of supplementary columns were plotted as ordinates against distances as abscissa. This synthetic descriptor provided a graphic tool for measuring the differences between the spatial dispersions of a same set of colors in several images.

• Chiodi, M., & Mineo, A. M. (1999).
  The Analysis of Auxological Data by Means of Nonlinear Multivariate Growth Curves. In M. Vichi, & O. Opitz (Eds.), Classification and Data Analysis. (pp. 247-254). Pescara: Proceedings of the Biannual Meeting of the Classification Group of Societa Italiana di Statistica (SIS).
  In this paper we treat the problem to analyse a data set constituted by multivariate growth curves for different subjects; thus in this context we deal with 3-way data tables. Nevertheless, it is not possible using factorial techniques proposed to deal with 3-way data matrices, because the observations are generally not equally spaced; moreover a multilevel approach founded on polynomial models is not suitable to deal with intrinsic nonlinear models. We propose a non-factorial technique to analyse auxological data sets using an intrinsic nonlinear multivariate growth model with autocorrelated errors. The application to a real data set of growing children gave easily interpretable results.

• Chojnicki, Z., & Czyz, T. (1976).
  Some problems in the application of factor analysis in geography. Geographical Analysis, 8, 416-427.
  Presents a general discussion of factor analysis, and cites possible advantages of T3 for spatial-temporal analysis of geographical data.

  

• Choulakian, V. (1988).
  Analyse factorielle des correspondances de tableaux multiples. [Factor correspondence analysis of multiway tables]. Revue de Statistique Appliquée, 36, 33-41.
  A method to analyse a multidimensional contingency table to flatten it into a binary table, then to carry out a correspondence analysis was proposed by Benzecri (1976). Lancaster's (1951, 1969) partitioning of K. Pearson's coefficient of contingency into additive terms is advocated; a simple criterion proposed by Choulakian (1985) will aid to select the appropriate flattened table. The procedure is illustrated on three-way and four-way contingency tables.

• Christensen, J., Povlsen, V. T., & Sorensen, J. (2003).
  Application of fluorescence spectroscopy and chemometrics in the evaluation of rocessed cheese during storage. Journal of Dairy Science, 86, 1101-1107.
  Front face fluorescence spectroscopy is applied for an evaluation of the stability of processed cheese during storage. Fluorescence landscapes with excitation from 240 to 360 nm and emission in the range of 275 to 475 nm were obtained from cheese samples stored in darkness and light in up to 259 d, at 5, 20 and 37 degreesC, respectively. Parallel factor (PARAFAC) analysis of the fluorescence landscapes exhibits four fluorophores present in the cheese, all related to the storage conditions. The chemometric analysis resolves the fluorescence signal into excitation and emission profiles of the pure fluorescent compounds, which are suggested to be tryptophan, vitamin A and a compound derived from oxidation. Thus, it is concluded that fluorescence spectroscopy in combination with chemometrics has a potential as a fast method for monitoring the stability of processed cheese.

• Christensen, J., Becker, E. M., & Frederiksen, C. S. (2005).
  Fluorescence spectroscopy and PARAFAC in the analysis of yogurt. Chemometrics and Intelligent Laboratory Systems, 75, 201-208.
  Parallel factor (PARAFAC) analysis and fluorescence spectroscopy were applied in the evaluation of yogurt during storage. Fluorescence landscapes with excitation wavelengths from 270 to 550 nm and emission wavelengths in the range 310–590 nm were obtained from front-face fluorescence measurements directly on yogurt samples during two storage experiments over a period of 5 weeks at 4 8C. PARAFAC analysis of the fluorescence landscapes exhibited three fluorophores present in the yogurt, all strongly related to the storage conditions. The fluorescence signal was resolved into excitation and emission profiles of the pure fluorescent compounds, which are suggested to be tryptophan, riboflavin and lumichrom. Thus, it is concluded that fluorescence spectroscopy in combination with chemometrics has a potential as a fast method for monitoring the oxidative stability and quality of yogurt. Regression models between fluorescence landscapes and riboflavin content, determined by the traditional chemical analysis, were performed, yielding a root mean square error of cross-validation of 0.09 ppm riboflavin, corresponding to 7% of the mean riboflavin content in the yogurt samples. Regression models based on PARAFAC scores, Partial Least Squares (PLS) and N-PLS were compared and yielded only minor differences with respect to prediction error. Several missing values appear in the fluorescence data matrices, for all emission wavelengths below the excitation wavelength. Substituting some of the missing values with zeros was observed to have a large impact on the model solution and the computation time. It is concluded that at least 43% of the missing values in the present data set need to be substituted in order to obtain meaningful PARAFAC models.

  
   

• Cinca, C. S., Molinero, C. M., & Larraz, J. L. G. (2002).
  A multivariate study of the economy of the European Union via financial statements analysis. Journal of the Royal Statistical Society Series D-the Statistician, 51, 335-354.
  To what extent do financial statements of firms reflect changes in the economy? We address this question by using European Union data. European countries have a service, normally under the name of the Central Balance Sheet Off ice, which is charged with the collection and aggregation of financial information from commercial firms. For members of the European Union this information has been harmonized and published under the 'Bank for the accounts of companies harmonised' project of the European Commission. The data contained in the project's database for the manufacturing sector are analysed by using weighted multidimensional scaling techniques. This approach fully exploits the cross-sectional time series nature of the data and results in statistical maps which can be interpreted by means of regression-based techniques. It is found that three characteristics, or dimensions, describe the movements that are observed in the database: business performance, capital structure and macroeconomic influences. These three characteristics do not impact on all countries in a similar way. Differences between countries are explored. Two cycles, a business performance cycle and an economic cycle, are observed in the data. A delay of about 2 years is observed between the two cycles, suggesting that changes in the economy are reflected in financial statements with a 2-year delay.

             

• Coello, J., Maspoch, S., & Villegas, N. (2000).
  Simultaneous kinetic-spectrophotometric determination of levodopa and benserazide by bi- and three-way partial least squares calibration. Talanta, 53,, 627-637.
  A procedure for the simultaneous kinetic-spectrophotometric determination of levodopa (I) and benserazide (II), from their oxidation reaction with KIO4 in an acidic medium, is described, Both species instantly oxidize, giving rise to compounds which present maximum values of absorbance close to 400 nm. In the presence of an excess of the oxidizing agent, the levodopa derivative evolves to form the corresponding aminochrome (lambda (m) = 480 nm), while the benserazide derivative decomposes to yield colorless compounds. The appearance of new compounds, with absorption bands in the region of 500-700 nm, is additionally seen upon adding the oxidizing agent to a mixture of I and II. These compounds also evolve decomposing and forming colorless products. In spite of the complexity of the system studied, the calibration by bi-lineal pal tial least squares (PLS) as well as by three-way partial least squares (nPLS) permit the quantification of both analytes with a precision on the order of 0.7% for levodopa and of 1.5% for benserazide. nPLS also allows for the qualitative interpretation of the phenomena which occur. The proposed method is applied to the quantification of I and II in the commercial, pharmaceutical preparation Madopar, using high performance liquid chromatography (HPLC) as the analytical reference technique.

• Cohen, H.S. (1974).
  Three-mode rotation to approximate INDSCAL structure (TRAIS). (Technical Memorandum no. 74-3433-14). Murray Hill, NJ: Bell Laboratories.
  It is proposed to rotate the extended core matrix from a three-mode scaling solution to approximate INDSCAL structure by using INDSCAL itself on the core matrix, and using the full rank 'object' space as transformation matrix for the real object space. Illustrated with subjects' judgements of switching network events.

  

• Cohen, J.E., & Rothblum, U.G. (1993).
  Nonnegative ranks, decompositions, and factorizations of nonnegative matrices. Linear Algebra and its Applications, 190, 149-168.**
  The nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively. Such decompositions are useful in diverse scientific disciplines. We obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm.

• Cole, H.W., & Ray, W.J. (1985)
  EEG correlates of emotional tasks related to attentional demands. International Journal of Psychophysiology, 3, 33-41.
  Forty males on each of two days were presented with tasks reflecting both attentional demands and affectual processing. Heart rate and bilateral EEG measures from frontal, parietal and temporal sites were recorded, and electrocortical activity was determined and analyzed. Differential hemispheric activity was found using PARAFAC with positively valenced tasks being associated with right temporal beta.

  

• Comas, E., Gimeno, R. A., Ferre, J., Marce, R. M., Borrull, F., & Rius, F. X. (2004).
  Quantification from highly drifted and overlapped chromatographic peaks using second-order calibration methods. Journal of Chromatography, 1035, 195-202.
  For determining low levels of pesticides and phenolic compounds in river and wastewater samples by high performance liquid chromatography (HPLC), solid phase extraction (SPE) is commonly used before the chromatographic separation. This preconcentration step is not necessarily selective for the analytes of interest and it may retain other compounds of similar characteristics as well. In this case, we present, humic and fulvic acids caused a large baseline drift and overlapped the analytes to be quantified. The inaccurate determinations of the area of the peaks of these analytes made it difficult to quantify them with univariate calibration. Here we compare three second-order calibration algorithms (generalized rank annihilation method (GRAM), parallel factor analysis (PARAFAC) and multivariate curve resolution-alternating least squares (MCR-ALS)) which efficiently solve this problem. These methods use second-order data, i.e., a matrix of responses for each peak, which is easily obtained with a high performance liquid chromatography-diode array detector (HPLC-DAD). With these methods, the area does not need to be directly measured and predictions are more accurate. They also save time and resources because they can quantify analytes even if the peaks are not resolved. GRAM and PARAEAC require trilinear data. Biased and imprecise concentrations (relative standard deviation, %R.S.D. = 34) were obtained without correcting the time-shift. Hence, a time-shift correction algorithm to align the peaks was needed to obtain accurate predictions. MCR-ALS was the most robust to the time-shift. All three algorithms provided similar mean predictions, which were comparable to those obtained when sulfite was added to the samples. However, the predictions for the different replicates were more similar for the second-order algorithms (%R.S.D. = 3) than the ones obtained by univariate calibration after the sulfite addition (%R.S.D. = 13).

• Commandeur, J. J. F., Kroonenberg, P. M., & Dunn III, W. J. (2004).
  A dedicated generalized Procrustes algorithm for consensus molecular alignment. Journal of Chemometrics, 18, 37-42.
  Recently the idea of using generalized Procrustes analysis for aligning sets of molecules was introduced using standard algorithms. In this paper it is shown that, by tailoring the algorithm to this specific problem, a great gain in computational speed and memory efficiency can be obtained, but even more importantly, by using rotations without reflection, changes in chirality of molecules can be prevented, which was not previously possible.

• Comon, P., (1996).
  Decompositions of quantics in sums of powers of linear forms. Signal Processing, 53, 96-107. Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of high-order statistics (HOS). However, very few special purpose tools are at our disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the eigenvalue decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms.

• Comon, P. (2002).
  Tensor decompositions. State of the art and applications. In J. G. McWhirter & I. K. Proudler (Eds.), Mathematics in signal processing V (pp. 1--24). Oxford, UK: Clarendon Press In this paper, we present a partial survey of the tools borrowed from tensor algebra which have been utilized recently in statistics and signal processing. It is shown why the decompositions well known in linear algebra can hardly be exteded to tensors. The concept of rank is itself difficult to define, and its calculation raises difficulties. Numerical algorithms have nevertheless been developed, and some are reported here, but their limitations are emphasized. The reports hopefully open research perspectives for enterprising readers.

• Comon, P., (2014).
  IEEE Signal Processing Magazine, 31(3), 43-53. Tensor decompositions are at the core of many Blind Source Separation (BSS) algorithms, either explicitly or implicitly. In particular, the Canonical Polyadic (CP) tensor decomposition plays a central role in identification of under- determined mixtures. Despite some similarities, CP and Singular Value Decomposition (SVD) are quite different. More generally, tensors and matrices enjoy different properties, as pointed out in this brief introduction

• Connor, S. C., Wu, W., Sweatman, B. C., Manini, J., Haselden, J. N., Crowther, D. J., & Waterfield, C. J. (2004).
  Effects of feeding and body weight loss on the H-1-NMR-based urine metabolic profiles of male Wistar Han rats: implications for biomarker discovery. Biomarkers, 9, 156-179.
  For almost two decades, H-1-NMR spectroscopy has been used as an 'open' system to study the temporal changes in the biochemical composition of biofluids, including urine, in response to adverse toxic events. Many of these in vivo studies have reported changes in individual metabolites and patterns of metabolites that correlated with toxicological changes. However, many of the proposed novel biomarkers are common to a number of different types of toxicity. These may therefore reflect non-specific effects of toxicity, such as weight loss, rather than a specific pathology. A study was carried out to investigate the non-specific effects on urinary metabolite profiles by administering four hepatotoxic compounds, as a single dose, to rats at two dose levels: hydrazine hydrate (0.06 or 0.08 g kg(-1)), 1,2-dimethylhydrazine (0.1 or 0.3 g kg(-1)), alpha-napthylisothiocyanate (0.1 or 0.15 g kg(-1)) and carbon tetrachloride (1.58 or 3.16 g kg(-1)). The study included weight-matched control animals along with those that were dosed, which were then 'pair-fed' with the treated animals so they achieved a similar weight loss. The urinary metabolite profiles were investigated over time using H-1-NMR spectroscopy and compared with the pathology from the same animals. The temporal changes were analysed statistically using multivariate statistical data analysis including principal component analysis, partial least squares, parallel factor analysis and Fisher's criteria. A number of metabolites associated with energy metabolism or which are partially dietary in origin, such as creatine, creatinine, tricarboxylic acid (TCA) cycle intermediates, phenylacetylglycine, fumarate, glucose, taurine, fatty acids and N-methylnicotinamide, showed altered levels in the urine of treated and pair-fed animals. Many of these changes correlated well with weight loss. Interestingly, there was no increase in ketone bodies ( acetate and beta-hydroxybutyrate), which might be expected if energy metabolism was switched from glycolysis to fatty acid beta-oxidation. In some instances, the metabolites that changed were considered to be nonspecific markers of toxicity, but were also identified as markers of a specific type of toxicity. For example, taurine was raised significantly in carbon tetrachloride-treated animals but reduced in the pair-fed group. However, raised urinary bile acid levels were only seen after alpha-napthylisothiocyanate treatment. The methodology, statistical analysis used and the data generated will help improve the identification of specific markers or patterns of urinary markers of specific toxic effects.

• Cooper, L.G. (1973).
  A multivariate investigation of preferences. Multivariate Behavioral Research, 8, 253-272.
  Three-mode scaling is used to analyse and compare the scalar products derived from preference and similarity judgments of 52 subjects. The stimuli were American soft drinks. The core matrices are heuristically used to form 'homogeneous' groups in the subject space. The method is, however, not necessarily transferrable to other data.

• Cooper, L.G. (1988).
  Competitive maps: The structure underlying asymmetric cross elasticities. Management Science, 34, 707-723.
  A special case of three-mode factor analysis is used to portray the systematic structure underlying asymmetric cross elasticities for a board class of market-share attraction models.

  

• Cooper, L.G., Klapper, D., & Inoue, A. (1996).
  Competitive-component analysis: A new approach to calibrating asymmetric market-share models. Journal of Marketing Research, 33, 224-238.
  Managers are unlikely to keep current with advanced developments in market-response analysis, and technical analysts often lack the market-place knowledge of the many product categories they must track through syndicated data; this is a recipe for bad decisions. The authors present methods based on three-mode factor analysis and multivariate regression that can help both analysts and managers make better decisions regarding whether UPCs should be aggregated into brand units and, if so, how should the aggregation be done; which marketing instruments to track; and how to disentangle correlated promotional strategies. The practicality of this approach is demonstrated by an application to UPC- level data (25 UPCs, seven marketing instruments, and 156 weeks). In this example, to aggregate UPCs within a manufacturer into brand units would distort the relations between the marketing instruments and market responses. A multivariate regression from the competitive- component scores provides a methodologically sound and practical method for calibrating market response in such cases.

• Cooper, M., & DeLacy, I. H. (1994).
  Relationships among analytical methods used to study genotypic variation and genotype-by-environment interaction in plant breeding multi-environment experiments. Theoretical and Applied Genetics, 88, 561-572.
  Abstract Following the recognition of the importance of dealing with the effects of genotype-by-environment (G x E) interaction in multi-environment testing of genotypes in plant breeding programs, there has been substantial development in the area of analytical methodology to quantify and describe these interactions. Three maj or areas where there have been developments are the analysis of variance, indirect selection, and pattern analysis methodologies. This has resulted in a wide range of analytical methods each with their own advocates. There is little doubt that the development of these methodologies has greatly contributed to an enhanced understanding of the magnitude and form of G x E interactions and our ability to quantify their presence in a multi-environment experiment. However, our understanding of the environmental and physiological bases of the nature of G x E interactions in plant breeding has not improved commensurably with the availability of these methodologies. This may in part be due to concentration on the statistical aspects of the analytical methodologies rather than on the complementary resolution of the biological basis of the differences in genotypic adaptation observed in plant breeding experiments. There are clear relationships between many of the analytical methodologies used for studying genotypic variation and G x E interaction in plant breeding experiments. However, from the numerous discussions on the relative merits of alternative ways of analysing G x E interactions which can be found in the literature, these relationships do not appear to be widely appreciated. This paper outlines the relevant theoretical relationships between the analysis of variance, indirect selection and pattern analysis methodologies, and their practical implications for the plant breeder interested in assessing the effects of G x E interaction on the response to selection. The variance components estimated from the combined analysis of variance can be used to judge the relative magnitude of genotypic and G x E interaction variance. Where concern is on the effect of lack of correlation among environments, the G x E interaction component can be partitioned into a component due to heterogeneity of genotypic variance among environments and another due to the lack of correlation among environments. In addition, the pooled genetic correlation among all environments can be estimated as the intraclass correlation from the variance components of the combined analysis of variance. Where G x E interaction accounts for a large proportion of the variation among genotypes, the individual genetic correlations between environments could be investigated rather than the pooled genetic correlation. Indirect selection theory can be applied to the case where the same character is measured on the same genotypes in different environments. Where there are no correlations of error effects among environments, the phenotypic correlation between environments may be used to investigate indirect response to selection. Pattern analysis (classification and ordination) methods based on standardised data can be used to summarise the relationships among environments in terms of the scope to exploit indirect selection. With the availability of this range of analytical methodology, it is now possible to investigate the results of more comprehensive experiments which attempt to understand the nature of differences in genotypic adaptation. Hence a greater focus of interest on understanding the causes of the interaction can be achieved.

• Cooper, M., Woodruff, D. R., Phillips, I. G., Basford, K. E., & Gilmour, A. R. (2001).
  Genotype-by-management interactions for grain yield and grain protein concentration of wheat. Field Crops Research, 69, 47-67.
  The magnitude of genotype-by-management (G x M) interactions for grain yield and grain protein concentration was examined in a multi-environment trial (MET) involving a diverse set of 272 advanced breeding lines from the Queensland wheat breeding program. The MET was structured as a series of management-regimes imposed at 3 sites for 2 years. The management-regimes were generated at each site-year as separate trials in which planting time, N fertiliser application rate, cropping history, and irrigation were manipulated. irrigation was used to simulate different rainfall regimes. From the combined analysis of variance, the G x M interaction variance components were found to be the largest source of G x E interaction variation for both grain yield (0.117 +/- 0.005 t(2) ha(-2); 49% of total G x E 0.238 +/- 0.028 t(2) ha(-2)) and grain protein concentration (0.445 +/- 0.020%(2); 82% of total G x E 0.546 +/- 0.057%(2)), and in both cases this source of variation was larger than the genotypic variance component (grain yield 0.068 +/- 0.014 t(2) ha(-2) and grain protein 0.203 +/- 0.026%(2)). The genotypic correlation between the traits varied considerably with management-regime, ranging from -0.98 to -0.31, with an estimate of 0.0 for one trial. Pattern analysis identified advanced breeding lines with improved grain yield and grain protein concentration relative to the cultivars Hartog, Sunco and Meteor. It is likely that a large component of the previously documented G x E interactions for grain yield of wheat in the northern grains region are in part a result of G x M interactions. The implications of the strong influence of G x M interactions for the conduct of wheat breeding METs in the northern region are discussed.

• Coppi, R. (1982).
  Theoretical and practical problems in the analysis of multidimensional contingency tables. Metron - International Journal of Statistics, 40, 3-35.
  Various techniques of analysis of multidimensional contingency tables are discussed, according to a set of methodological criteria. These concern, basically, the type of data matrix to be examined, the mathematical structure which is to be used (e.g. probability space or just euclidean space, without introducing a probability on it), the nature of the adopted technique (mainly with reference to the existence and type of the objective function possibly associated with the technique). In the above framework the following problems are dealt with: measures of association and model selection criteria (with specific reference to loglinear models), general inference and choice of appropriate objective functions (in the context of non probabilistic techniques of analysis). In the final part of the paper a comparison is carried out between the loglinear model and correspondence analysis approaches. Particular emphasis is given to comparing the definitions and analyses of interaction (among several characters), which are related to each of these techniques.

• Coppi, R. (1986a).
  Analysis of three-way data matrices based on pairwise relation measures. In F. de Antoni, N. Lauro, & A. Rizzi (Eds.) Compstat 1986 (pp. 129-139). Heidelberg: Physica-Verlag.
  The basic types of three-way data matrices are described. The special case of one or more sets of qualitative characters observed on one or more groups of individuals is then examined. Several exploratory techniques of analysis are suggested, based on appropriately defined measures of relationships between individuals or between categories of the investigated characters.

• Coppi, R. (1994).
  An introduction to multiway data and their analysis. Computational Statistics & Data Analysis, 18, 3-13.
  This paper provides a presentation of the main types of multiway data and the most known methods proposed in the literature. General guidelines are provided concerning the various aspects of statistical information conveyed by different data arrays or sets and the methods of data analysis enabling the researcher to extract this information. The description of the techniques focuses on the different methodological approaches in order to enable the reader to find out links among the methods.

  

• Coppi, R., & D'Urso, P. (2003).
  Three-way fuzzy clustering models for LR fuzzy time trajectories. Computational Statistics & Data Analysis, 43, 149-177.
  Fuzzy multivariate time trajectories are defined. For a suitable class, called LR time trajectories, three types of dissimilarity measures are introduced: the instantaneous, the velocity and the simultaneous measures, respectively. Correspondingly, three different kinds of dynamic fuzzy clustering models are suggested, based on a generalization of the Bezdek and Yang and Ko objective functions for fuzzy clustering. The solutions and characteristics of the three models are then illustrated. A comparative appraisal of their practical meaning is proposed by means of an application to the time pattern of the subjective judgments expressed by a sample of web navigators on different types of banners. Some indications for future research in this methodological domain are finally provided.

• Cornelius, E.T., III, Hakel, M.D. & Sackett, P.R. (1979).
  A methodological approach to job classification for performance appraisal purposes. Personnel Psychology, 32, 283-297.
  Data from a job inventory containing 153 worker-oriented ele- ments were obtained from 2023 incumbents across 18 jobs and 5 levels of responsibility (ranks) in the US Coast Guard. The mean values of the 153 scores of the respondents for each of the job x responsibility combinations were used for a T3 analysis. No scaling of the data; varimax rotations. The elements of the core matrix were interpreted in relation to the average value of each plane. The relatively clear solution was used to decide on the number and contents of forms necessary for evaluating personnell in the US Coast Guard.

  

• Courcoux, P., Devaux, M.F., & Bouchet, B. (2002).
  Simultaneous decomposition of multivariate images using three-way data analysis: Application to the comparison of cereal grains by confocal laser scanning microscopy. Chemometrics and Intelligent Laboratory Systems, 62, 103-113.
  A three-way data analysis method is presented which is adapted to compare multispectral images such as those acquired by fluorescence microscopy. Spectral images are multivariate and can be considered as sets of pixels for which different spectral intensities have been measured. The algorithm proposed is based on the simultaneous decomposition of the covariance matrices calculated, in the present work, for fluorescence variables. This method allows the extraction of components common to each image and reveals their specificities by means of specific weights. The technique is illustrated by the analysis of microscopic image sequences acquired for five varieties of wheat and barley grains in 19 fluorescence conditions. The method made it possible to compare the fluorescence behaviors observed in the images and the strong similarities of the external tissues for the five cereals were highlighted.

       

• Crossa, J., Basford, K., Taba, S., DeLacy, I., & Silva, E. (1995).
  Three-mode analyses of maize using morphological and agronomic attributes measured in multilocational trials. Crop Science, 35, 1483-1491.
  This study provides examples of the application of three-mode methods to agronomic and morphological data from multiattribute and multilocational trials of accessions from two Mexican maize (Zea mays L.) races, Cónico and Tuxpeño. Initially, Cónico accessions from the state of México were classified into five subgroups, whereas Tuxpeño accessions were classified into three subgroups based on the ecogeographical region of origin. A three-mode classification method reassigned accessions to form more homogeneous subgroups. The new Cónico groups were called C1, C2, C3, C4, and C5 and the new Tuxpeño grups were named T1, T2, and T3. Intra- racial genetic diversity was investigated by three-mode principal component analysis. Most Cónico Group C3 accessions had low grain yield, early maturity, short plants, and short ears. Group C2 accessions had the tallest plants, the shortest kernels, and the narrowest ears; whereas Group C4 accessions had the longest kernels and the widest ears. Accessions in Groups C1 and C5 were the highest yielders and had the longest ears. A core subset would include accessions from all five subgroups with an approximate average response for all attributes, plus some accessions with extreme responses. Tuxpeño Group T2 accessions were earlier and had shorter plants than the accessions from Group T1. Group T3 accessions had higher grain yield and longer kernel length. A core subset of Tuxpeño accessions could be formed as suggested for Cónico. If a Tuxpeño core subset with high grain yield and shorter plant type is desired, more accessions from Group T2 and T3 should be included.

• Crouch, S. R., Coello, J., Maspoch, S., & Porcel, M. (2000).
  Evaluation of classical and three-way multivariate calibration procedures in kinetic-spectrophotometric analysis. Analytica Chimica Acta, 424, 115-126.
  The bidimensional multivariate regression procedures: multiple linear regression (MLR), principal component regression (PCR), partial least squares regression (PLS) and continuum regression (CR), and several N-way methods such as N-way PLS (nPLS) and parallel factor analysis (PARAFAC) are tested as calibration methods for the kinetic-spectrophotometric determination of ternary mixtures in a pseudo first-order kinetic system. The different calibration procedures were first applied to computer simulated kinetic-spectrophotometric data where the effect of spectral overlap and the differences in the kinetic constants were evaluated at a low level of experimental noise. Later they were applied to the stopped-flow kinetic-spectrophotometric simultaneous resolution of Co(II), Ni(II) and Ga(III) using 4-(2-pyridylazo)resorcinol (PAR) as a chromogenic reagent. Accurate estimations of concentrations with relative standard errors of prediction of about 8% were obtained even though a high degree of spectral overlap and similar rate constants were present. The study of the influence of experimental noise on the 3-component system justifies the difference between the simulations and the experimental results for the different calibration procedures. PARAFAC and MLR did not allow the resolution of the proposed 3-component system. CR provided slightly better results than those obtained by PLS, PCR and nPLS.

       

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

• Cullen, T.F., & Crouch, S.R. (1997).
  Multicomponent kinetic determinations using multivariate calibration techniques. Mikrochimica Acta, 126, 1-9.**
  Multivariate calibration techniques for use in multicomponent kinetic-based determinations are reviewed. Multivariate calibration is a chemometric tool that continues to grow in popularity among analytical chemists. Multicomponent kinetic methods depend on differences in rates of reactions or processes to distinguish among the components. Kinetic profiles or a combination of kinetic profiles and spectra are commonly used. Because of their ability to process large quantities of data, multivariate calibration techniques are well suited for kinetic-based determinations. The concepts and principles of multivariate calibration are discussed first. Classical least squares regression, principal component regression, partial least squares regression and artificial neural networks are the multivariate calibration techniques considered here in detail. Recent examples of the application of these techniques to multicomponent kinetic determinations are reviewed. Both single and multiwavelength kinetic data are considered.

           

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