Three-Mode Abstracts, Part K

INDEX

•  
• Kapteyn, A., Neudecker, H., & Wansbeek, T. (1982).
  An approach to n-mode components analysis. Psychometrika, 51, 269-275.
  As an extension of Lastovicka's four-mode components analysis an n-mode components analysis is developed. Using a convenient notation, both a canonical and a least squares solution are derived. The relation between both solutions and their computational aspects are discussed. An n-mode extension of the three-mode model and algorithm developed by Kroonenberg & De Leeuw (1980) is proposed. By using some new notation an elegant and concise presentation of the theory is given. Their model is compared with the standard Tucker (1966) approach and its extension to four modes by Lastovicka (1981).

  

• Karnas, G. (1975).
  Note sur une procédure d'analyse de données relatives à une correspondance ternaire ou pseudo-ternaire par la méthode d'analyse binaire de Faverge. Le Travail Humain, 38, 287-300.
  Proposal to string out a three-mode matrix in one of three ways with the object to perform on the resultant (two-mode) matrix a variant of the singular value decomposition (called the binary method of Faverge). One of the stringing-out proposals is identical to forming a combination-mode (Tucker, 1966). The other two are variants of the same idea. Illustrated with beer appreciation data, and data of tram conductors judging aspects of their profession.

• Karukstis, K.K., Suljak, S.W., Waller, P.J., Whiles, J.A., & Thompson, E.H.Z. (1996).
  Fluorescence analysis of single and mixed micelle systems of SDS and DTAB. Journal of Physical Chemistry, 100, 11125-11132.
  Structural features of the single and mixed micellar systems of sodium dodecyl sulfate (SDS) and dodecyltrimethylammonium bromide (DTAB) were characterized using the fluorescence probe 6-propionyl-2-(dimethylamino)naphthalene (Prodan). In particular, our investigations capitalized on the spectral sensitivity of Prodan to its environment as well as the extensive solubility of Prodan in solvents of varying polarity and/or hydrophobicity to effectively use a three-mode factor analysis technique to resolve the coincident emission from Prodan in multiple microenvironments of single and mixed micelle systems.

  
         

• Kelly, E.F., Lenz, J.E., Franaszczuk, P.J., & Truong, Y.K. (1997).
  A general statistical framework for frequency-domain analysis of EEG topographic structure. Computers and Biomedical Research, 30, 129-164.**
  A wide variety of rhythmic electrophysiological phenomena-including driven, induced, and endogenous activities of cortical neuronal masses-lend themselves naturally to analysis using frequency-domain techniques applied to multichannel recordings that discretely sample the overall spatial pattern of the rhythmic activity. For such cases, a large but so far poorly utilized body of statistical theory supports a third major approach to topographic analysis, complementing the more familiar mapping and source-recovery techniques. These methods, many of which have only recently become computationally feasible, collectively provide general solutions to the problem of detecting and characterizing systematic differences that arise-not only in the spatial distribution of the activity, but also in its frequency-dependent between-channel covariance structure-as a function of multiple experimental conditions presented in conformity with any of the conventional experimental designs. This application- oriented tutorial review provides a comprehensive outline of these resources, including: (1) real multivariate analysis of single-channel spectral measures (and measures of between-channel relationships such as coherence and phase), (2) complex multivariate analysis based on multichannel fourier transforms, and (3) complex multivariate analysis based on multichannel parametric models. Special emphasis is placed on the potential of the multichannel autoregressive model to support eeg (and meg) studies of perceptual and cognitive processes. (c) 1997 academic press.

  ---

• Kettenring, J. R. (1971).
  Canonical analysis of several sets of variables. Biometrika, 58, 433-460.
  Five extensions of the classical two-set theory of canonical correlation analysis to three or more sets are considered. For each one, a model of the general principal component type is constructed to aid in motivating, comparing and understanding the methods. Procedures are developed for finding the canonical varialbles associated with the different approaches. Some practical considerations and an example are also included.

• Keramidas, E.M., Devlin, S.J., & Gnanadesikan, R. (1987).
  A graphical procedure for comparing the principal components of several covariance matrices. Communications in Statistics, Part B - Simulation and Computation, 16, 161-191.
  Principal components analysis is an extensively used tool for reduction of dimensionality in multivariate analyses. In many applications, however, little attempt is made to compare principal components solutions (i.e., eigenvectors) across many samples. Methods are needed for assessing the degree of similarity of corresponding eigenvectors, a problem that is meaningful in the presence of clearly separated eigenvalues. This paper proposes a gamma probability plotting procedure for a measure of the angle between a pair of eigenvectors, or equivalently, the distance between points on the unit sphere defined by such vectors. One of the vectors in the pair is the principal component of a sample, and the other can be either a prespecified vector or a "typical" vector obtained from the corresponding eigenvectors in all samples. Simulations, as well as real-data examples, are presented.

• Kettenring, J.R. (1983a).
  A case study in data analysis. Proceedings of Symposia in Applied Mathematics, 28, 105-139.
  A set of data consisting of annual revenues of Bell operating companies broken down into several components is analyzed using two different models. The intent is not only to describe the results but also to convey as many facets of the data analytic process as possible.

  

• Kettenring, J.R. (1983b).
  Components of interaction in analysis of variance models with no replications. In P.K. Sen (Ed.), Contributions to Statistics: Essays in Honour of Norman L. Johnson (pp. 283-297). Amsterdam: North-Holland.
  An old problem in statistics is how to properly analyze data in a multi- way table with only one observation per cell. Standard analysis of variance models for such data, which include all possible "main effect" and "interaction" terms, are unsatisfactory because the number of independent parameters involved equals the number of data points.
  For two-way tables, a popular antidote is to impose a particular structural form on the interactions. One approach, suggested by Gollob (1968) and Mandel (1971), is based on the singular value decomposition of the matrix of two-way interaction terms estimated from the data. Probability plots of the singular values from this decomposition vs. computer-generated null values provide an effective way of informally implementing this approach.
  Extensions of these ideas to deal with three-way interactions are also feasible. In some contexts, it makes sense to treat these interactions as if they were layers of two-way interactions and to apply methods which exploit this viewpoint. More generally, an extended form of the singular value decomposition, based on the work of Harshman (1970) and Carroll and Chang (1970), can be applied to these interactions although this method is not as simple as in the two-way case. Each of these approaches could be broadened to treat higher-way interactions as well.
  This paper develops all of the methods mentioned and illustrates them using a three- way array of revenue data for Bell System operating telephone companies.

  
         

• Kiers, H.A.L. (1988).
  Comparison of "Anglo-Saxon" and "French" three-mode methods. Statistique et Analyse des Données, 13, 14- 32.
  Seven methods for the analysis of three-mode data are described in terms of minimizing loss functions. On the basis of this description global and specific comparisons are made between a number of the methods presented here. Special attention is paid to the comparison of french methods and anglo-saxon methods.

• Kiers, H.A.L. (1989a).
  INDSCAL for the analysis of categorical data. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 155-167).
  Amsterdam: Elsevier.
  A method for PCA of categorical variables is proposed which does not only yield loadings for the categorical variables, but also coordinates for the observation units. The method is based on INDSCAL applied to a set of matrices whose entries are similarities between the observation units with respect to the categorical variables. The resulting methods is shown to be a compromise between Multiple Correspondence Analysis and a method proposed by Cazes, Bonnefous, Baumerder and Pagès. Some properties of the method proposed here are described.

• Kiers, H.A.L. (1989c).
  An alternating least squares algorithm for fitting the two- and three- way DEDICOM model and the idioscal model. Psychometrika, 54, 515-521.
  The DEDICOM nodel is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions. The present paper offers an alternating least squares algorithm for fitting the DEDICOM model. The model can be generalized to represent any number of sets of relations among the same set of objects. An algorithm for fitting this three-way DEDICOM model an algorithm is developed for fitting the IDIOSCAL model in the least squares sense.

• Kiers, H.A.L. (1989d).
  A computational short-cut for INDSCAL with orthonormality constraints on positive semi-definite matrices of low rank. Computational Statistics Quarterly, 5, 119-135.
  When INDSCAL with orthonormality constraints (INDORT) is to be applied to a set of very large similarity matrices, one is confronted with huge computational problems. When those very large matrices are positive semi-definite (p.s.d.) matrices of low rank, computations can be facilitated to a large extent. The present paper describes a simplified algorithm for the INDORT analysis of such matrices. One of the applications of INDORT on low-rank p.s.d. matrices is that of INDORT on a set of quantification matrices for qualitative and/or quantitative variables. The implications of using the simplified algorithm for this INDORT analysis are worked out. It is concluded that, in case INDORT is applied to quantification matrices for the qualitative variables exclusively, the computation of the solution uses only the categories frequencies and the bivariate frequencies of pairs of categories from two different variables. In this way INDORT for qualitative data can be applied when one only has the total bivariate contingency table for all variables.

• Kiers, H.A.L. (1990a).
  Procrustes-pc v2.0: Een programma voor gegeneraliseerde Procrustes-analyse. [Procrustes-pc v2.0: A program for generalized Procrustes-analysis.] Kwantitatieve Methoden, 11, 177-188.
  In this paper a description and evaluation are given of a pc-program for generalized Procrustes-analysis. After a short discussion of the aim of generalized Procrustes-analysis, the possibilities and restrictions of the program are discussed. The operation and user-friendliness of the program are looked into using several simple test-analyses.

• Kiers, H.A.L. (1991a).
  Hierarchical relations among three-way methods. Psychometrika, 56, 449-470.
  Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.

• Kiers, H.A.L. (1991b).
  Simple structure in component analysis techniques for mixtures of qualitative and quantitative variables. Psychometrika, 56, 197-212.
  The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitaive variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.

  

• Kiers, H.A.L. (1992).
  TUCKALS core rotations and constrained TUCKALS modelling. Statistica Applicata, 4, 659-667.
  TUCKALS3 is a method for analyzing a three-way data set. The method decomposes the data into component matrices for all three sets of components. The solution for the component matrices and the core array are determined up to a (possibly oblique) rotation only. In the present paper some procedures are proposed for rotating the solution such that the core becomes optimally simple in a particular way. An alternative procedure for obtaining a simple core is to fit the TUCKALS3 model subject to the constraint that the core has zeros at specified places. An algorithm for this procedure is sketched and the procedure is illustrated by means of an example.

• Kiers, H.A.L. (1993a).
  An alternating least squares algorithm for PARAFAC2 and three-way DEDICOM. Computational Statistics and Data Analysis, 16, 103-118.
  PARAFAC2 is a method for analyzing three-way data consisting of symmetric frontal slices. Three-way DEDICOM can be considered a generalization of PARAFAC2 in that it fits essentially the same model to three-way data consisting of square frontal slices that may be asymmetric. In the present paper, an alternating least squares algorithm is developed for three-way DEDICOM, and an algorithm for PARAFAC2 is derived from it. The performance of the algorithms is studied for some empirical and synthetical data.

  

• Kiers, H.A.L. (1993b).
  A comparison of techniques for finding components with simple structure. In C.M. Cuadras & C.R. Rao (Eds). Multivariate Analysis: Future Directions 2 (pp. 67-86). Amsterdam: Elsevier.
  Principal component analysis (PCA) is usually followed by rotation to simple structure to facilitate the interpretation of the components. Recently, some alternatives to PCA have been developed, in which simple structure is part of the criterion optimized, and is no longer seen as a secondary objective. These methods, Principal cluster components analysis (PCCA), INDOMIX, 'Varimax Optimization' and INDSCAL applied to quantification matrices for quantitative variables, are designed for situations where, apart from fit, parsimony of the solution is deemed valuable as well. In the present study, these techniques are described in some detail, and compared on theoretical grounds. Next, in a simulation study, the methods are compared to PCA followed by three different simple structure rotations (Varimax, Promax, and Orthoblique rotation). All techniques have been applied to artificial data sets with a known simple structure, to which noise had been added. It is studied to what extent the original structure is recovered, how 'simple' the different solutions are, and how a possible gain in simplicity is offset by a loss of fit. The main conclusion is that, if it is desired to gain simplicity and one is prepared to incur a small loss of fit, then INDOMAX and Varimax Optimization should not be used, but both PCCA and INDSCAL can be used, with a slight preference for INDSCAL because it gives a better ratio of gain of simplicity versus loss of fit. The paper is concluded by the INDSCAL analysis of an empirical data set.

• Kiers, H.A.L. (1993c).
  Handling ordinal variables in three-way analysis of quantification matrices for variables of mixed measurement levels. British Journal of Mathematical and Statistical Psychology, 46, 135-152.
  For the analysis of variables of mixed measurement levels a class of methods can be used that is based on three-way analysis of quantification matrices for nominal or quantitative variables. This class of methods incorporated some well-known techniques but also offers a series of interesting new alternatives for the analysis of nominal or quantitative variables. Ordinal variables have received hardly any attention in this class of methods, and are usually treated as if they are quantitative variables. In the present paper this gap is filled by constructing quantification matrices for ordinal variables via optimal scaling of the ordinal variables, thus yielding optimal quantification matrices for these variables. Algorithms for this optimal scaling procedure are developed, and the optimal scaling procedures are compared to optimal scaling of ordinal variables in the context of principal components analysis and multidimensional scaling.

• Kiers, H.A.L. (1993d).
  The exploratory analysis of qualitative variables by means of three-way analysis of two types of quantification matrices. Applied Stochastic Models and Data Analysis, 9, 301-317.
  A comparison is made between a number of techniques for the exploratory analysis of qualitative variables. The paper mainly focuses on a comparison between multiple correspondence analysis (MCA) and Gower's principal co-ordinates analysis (PCO), applied to qualitative variables. The main difference between these methods is in how they deal with infrequent categories. It is demonstrated that MCA solutions can be dominated by infrequent categories, and that, especially in such cases, PCO is a useful alternative to MCA, because it tends to downweight the influence of infrequent categories. Apart from studying the difference between MCA and PCO, other alternatives for the analysis of qualitative variables are discussed, and compared to MCA and PCO.

• Kiers, H. A. L. (1995).
  Maximization of sums of quotients of quadratic-forms and some generalizations. Psychometrika, 60, 221-245.
  Monotonically convergent algorithms are described for maximizing six (constrained) functions of vectors x, or matrices X with columns x(1),..., x(r). T hese functions are h(1)(x) = Sigma(k) (x'A(k)x)(x'C(k)x)(-1), H-1(X) = Sigma(k) tr (X'A(k)X)(X'C(k)X)(-1), (h) over tilde(1)(X) = Sigma(k) Sigma(l)(x'(l)A(k)x(l))(x'(l)C(k)x(l))(-1) with X constrained to be columnwise orthonormal, h(2)(x) = Sigma(k) (x'A(k)x)(2)(x'C(k)X)(-1) subject to x'x = 1, H-2(X) = Sigma(k) tr (X'A(k)X)(X'A(k)X)'(X'C(k)X)(-1) subject to X'X = I, and (h) over tilde(2)(X) = Sigma(k) Sigma(l) (x'(l)A(k)x(l))(2)(x'(l)C(k)x(l))(-1) subject to X'X = I. In these functions the matrices C-k are assumed to be positive definite. The matrices A(k) can be arbitrary square matrices. The general formulation of the functions and the algorithms allows for application of the algorithms in various problems that arise in multivariate analysis. Several applications of the general algorithms are given. Specifically, algorithms are given for reciprocal principal components analysis, binormamin rotation, generalized discriminant analysis, variants of generalized principal components analysis, simple structure rotation for one of the latter variants, and set component analysis. For most of these methods the algorithms appear to be new, for the others the existing algorithms turn out to be special cases of the newly derived general algorithms.

• Kiers, H.A.L. (1997a).
  A modification of the SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models Journal of Classification, 14, 297-310.
  The SINDCLUS algorithm for fitting the ADCLUS and INDCLUS models deals with a parameter matrix that occurs twice in the model by considering the two occurrences as independent parameter matrices. This procedure has been justified empirically by the observation that upon convergence of the algorithm to the global optimum, the two independently treated parameter matrices turn out to be equal. In the present paper, results are presented that contradict this finding, and a modification of SINDCLUS is presented which obviates the need for independently treating two occurrences of the same parameter matrix.

• Kiers, H.A.L. (1997b).
  Three-mode orthomax rotation. Psychometrika, 62, 579-598.
  Factor analysis and principal components analysis (pca) are often followed by an orthomax rotation to rotate a loading matrix to simple structure. The simple structure is usually defined in terms of the simplicity of the columns of the loading matrix. In three-mode pca, rotational freedom of the so called core (a three-way array relating components for the three different modes) can be used similarly to find a simple structure of the core. Simple structure of the core can be defined with respect to ail three modes simultaneously, possibly with different emphases on the different modes. The present paper provides a fully flexible approach for orthomax rotation of the core to simple structure with respect to three modes simultaneously. Computationally, this approach relies on repeated (two-way) orthomax applied to supermatrices containing the frontal, lateral or horizontal slabs, respectively. The procedure is illustrated by means of a number of exemplary analyses. As a by-product, application of the three-mode orthomax procedures to two-way arrays is shown to reveal interesting relations with and interpretations of existing two-way simple structure rotation techniques.

• Kiers, H.A.L. (1997c).
  Weighted least squares fitting using ordinary least squares algorithms. Psychometrika, 62, 251-266.
  A general approach for fitting a model to a data matrix by weighted least squares (wls) is studied. This approach consists of iteratively performing (steps of) existing algorithms for ordinary least squares (ols) fitting of the same model. The approach is based on minimizing a function that majorizes the wls loss function. The generality of the approach implies that, for every model for which an ols fitting algorithm is available, the present approach yields a wls fitting algorithm. In the special case where the wls weight matrix is binary, the approach reduces to missing data imputation.

• Kiers, H. A. L. (1997d).
  Discrimination by means of components that are orthogonal in the data space. Jounal of Chemometrics, 11, 533-545.
  Krzanowski (J. Chemometrics, 9, 509 (1995)) proposed a method for obtaining so-called orthogonal canonical variates (henceforth called components) for discrimination purposes. In contrast with ordinary discriminant analysis, this method employs components that are orthogonal in the original data space. These components are derived in a successive way, thus optimizing discrimination of a component given the previously extracted components. Two alternative procedures are proposed to extract the desired number of components simultaneously, yielding a better overall discrimination. The simultaneous approaches are applied to the same two data sets as analysed by Krzanowski, as well as to Anderson's Iris data, and a comparison of discriminatory quality of the solutions is presented.

• Kiers, H.A.L. (1998a).
  Recent developments in three-mode factor analysis: Constrained three-mode factor analysis and core rotations. In C. Hayashi, N. Ohsumi, K. Yajima, Tanaka, Y., H.-H. Bock, & Y. Baba, Data Science, Classification, and Related Methods (pp. 563-574). Tokyo: Springer- Verlag.
  A review is presented of some recent developments in three-mode factor analysis, that are all aimed at reducing the difficulties in interpreting three-mode factor analysis solutions. First, variants of three-mode factor analysis with zero constraints on the core are described, and attention is paid to algorithms for fitting these models, as well as to uniqueness of the representations. Next, various methods for rotation of the core to simple structure are discussed and related to two-way simple structure rotation techniques. In the concluding section, new perspectives for simplification of the interpretation of three-mode factor analysis solutions are discussed.

• Kiers, H.A.L. (1998b)
  An overview of three-way analysis and some recent developments. In A. Rizzi, M. Vichi & H.-H. Bock (Eds.), Advances in data science and classification (pp. 593-602). Berlin: Springer.
  A brief overview is presented of techniques for the analysis of three-way data. Techniques will be distinguished in terms of the type of data they are meant to analyze, on the basis of the type of analysis pursued, and the three-way models are distinguished with respect to their uniqueness. Next, recent approaches are discussed for eliminating nonuniqueness by imposing constraints and rotation to simple structure.

• Kiers, H.A.L.(1998c).
  A three-step algorithm for CANDECOMP/PARAFAC analysis of large data sets with multicollinearity. Journal of Chemometrics, 12, 155-171.
  Fitting the CANDECOMP/PARAFAC model by the standard alternating least squares algorithm often requires very many iterations. One case in point is that of analysing data with mild to severe multicollinearity. If, in addition, the size of the data is large, the computation of one CANDECOMP/PARAFAC solution is very time-consuming. The present paper describes a three-step procedure which is much more efficient than the ordinary CANDECOMP/PARAFAC algorithm, by combining the idea of compression with a form of regularization of the compressed data array.

• Kiers, H.A.L.(1998d).
  Joint orthomax rotation of the core and component matrices resulting from three-mode principal components analysis. Journal of Classification, 15, 245-263.
  The analysis of a three-way data set using three-mode principal components analysis yields component matrices for all three modes of the data, and a three-way array called the core, which relates the components for the different modes to each other. To exploit rotational freedom in the model, one may rotate the core array (over all three modes) to an optimally simple form, for instance by three-mode orthomax rotation. However, such a rotation of the core may inadvertently detract from the simplicity of the component matrices. One remedy is to rotate the core only over those modes in which no simple solution for the component matrices is desired or available, but this approach may in turn reduce the simplicity of the core to an unacceptable extent. In the present paper, a general approach is developed, in which a criterion is optimized that not only takes into account the simplicity of the core, but also, to any desired degree, the simplicity of the component matrices. This method (in contrast to methods for either core or component matrix rotation) can be used to find solutions in which the core and the component matrices are all reasonably simple.

• Kiers, H.A.L.(1998e).
  Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure. Computational Statistics & Data Analysis, 28, 307-324.
  SIMPLIMAX is proposed as a procedure for oblique rotation of a factor loadings matrix to simple structure. The distinguishing feature of this method is that it rotates the loading matrix so that after rotation the m smallest elements have a minimal sum of squares (where m is specified in advance). In the present paper, the SIMPLIMAX method is generalized to handle three-way arrays: Three-way SIMPLIMAX finds oblique simple structure rotations of the core matrix that results from a three-mode factor analysis. Specifically, three-way SIMPLIMAX minimizes the sum of the m smallest elements of the rotated core array. An algorithm for three-way SIMPLIMAX is presented, the performance of the algorithm is discussed, some applications are shown, and it is indicated how the method can be used for rotation of solutions of N-mode factor analysis, and for rotation over a subset of the modes of an N-mode core array.

• Kiers, H.A.L. (2000a).
  Some procedures for displaying results from three-way methods. Journal of Chemometrics, 14, 151-170.
  Three-way Tucker analysis and CANDECOMP/PARAFAC are popular methods for the analysis of three-way data (data pertaining to three sets of entities). To interpret the results from these methods, one can, in addition to inspecting the component matrices and the core array, inspect visual representations of the outcomes. In this paper, first an overview is given of plotting procedures currently in use with three-way methods. Not all of these optimally correspond to the actual approximation of the data furnished by the three-way method at hand. Next it is described how plotting procedures can be designed that do correspond exactly to the low-dimensional description of the data by means of the three-way method at hand, and it is indicated to what extent these correspond to the ones currently in use. Specifically, procedures are described for displaying either one set of entities (e.g. a set of chemical samples) in two- or three-dimensional plots, or a set of combinations of entities (e.g. pertaining to each object at each time point, thus providing "trajectories" for each object). Furthermore, it is shown how, in these plots, the other entities can be plotted simultaneously (e.g. superimposing the variables on a plot with trajectories for objects). Both procedures are summarized in an appendix.

• Kiers, H.A.L. (2000b).
  Towards a standardized notation and terminology in multiway analysis. Journal of Chemometrics, 14, 105-122.
  This paper presents a standardized notation and terminology to be used for three- and multiway analyses, especially when these involve (variants of) the CANDECOMP/PARAFAC model and the Tucker model. The notation also deals with basic aspects such as symbols for different kinds of products, and terminology for three- and higher-way data. The choices for terminology and symbols to be used have to some extent been based on earlier (informal) conventions. Simplicity and reduction of the possibility of confusion have also played a role in the choices made.

• Kiers, H. A. L. (2004).
  Bootstrap confidence intervals for three-way methods. Journal of Chemometrics, 18, 22-36.
  Results from exploratory three-way analysis techniques such as CANDECOMP/PARAFAC and Tucker3 analysis are usually presented without giving insight into uncertainties due to sampling. Here a bootstrap procedure is proposed that produces percentile intervals for all output parameters. Special adjustments are offered for handling the non-uniqueness of the solutions. The percentile intervals indicate the instability of the sample solutions. By means of a simulation study it is demonstrated that the percentile intervals can fairly well be interpreted as confidence intervals for the output parameters.

• Kiers, H.A.L., Cléroux, R., & Tenberge, J.M.F. (1994).
  Generalized canonical-analysis based on optimizing matrix correlations and a relation with idioscal. Computational Statistics & Data Analysis, 0, 0.**
  Carroll's method for generalized canonical analysis of two or more sets of variables is shown to optimize the sum of squared inner-product matrix correlations between a consensus matrix and matrices with canonical variates for each set of variables. In addition, the method that analogously optimizes the sum of squared rv matrix correlations (proposed by escoufier, 1973) between a consensus matrix and matrices with canonical variates, can be interpreted as an application of carroll and chang's idioscal. A simple algorithm is developed for this and other applications of idioscal where the similarity matrices are positive semi-definite.

  ---

• Kiers, H.A.L., & Der Kinderen, A. (2003).
  A fast method for choosing the numbers of components in Tucker3 analysis British Journal of Mathematical and Statistical Psychology, 56, 119-125.
  Recently, Timmerman and Kiers proposed an effective procedure for choosing the numbers of components in Tucker3 analysis, a kind of component analysis of three-way data. The procedure, however, is rather time-consuming, relying on very many complete Tucker3 analyses. Here, an alternative procedure is proposed, which basically relies on a single, quick analysis of the three-way data set. In a simulation study it was found that the new procedure is comparable in its effect to the original procedure.
  
  
• Kiers, H.A.L., & Harshman, R.A. (1997).
  Relating two proposed methods for speedup of algorithms for fitting two- and three-way principal component and related multilinear models. Chemometrics and Intelligent Laboratory Systems, 36, 31-40.
  Multilinear analysis methods such as component (and three-way component) analysis of very large data sets can become very computationally demanding and even infeasible unless some method is used to compress the data and/or speed up the algorithms. We discuss two previously proposed speedup methods. (a) Alsberg and Kvalheim have proposed use of data simplification along with some new analysis algorithms. We show that their procedures solve the same problem as (b) the more general approach proposed (in a different context) by Carroll, Pruzansky, and Kruskal. In the latter approach, a speed improvement is attained by applying any (three-mode) PCA algorithm to a small (three-way) array derived from the original data. Hence, it can employ the new algorithms by Alsberg and Kvalheim, but, as is shown in the present paper, it is easier and often more efficient to apply standard (three-mode) PCA algorithms to the small array. Finally, it is shown how the latter approach for speed improvement can also be used for other three-way models and analysis methods (e.g., PARAFAC/CANDECOMP and constrained three-mode PCA).

• Kiers, H.A.L., & Krijnen, W.P. (1991).
  An efficient algorithm for PARAFAC of three-way data with large numbers of observation units. Psychometrika, 56, 147-152.
  The CANDECOMP algorithm for the PARAFAC analysis of n x m x p three- way arrays is adapted to handle arrays in which n > mp more efficiently. For such arrays, the adapted algorithm needs less memory space to store the data during the iterations, and uses less computation time than the original CANDECOMP algorithm. The size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units (n) in the data.

• Kiers, H.A.L., Kroonenberg, P.M., & Ten Berge, J.M.F. (1992).
  An efficient algorithm for TUCKALS3 on data with large numbers of observation units. Psychometrika, 57, 415-422
  A modification of the TUCKALS3 algorithm is proposed that handles three-way arrays of order I x J x K for any I. When I is much larger than JK, the modified algorithm needs less work space to store the data during the iterative part of the algorithm than does the original algorithm. Because of this and the additional feature that execution speed is higher, the modified algorithm is highly suitable for use on personal computers.

  

• Kiers, H.A.L., & Marchetti, G.M. (1994).
  Handling large numbers of observation units in three-way methods for the analysis of qualitative and quantitative two-way data. Computational statistics, 9, 41-64.
  Recently, a number of methods have been proposed for the exploratory analysis of mixtures of qualitative and quantitative variables. In these methods for each variable an object by object similarity matrix is constructed, and these are consequently analyzed by means of three-way methods like indscal, idioscal and tuckals-3. When the number of observation units (objects) is large, algorithms for indscal, idioscal and tuckals-3 become. Inefficient or even infeasible. The present paper offers variants of these algorithms that can handle large numbers of objects in case the similarity matrices are of rank much smaller than the number of objects, which is usually the case. In addition, it is shown that results of the three-way methods at hand are essentially based only on certain aggregate measures for the variables, like variances and covariances for numerical variables, and bivariate and marginal frequencies for nominal variables.

• Kiers, H.A.L., & Smilde, A.K. (1995).
  Some theoretical results on second-order calibration methods for data with and without rank overlap. Journal of Chemometrics, 9, 179-195.
  Gram, a method for second-order calibration, has been introduced by Sánchez and kowalski and later modified by wilson, Sánchez and kowalski. The methods are based on the claim that, in cases without measurement error they yield correct estimates for the concentration ratios and profiles of (rank-one) analytes present in sample and mixture. This claim has not been proven rigorously. In the present paper, rigorous proofs are given for situations where the claims are valid indeed. In addition, it is shown that parafac, an alternative method for second-order calibration, can be used to obtain the same results. Next it is shown that the claims do not hold in cases with 'rank overlap' (partly overlapping profiles) and it is proven that a procedure by wang et al. Can still be used to assess some of the concentration ratios. A general framework is provided for a variety of second-order calibration problems and the extent to which quantitative and qualitative information can be expected is given.

• Kiers, H.A.L., & Smilde, A.K. (1998).
  Constrained three-mode factor analysis as a tool for parameter estimation with second-order instrumental data Journal of Chemometrics, 12, 125-147.
  Three-mode factor analysis models are often used in exploratory analysis of three-way data. However, in some situation it is a priori known that a particular constraint three-mode factor analysis (C3MFA) model describes an underlying process exactly. In such situations, fitting C3MFA model to a data set can be used for both quantitative analysis (e.g. estimating concentrations of a chemical substance in a mixture) and qualitative analysis (e.g. on the basis of certain subsets of parameters one can identify the substance in a mixture). In this paper a general algorithm for fitting a range of such C3MFA models is proposed. Whether C3MFA is used for qualitative or quantitative analyses, in both cases it is crucial that the relevant parameter estimates are uniquely determinable. In the present paper it is discussed how and to what extent uniqueness of certain model parameters can be assessed.

• Kiers, H.A.L., & Ten Berge, J.M.F. (1989).
  Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations. Psychometrika, 54, 467-473.
  Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Merdith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.

• Kiers, H.A.L., Ten Berge, J.M.F., & Bro, R. (1999).
  PARAFAC2 - Part I. A direct fitting algorithm for the PARAFAC2 model. Journal of Chemometrics, 13, 275-294.
  PARAFAC is a generalization of principal component analysis (PCA) to the situation where a set of data matrices is to be analysed. If each data matrix has the same row and column units, the resulting data are three-way data and can be modelled by the PARAFAC1 model. If each data matrix has the same column units but different (numbers of) row units, the PARAFAC2 model can be used. Like the PARAFAC1 model, the PARAFAC2 model gives unique solutions under certain mild assumptions, whereas it is less severely constrained than PARAFAC1. It may therefore also be used for regular three-way data in situations where the PARAFAC1 model is too restricted. Usually the PARAFAC2 model is fitted to a set of matrices with cross-products between the column units. However, this model- fitting procedure is computationally complex and inefficient. In the present paper a procedure for fitting the PARAFAC2 model directly to the set of data matrices is proposed. It is shown that this algorithm is more efficient than the indirect fitting algorithm. Moreover, it is more easily adjusted so as to allow for constraints on the parameter matrices, to handle missing data, as well as to handle generalizations to sets of three- and higher-way data. Furthermore, with the direct fitting approach we also gain information on the row units, in the form of 'factor scores'. As will be shown, this elaboration of the model in no way limits the feasibility of the method. Even though full information on the row units becomes available, the algorithm is based on the usually much smaller cross-product matrices only.

• Kiers H.A.L., Ten Berge, J.M.F., & Rocci, R. (1997).
  Uniqueness of three-mode factor analysis models with sparse cores: The 3×3×3 case. Psychometrika, 62, 349-374.
  Three-mode PCA and PARAFAC are methods to describe three-way data. The three-mode PCA model also uses a three-way core array for linking all components to each other, which makes it far more general than the PARAFAC model, but also more complicated. In contrast with PARAFAC, the three-mode PCA model has non-unique components and it seems hard to choose among all possible solutions. The present paper introduces a class of three-mode PCA models in between three-mode PCA and PARAFAC that share good properties of both models. They are relatively simple and they fit (almost) as well as the former model and they have the same uniqueness properties as the latter model.

• Kiers, H.A.L., & Van Mechelen, I. (2001).
  Three-way component analysis: Principles and illustrative application. Psychological Methods, 6, 84-110.
  Three-way component analysis techniques are designed for descriptive analysis of 3-way data, for example, when data are collected on individuals, in different settings, and on different measures. Such techniques summarize all information in a 3-way data set by summarizing, for each way of the 3-way data set, the associated entities through a few components and describing the relations between these components. In this article, 3-mode principal component analysis is described at an elementary level. Guidance is given concerning the choices top be made in each step of the process of analyzing 3-way data by this technique. The complete process is illustrated with a detailed description of the analysis of an empirical 3-way data set.

  
   

• Kjerulff, K. & Wiggins, N.H. (1976).
  Graduate student styles for coping with stressful situations. Journal of Educational Psychology, 68, 247-254.
  34 graduate students were asked to rate 26 stressful situations encountered since entering graduate school on 11 characteristics. Data centred by subtracting the grand mean per rating scale. T3 was applied with varimax rotation for situations, scales and the two subject dimensions. Reasonable amount of detail presented. Validation with outside variables.

  
     

• Klauer, K.C., & Carroll, J.D. (1995).
  Network models for scaling proximity 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. 319-342). Mahwah, NJ: Erlbaum.
  Network models aim at representing proximity data by means of the minimum- path-length function of connected and weighted graphs. Fundamental representation and uniqueness results underlying network models as psychological representations of stimuli, given both ordinal-scale as well as interval-scale proximity measures, are discussed. In addition, computational methods for network analyses are reviewed and compared. Methods now exist to scale metric as well as nonmetric data, symmetric and nonsymmetric proximity measures, and two-way and three-way data. They are compared with respect to the factors of (a) computational cost, (b) accuracy of recovery of an underlying network, and (c) goodness of fit to the observed proximity data.

     

• Knobloch, E.M. (1972).
  Einschätzung von leistungsrelevanten Begriffen. Unpublished master thesis, University of Hamburg, Hamburg, FRG., 1972.

  
   

• Kohler, A. (1980).
  Das Trimod-Programm-System (TRIPSY) zur Berechnung der dreimodalen Faktorenanalyse nach Orlik (manuscript in preparation).
  Description of a computer program implementing Orlik's (1981) Summax model.

• Kofides, E., & Regalia, P. A. (2002).
  On the best rank-1 approximation of higher-order supersymmetric tensors. Siam Journal of Matrix Analysis and Applications,23, 863-884.
  Recently the problem of determining the best,in the least-squares sense,rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the well- known power method for matriceswasproposed for itssolution.Thishigher-order power method is also proposed for the special but important class of supersymmetric tensors,with no change. simpli .ed version,adapted to the special structure of the supersymmetric problem,is deemed unreli- able,asitsconvergence isnot guaranteed.The aim of thispaper isto show that a symmetric version of the above method converges under assumptions of convexity (or concavity)for the functional in- duced by the tensor in question,assumptions that are very often satis .ed in practical applications. The use of this version entails signi .cant savings in computational complexity as compared to the unconstrained higher-order power method.Furthermore,a novel method for initializing the iterative processisdeveloped which hasbeen observed to yield an estimate that liescloser to the global op- timum than the initialization suggested before.Moreover,its proximity to the global optimum is a priori quanti .able.In the course of the analysis,some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied.

• Kolda, T.G. (2001).
  Orthogonal tensor decompositions.Siam Journal of Matrix Analysis and Applications, 23, 243-255.
  We explore the orthogonal decomposition of tensors (also known as multidimensional arrays or n-way arrays) using two di erent de nitions of orthogonality. We present numerous examples to illustrate the diculties in understanding such decompositions. We conclude with a counterexample to a tensor extension of the Eckart-Young SVD approximation theorem by Leibovici and Sabatier.

• Kolda, T. G. (2003).
  A couterexample to the possibility of an extension of the eckart-young low-rank approximation theorem for the orthogonal rank tensor decomposition. Siam Journal of Matrix Analysis and Applications,24, 762-767.
  Earlier work has shown that no extension of the Eckart –Young SVD approximation theorem can be made to the strong orthogonal rank tensor decomposition.Here,we present a counterexample to the extension of the Eckart –Young SVD approximation theorem to the orthogonal rank tensor decomposition, answering an open question previously posed by Kolda [SIAM J.Matrix Anal.Appl.,23 (2001),pp.243–355 ].

• Korzhnev, D. M., Ibraghimov, I. V., Billeter, M., & Orekhov, V. Y. (2001).
  MUNIN: Application of three-way decomposition to the analysis of heteronuclear NMR relaxation data. Journal of Biomolecular NMR,21, 263-268.
  MUNIN (Multidimensional NMR Spectra Interpretation), a recently introduced approach exploiting the mathematical concept of three-way decomposition, is proposed for separation and quantitative relaxation measurements of strongly overlapped resonances in sets of heteronuclear two-dimensional spectra that result from typical relaxation experiments. The approach is general and may also be applied to sets of two-dimensional spectra with arbitrary modulation along the third dimension (e.g., J-coupling, diffusion). Here, the method is applied for the analysis of 15N rotating frame relaxation data.

• Kosanovich, K.A., Dahl, K.S., & Piovoso, M.J. (1996).
  Improved process understanding using multiway principal component analysis. Industrial & Engineering Chemistry Research, 35, 138-146.
  Producing a uniform polymer by batch processing is important for the following reasons: To improve the downstream processing performance, to enable material produced at one site to be used by another, and to remain competitive. Eliminating the sources of batch-to-batch variability and tightening the control of key variables are but two ways to accomplish these objectives. In this work, it is shown that multiway principal component analysis (mpca) can be used to identify major sources of variability in the processing steps. The results show that the major source of batch-to-batch variability is due to reactor temperature variations arising from disturbances in the heating system and other heat-transfer limitations. Correlations between the variations in the processing steps and the final product properties are found, and recommendations to reduce the sources of variations are discussed.

• Kourti, T., & MacGregor, J.F. (1995).
  Process analysis, monitoring and diagnosis, using multivariate projection methods. Chemometrics and Intelligent Laboratory Systems, 28, 3-21.
  Multivariate statistical methods for the analysis, monitoring and diagnosis of process operating performance are becoming more important because of the availability of on-line process computers which routinely collect measurements on large numbers of process variables. Traditional univariate control charts have been extended to multivariate quality control situations using the hotelling t-2 statistic. Recent approaches to multivariate statistical process control which utilize not only product quality data (y), but also all of the available process variable data (x) are based on multivariate statistical projection methods (principal component analysis, (pca), partial least squares, (pls), multi-block pls and multi-way pca). An overview of these methods and their use in the statistical process control of multivariate continuous and batch processes is presented. Applications are provided on the analysis of historical data from the catalytic cracking section of a large petroleum refinery, on the monitoring and diagnosis of a continuous polymerization process and on the monitoring of an industrial batch process.

  ---

• Kourti, T., Nomikos, P., & MacGregor, J.F. (1995).
  Analysis, monitoring and fault diagnosis of batch processes using multiblock and multiway PLS. Journal of Process Control, 5, 277-284.
  Multivariate statistical procedures for the analysis and monitoring of batch processes have recently been proposed. These methods are based on multiway principal component analysis (pca) and partial least squares (pls), and the only information needed to exploit them is a historical database of past batches. In this paper, these procedures are extended to allow one to use not only the measured trajectory data on all the process variables and information on measured final quality variables but also information on initial conditions for the batch such as raw material properties, initial ingredient charges and discrete operating conditions. Multiblock multiway projection methods are used to extract the information in the batch set-up data and in the multivariate trajectory data, by projecting them onto low dimensional spaces defined by the latent variables or principal components. This leads to simple monitoring charts, consistent with the philosophy of spc, which are capable of tracking the progress of new batch runs and detecting the occurrence of observable upsets. Powerful procedures for diagnosing assignable causes for the occurrence of a fault by interrogating the underlying latent variable model for the contributions of the variables to the observed deviation are also presented. The approach is illustrated with databases from two industrial batch polymerization processes.

• Kouwer, B.J. (1967).
  Drie-modale faktoranalyse. Programmabeschrijving (GRON. PSYCH. 07+07BIS). Orthogonale rotaties (GRON.PSYCH.12). Reports, Institute of Psychology, University of Groningen, Groningen, The Netherlands.

  
       

• Kreiman, J., & Gerratt, B. R. (1996).
  The perceptual structure of pathologic voice quality. Journal of the Acoustical Society of America, 100, 1787-1795.
  Although perceptual assessment is included in most protocols for evaluating pathologic voices, a standard set of valid scales for measuring voice quality has never been established. Standardization is important for theory and for clinical acceptance, and also because validation of objective measures of voice depends on valid perceptual measures. The present study used large sets (n=80) of male and female voices, representing a broad range of diagnoses and vocal severities. Eight experts judged the dissimilarity of each pair of voices, and responses were analyzed using nonmetric individual differences multidimensional scaling. Results indicate that differences between listeners in perceptual strategy are so great that the fundamental assumption of a common perceptual space must be questioned. Because standardization depends on the assumption that listeners are similar, it is concluded that efforts to standardize perceptual labels for voice quality are unlikely to succeed. However, analysis by synthesis may provide an alternate means of modeling quality as a function of both voices and listeners, thus avoiding this problem. (C) 1996 Acoustical Society of America.

• Krijnen, W.P. (1993).
  The analysis of three-way arrays by constrained PARAFAC methods. Leiden: DSWO Press.
  In this book - a companion volume to Kroonenberg's Three-mode Principal Component Analysis - much attention is directed to the most salient property of the PARAFAC model, the uniqueness of its components. It turns out that the theoretical property of uniqueness does not exclude the existence of alternative representations that fit the data almost as well as the standard PARAFAC solution. In such cases, the uniqueness is called weak. A constrained PARAFAC variant is developed specifically to determine the degree of uniqueness of PARAFAC components. For data having weak uniqueness, three new constrained PARAFAC methods are introduced that allow for easier interpretations. The first one employs constant relative importances of the components across occasions, the second one determines contrast-free components, and the third one determines components that correspond to non-overlapping clusters of variables. The usefulness of the constrained PARAFAC methods is illustrated by various analyses of empirical three-way data, and by simulations.

• Krijnen, W.P. (1993).
  Non-contrast components according to the PARAFAC model. In R. Steyer, K.F. Wender, & K.F. Widaman (Eds.), Psychometric Methodology, Proceedings of the 7th European Meeting of the Psychometric Society in Trier (pp. 237-241). Stuttgart and New York: Gustav Fischer Verlag.
  If a three-way array with scores of for instance a number of persons on a number of variables that measure intelligence on a number of occasions is analyzed with the PARAFAC method, contrast components may be found. That is, one may find components that correlate positively with some variables and negatively with other variables and/or have positive and negative component regression weights for the variables (pattern elements). It is illustrated with results from a PARAFAC analysis of empirical data that these contrast components can be less nicely interpreted than non-contrast components. In case one finds contrast components with the PARAFAC method, the uniqueness property prevents using a rotation to find non-contrast components. A restricted PARAFAC method is presented that identifies non-contrast components that optimally represents the variables. In case the fit of the non-contrast components is close to the fit of he ordinary PARAFAC components the non-contrast solution may be preferred because its components are more simple to interpret than the PARAFAC components.

• Krijnen, W.P., & Kiers, H.A.L. (1993).
  Clustered variables in PARAFAC. In J.H.L. Oud & R.A.W. van Blokland- Vogelesang (Eds.), Advances in longitudinal and multivariate analysis in the behavioral sciences: Proceedings of the SMABS 1992 conference. (pp. 165-177). Nijmegen: ITS.
  In this paper a constrained PARAFAC method is proposed with which it can be verified whether or not the variables can be partitioned into non-overlapping clusters. In addition a second variant of PARAFAC is proposed that identifies an optimal PARAFAC representation which has components corresponding to non-overlapping clusters of variables. Both methods are illustrated by an analysis of empirical data, and some relationships with other methods are outlined.

  

• Krijnen, W.P., & Kiers, H.A.L. (1995).
  An efficient algorithm for weighted PCA. Computational Statistics, 10, 299-306.
  The method for analyzing three-way data where one of the three components matrices in TUCKALS3 is chosen to have one column is called Replicated PCA. The corresponding algorithm is relatively inefficient. This is shown by offering an alternative algorithm called Weighted PCA. Specifically, it is proven that the algorithms Replicated PCA and Weighted PCA produce identical convergent sequences of loss-function values, and that Weighted PCA attains these values using fewer floating point operations.

• Krijnen, W.P., & Ten Berge, J.M.F. (1991).
  Contrastvrije oplossingen van het CANDECOMP/PARAFAC-model [Contrast-free solutions of the CANDECOMP/PARAFAC-model]. Kwantitatieve Methoden, 12, 87-96.
  The CANDECOMP/PARAFAC decomposition of three-way arrays produces components and loadings that are unique in the sense that rotation results in a worse fit and is therefore not allowed. This paper investigates whether the absence of rotational freedom causes problems in contrast with ordinary principal components for data which can be beautifully analysed using rotational freedom. This is especially true for variables which correlate positively with each other such that they can be rotated to components with contrast-free loadings. CANDECOMP/PARAFAC is indeed able to produce contrast components; the lack of rotational freedom is a problem in that case. The solution can be found by imposing non-negativity constraints. It turns out that with a minimal loss, constrast-free components can be found. This result calls into question the uniqueness of the CANDECOMP/PARAFAC solution.

• Krijnen, W.P., & Ten Berge, J.M.F. (1992).
  A constrained PARAFAC method for positive manifold data. Applied Psychological Measurement, 16, 295-305.
  A set of non-negativity correlated variables, referred to as positive manifold data, display a peculiar pattern of loadings in principal components analysis (PCA). If a small set of principal components is rotated to a simple structure, the variables correlate positively with all components, thus displaying positive manifold. However, this phenomenon is critically dependent on the freedom of rotation, as is evident from the unrotated loadings. That is, although the first principal component is without contrast (which means that all variables correlate either positively or negatively with the first component), subsequent components have mixtures of positive and negative loadings - which means that positive manifold is absent. PARAFAC is a generalization of PCA that has unique components, which means that rotations are not allowed. This paper examines how PARAFAC behaves when applied to positive manifold data. It is shown that PARAFAC does not always produce positive manifold solutions. For cases in which PARAFAC does not produce a positive manifold solution, a constrained PARAFAC method is offered that restores positive manifold by introducing non-negativity constraints. Thus, noncontrast PARAFAC components can be found that explain only a negligible amount of variance less than the PARAFAC components. These noncontrast components cannot be degenerate and cannot be partially unique in the traditional sense.

• Krolak-Schwerdt, S. (1991a).
  Modelle der dreimodalen Faktorenanalyse. Frankfurt am Main: Peter Lang.
  Contents:
  1. Introduction.
  2. From the co-variation chart to three-mode factor analysis: A historic milestone.
  3. Three-mode factor analytical models: Basic equations, algorithms and interpretations.
  4. Formal characteristics and specifics of the three-mode factor analytical model: A classification.
  5. Coherence and theoretical relations between three-mode factor analytical models.
  6. Application of the three-mode factor analytical models: Preliminary remarks.
  7. Re-analysis of a model-experiment by Orlik concerning the psychophysics of polarity profiles.
  8. Dimensions of colour perception: A comparison between MDS-models and individual differences structure-analysis.
  9. Different types of sorting techniques: Rosenberg and Kim's study of methods.
  10. Osgood and Luria: A case of multiple personalities - an empirical comparison of three-mode data models.
  11. Discussion.

• Krolak-Schwerdt, S. (1991b).
  Modelle der dreimodalen Faktorenanalyse: Formale Eigenschaften, theoretische Zusammenhänge und ihre Implikationen für das Konzept individueller Differenzen. Psychologische Beiträge, 33, 314-346.
  The present paper is concerned with methods of three-mode factor analysis to obtain a dimensional representation of three-way data. Classifying the methods by the number of derived spaces and their interrelations yields two distinct classes of models: CANDECOMP (Carroll & Chang, 1970), PARAFAC (Harshman, 1976) and SUMMAX (Orlik, 1980) rest on a basic trilinear decomposition for the data defining a separate space for each mode, whereas Tucker's (1964a) three-mode factor analysis and SUMMAX in its extended form use a quadrilinear model specifying an additional core matrix. Associated with the current classification are different properties of the two types of models which refer to the number of substantial dimensions, their interpretation and the orientation of dimensions which is subject to rotations within the quadrilinear class and uniquely determined by trilinear methods. Considering the different characteristics of the methods, formal relations between the models have been found under very restrictive conditions only. However, there exist some general connections between trilinear and quadrilinear models. CANDECOMP and PARAFAC derive from the trilinear SUMMAX model by rescaling and permutation of axes, and the methodological link between the Tucker model and SUMMAX is given by orthogonal rotations of the SUMMAX configuration. These relationships are shown in an empirical example and their implications for the distinct concepts of individual differences within the two classes of methods are discussed.

• Krolak-Schwerdt, S., Orlik, P., & Ganter, B. (1994).
  TRIPAT: A model for analyzing three-mode binary data. In H.H. Bock, W. Lenski, & M.M. Richter (Eds.), Information Systems and Data Analysis: Prospects-Foundations-Applications (pp. 298-307). Berlin: Springer.
  A discrete, categorical model is presented for three-mode (conditions by objects by attributes) data arrays with binary entries x_ijk element of {0,1}. Basically, the model attempts a simultaneous classification of the elements of the three modes in a number of common clusters. Clusters are defined by three-mode submatrices of maximum size with entries x_ijk = 1. In performing a discrete representation of the data structure, the model may be classified as a non-hierarchical clustering procedure. It involves a reorganization of the data array such that the final clustering solution is interpreted directly on the data, and it allows for overlapping as well as nonoverlapping clusters. The method is similar to three-mode component models such as CANDECOMP and SUMMAX in the model function to predict the data. An application concerning recall data in a study of social perception is provided.

• Kroonenberg, P.M. (1981).
  User's Guide to TUCKALS3. A program for three- mode principal component analysis. WEP-reeks, WR 81-6-RP, Vakgroep W.E.P., University of Leiden, Leiden, The Netherlands, 1979(a).
  A description of the implementation of the algorithm described in Kroonenberg & De Leeuw (1980). Includes a detailed example from Dutch politics.

  

• Kroonenberg, P.M. (1981b).
  Scaling of input data for three-mode principal component analysis. WEP-reeks, WR 81-21-EX, Vakgroep W.E.P., University of Leiden, Leiden, The Netherlands.
  A number of proposals for scaling of input data are collected within one framework. Examples of some of the more common scaling procedures are given, and some effects on three-mode component analysis are considered.

• Kroonenberg, P.M. (1981c).
  User's guide to TUCKALS2. A program for three- mode principal component analysis with extended core matrix. WEP- reeks, WR-81-35-RP, Vakgroep W.E.P., University of Leiden, Leiden, The Netherlands.
  Description of the implementation of the algorithm described in Kroonenberg & De Leeuw (1977, 1978). Includes an example from the Dutch political scene.

  

• Kroonenberg, P.M. (1982).
  TUCKALS3: A program for three-mode principal component analysis. Kwantitatieve Methoden, 3, 65-94.
  After a relatively non-technical account of three-mode component analysis of three-way data, several features of a computer program to perform such an analysis, TUCKALS3, are described. A detailed analysis of data on the similarities between Dutch political parties is presented to illustrate how three-mode principal component analysis may be used to unravel complex relationships.

• Kroonenberg, P.M. (1983a).
  Annotated bibliography of three-mode factor analysis. British Journal of Mathematical and Statistical Psychology, 36, 81-113.
  Published and unpublished theoretical and applied papers on three-mode principal component analysis and factor analysis have been annotated. In addition, the applications have been classified according to subject matter, data type and language. Theoretical papers have been classified according to problem, model, method or computer program treated.

• Kroonenberg, P.M. (1983b).
  Correlational structure of the subtests of the Snijders-Oomen non-verbal intelligence scale. Kwantitatieve Methoden, 4, 40-51.
  Using three-mode principal component analysis on correlation matrices for three age groups of both hearing and deaf children, it is shown that the structure of the subtests is virtually the same in all six groups. This structure might be described by a component shared by all tests and two other components of almost equal importance.

• Kroonenberg, P.M. (1983c).
  Three-Mode Principal Component Analysis: Theory and Applications. Leiden: DSWO Press.
  Contents:
  1. Preliminaries
  2. Survey
  Part I: THEORY
  3. Models
  4. Methods and algorithms
  5. Transformations of core matrices
  Part II: THEORY FOR APPLICATIONS
  6. Scaling and interpretations
  7. Residuals
  Part III: APPLICATIONS
  8. Standard three-mode data: Attachment study
  9. Semantic differential data: Triple personality study
  10. Asymmetric similarity data: ITP study
  11. Similarities and adjective ratings: Cola study
  12. Correlation matrices: Four ability-factors study
  13. Multivariate longitudinal data: Hospital study
  14. Growth curves: Learning-to-read study
  15. Three-mode correspondence analysis: Leiden electorate study
  (Errata, 1989; available from author).

• Kroonenberg, P.M. (1984a).
  Centring three-mode data: Views, problems, and queries. A discussion with Harshman and Lundy. (WEP Reeks, WR 84-54- IN), Leiden: University of Leiden, Department of Education.
  This contains a number of comments on the question of "appropriate" centring and normalization. The discussion is continued in Harshman & Lundy (1985b).

• Kroonenberg, P.M. (1984b).
  Three-mode principal component analysis: Illustrated with an example from attachment theory. In H.G. Law, C.W. Snyder Jr, J.A. Hattie & R.P. McDonald (Eds.), Research methods for multimode data analysis (pp. 64-103). New York: Praeger.
  In this chapter, the three-mode principal component model is presented on a conceptual level by providing various informal ways of looking at it. Secondly, an outline is provided of some technical aspects connected with analyzing this type of model. Finally, an example treating data from attachment theory is used to illustrate some of the major aspects and possibilities of analyzing three-mode data with the three-mode principal component model.

• Kroonenberg, P.M. (1985).
  Three-mode principle components analysis of semantic differential data: The case of a triple personality. Applied Psychological Measurement, 9, 83-94.
  This paper shows how three-mode principal components analysis can be useful for the analysis of semantic differential ratings, in particular because no summation is necessary over any one mode. The use of "joint plots" (a variant of the biplot) and sums-of- squares interpretations is explained and illustrated.

• Kroonenberg, P.M. (1986).
  The three-mode world. (WEP Reeks, WR 86-01-LE), Leiden: University of Leiden, Department of Education.
  This report is a rough attempt to show Who is Who in three-mode land. The emphasis is on three-mode factor and component analysis (i.e. three-mode three-way data), rather than on individual differences multidimensional scaling (i.e. two-mode three-way data).

• Kroonenberg, P.M. (1988b).
  TUCKALS2. Three-mode principal component analysis with extended core matrix. In A. di Ciaccio & G. Bove (Eds.), Multiway '88. Software Guide (pp. 93-103). Roma: Università di Roma "La Sapienza".
  The main principles of three-mode PCA with extended core array (or Tucker2- model) are described, as well as the capabilities and features of the computer program TUCKALS2 based on this model. Relations with PARAFAC and INDSCAL are pointed out.

• Kroonenberg, P.M. (1988c).
  TUCKALS3. Three-mode principal component analysis. In A. di Ciaccio & G. Bove (Eds.), Multiway '88. Software guide (pp. 105-114). Roma: Università di Roma "La Sapienza".
  The main principles of three-mode PCA with full core array (or Tucker2-model) are described, as well as the capabilities and features of the computer program TUCKALS3 based on this model.

• Kroonenberg, P.M. (1989b).
  Singular value decompositions of interactions in three-way contingency tables. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 169-184). Amsterdam: Elsevier.
  In this paper generalizations of the singular value decomposition are used to analyze interactions from three-way contingency tables. These decompositions are primarily applied to standardized residuals from various loglinear models to produce three-way generalizations of correspondence analysis.

• Kroonenberg, P.M. (1989c).
  The analysis of multiple tables in factorial ecology. III. Three-mode principal component analysis: "Analyse triadique complète". Acta Oecologica. Oecologica Generalis, 10, 245-256.
  Thioulouse and Chessel's (1987) "partial" triadic analysis to handle multiple tables in ecology can be extended to a complete triadic analysis. This method was already developed by Tucker (1966) under the name of three-mode factor (or principle components) analysis. This technique is applied to Thioulouse and Chessel's data on the water quality of the Méaudret. The compact and efficient data condensation of the method is emphasized and illustrated.

• Kroonenberg, P.M. (1990b).
  Three-mode analysis by example. In Metodoloxía da Investigacíon Científica (pp. 105-126). Santiago de Compostela, Spain: Universidade. Servicio de Publicacíons e Intercambio Científico.
  In this paper three-mode analysis, in particular, three-mode principal component analysis and, to a lesser extent, parallel factor analysis are presented. The level of explanation is exclusively on a conceptual level, and formulas are entirely avoided. The example is based on the data from a psychophysiological experiment. Twin pairs were given an acute dose of alcohol and several measures were taken before and three times after the drinking. Many other domains of enquiry also yield data which have been fruitfully handled by three-way techniques. For instance, the plant breeders' problem of evaluating genotypes of soy beans in different locations on various attributes for further selection has been examined with three-mode techniques, as well as, intelligence scores from normal and retarded children. In the latter case, only the correlation matrices were available, but not the original scores. Thus both cross-sectional data bases and repeated measures data can be analysed fruitfully with three-way methods.

• Kroonenberg, P.M. (1992a).
  PARAFAC in three-way land. [Comment on the article "Multilinear models: applications in spectroscopy"]. Statistical Science, 7, 312-314.
  The purpose of this comment is to provide a somewhat wider background to the PARAFAC model discussed in Leurgans and Ross' paper on three-way methods in spectroscopy. The literature on the PARAFAC model is briefly sketched.

• Kroonenberg, P.M. (1992b).
  Three-mode component models: A survey of the literature. Statistica Applicata, 4, 619-633.
  This paper is a part of a larger review paper on three-way techniques. In particular, component models are reviewed, with special emphasis on PARAFAC and the Tucker models.

• Kroonenberg, P.M. (1993).
  Three-way methods for multivariable-multioccasion matrices. Paper presented at the 8th European Meeting of the Psychometric Society, Barcelona, Spain.
  A preliminary investigation is made into the usefulness of three-mode PCA for the analysis of multivariable-multioccasion (or multitrait-multimethod) matrices. A brief comparison is made with stochastic three-mode models (see Bentler & Lee, 1978, 1979; Browne, 1984; McDonald, 1984.)

• Kroonenberg, P.M. (1994).
  The Tuckals line: A suite of programs for three-way data analysis. Computational Statistics & Data Analysis, 18, 73-96.
  This paper describes two programs (tuckals2 en tuckals3) with which three-way data can be analysed. Both are based on generalisations of standard (two-way) principal component analysis. The working of the programs, and the basic theory behind them, is explained, and is illustrated with data on the influence of alcohol on the behaviour of Australian twins.

• Kroonenberg, P.M. (1995).
  Introduction to biplots for G×E tables. (Research report, no. 51). Brisbane: University of Queensland, Centre for Statistics.
  This report contains an introduction to biplots, a technique to display large tables in a graph. The construction and interpretation is explained at a fairly basic level and is directed at plant breeders. The technique is illustrated with several artificial data sets as well as a real one from maize breeding in drought conditions.

• Kroonenberg, P.M. (1996a).
  3WAYPACK Menu Structure. (Technical report+software), Leiden: University of Leiden, Education and Child Studies.
  This document is primarily an annotated overview of all menu entries in INTERFACE3, and as such it may serve as a manual for the program package. In essence the document is a formatted version of the Help File with the interface.

• Kroonenberg, P.M. (1996b).
  3WAYPACK User's manual: A package of three-way programs. (Technical report+software), Leiden: University of Leiden, Department of Education.
  The collection of programs which constitute 3WAYPACK have been designed for the analysis of three-way data. The package consists of three analysis programs, i.e., TUCKALS3, TUCKALS2, and TRILIN and four additional programs: PREPROC3, RESIDUAL, ROTATE, and JOINTPLT. All of these programs can be accessed through of a user-friendly interface, INTERFACE3.

• Kroonenberg, P.M. (1997a).
  Recent developments in three-way data analysis: A showcase of methods and examples. In R. Klar & O. Opitz (Eds.), Classification and Knowledge Organization (pp. 44-62). Berlin: Springer.
  In this paper a compact idiosyncratic overview will be provided of the areas into which three-way data analysis has expanded. The historical introduction is followed by a scheme presenting an indication of the techniques involved. Then four condensed examples give a feel of the scope of applications, while the final section is devoted to publicly available programs to perform the analyses.

• Kroonenberg, P.M. (1998)
  Studying the diffusion of three-mode analysis in chemistry: Design considerations. In A. Rizzi, M. Vichi & H.-H. Bock (Eds.), Advances in data science and classification (pp. 603-611). Berlin: Springer.
  The paper presents an example of studying the introduction of methodological innovation in a science, in particular that of three-mode (component) analysis into chemistry, especially in such areas as chromatography, fluorescence, spectrometry, and analytical chemistry in general.

• Kroonenberg, P.M. (2001).
  Three-mode correspondence analysis: An illustrated exposé. In: Actes des XXXIIIèmes Journée de Statistique (pp. 101-108). Nantes, France: 14-18 May.
  In this presentation, André Carlier's contributions to three-mode correspondence analysis will be presented. The special merit of his work is that such tables are treated as genuine three-way arrays, rather than as a matriced (or flattened) two-way matrix. The procedure will be illustrated with an example describing the changes in play quality of young children.

• Kroonenberg, P.M., & Basford, K.E. (1989).
  An investigation of multi-attribute genotype response across environments using three-mode principal component analysis. Euphytica, 44, 109-123.
  The usefulness of three-mode principal component analysis to explore multi- attribute genotype-environment interaction is investigated. The technique provides a general description of the underlying patterns present in the data in terms of interactions of the three quantities (attributes, genotypes, and environments) involved. As an example, data from an Australian experiment on the breeding of soybean lines are treated in depth.

• Kroonenberg, P.M., & Basford, K.E. (2002).
  Applied Three-Mode Data Analysis. Chapter: Three-Mode Clustering. Research Report No. 104, Centre of Statistics University of Queensland, Brisbane Australia.
  In this chapter the question is addressed of how to group individuals given that we have measured three-mode profiles, i.e. individuals having scores on variables under different conditions. We require the same groups to exist in all conditions under consideration. After the theoretical introduction, the search for groups is illustrated with a small example taken from a study of the effect of pollution on blue crabs (Gemperline et al., 1992). The subsequent sections will provide guidance for applying the clustering technique in practical problems and a fully fledged illustration will deal with the attachment relations between mother and infant and how different infants have different types of relationships with their mothers.

• Kroonenberg, P.M., Basford, K.E., & Ebskamp, A.G.M. (1995).
  Three-way cluster and component analysis of maize variety trials. Euphytica, 84, 31-42.
  Data from the Dutch variety list trials for maize were analysed with three-way mixture method clustering and three-mode component analysis. The main objective of the paper is to demonstrate the usefulness of such multivariate analysis techniques for plant breeding data. In particular, attention is paid how one may gain insight into the complex patterns that are embodied in this type of data sets.

• Kroonenberg, P.M., Basford, K.E., & Van Dam, M. (1995).
  Classifying infants in the Strange Situation with three-way mixture method clustering. British Journal of Psychology, 86, 397-418.
  The quality of the attachment relationship between mother and infant is typically determined in the Strange Situation. The assignments of infants to the A (avoidant), B (secure), and C (resistant) attachment classes are largely but not exclusively based on measurements during the reunion episodes. In this paper, the measurements in the reunion episodes are used to derive a clustering of the infants via three-way mixture method of clustering, a technique especially designed for clustering three-way data (here: infants, variables and episodes). The results are compared with the A-B-C classification, and the relevance of the outcomes for attachment research are discussed. At the same time, the paper aims to demonstrate the use and usefulness of the three-way clustering procedure for data from the social and behavioural sciences.

• Kroonenberg, P.M., Basford, K.E., & Van Dam, M. (1992).
  Three-way mixture method clustering. Annual Meeting Classification Society of North America, 12-13 June, 1992, East Langsing.
  This paper briefly explains the data - from 326 Dutch infants observed in the Strange Situation - and three-way mixture method clustering. It also shortly dwells upon the ordination technique to portray the cluster results. The major objective is to show three-way mixture method of clustering at work.

• Kroonenberg, P.M. & De Leeuw, J. (1977).
  TUCKALS2: A principal component analysis of three mode data. Res. Bull. RB. 001-77, Department of Data Theory, University of Leiden, Leiden, The Netherlands.
  An ALS method to estimate the T2 is presented, in which the principle components are computed for two of the three modes, resulting in an extended core matrix. Two examples from the 1968 Dutch political scene, i.e. 11 psychologists indicating which of 12 parties had which of 17 aspects, and 100 psychology students rating the similarity on a rating scale of the nine major Dutch parties. A method for producing joint plots of two modes is introduced, as well as an algorithm for orthonormally rotating an extended core matrix.

  

• Kroonenberg, P.M. & De Leeuw, J. (1978).
  TUCKALS2: Een hoofdassenanalyse voor drieweggegevens. Methoden en Data Nieuwsbrief (vd SWS vd VVS), 3 (3), 30-53.
  A condensed (Dutch) version of Kroonenberg & De Leeuw (1977).

• Kroonenberg, P.M. & De Leeuw, J. (1980).
  Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69-97.
  A new method to estimate T3 is discussed, and the convergence properties of the ALS algorithm are considered. A special case of T3, using an extended core matrix, i.e. T2 (which was treated extensively in Kroonenberg & De Leeuw, 1977), is outlined as well. The Miller & Nicely data on the confusion of English consonants (16 consonant spoken, 16 consonants heard and 17 degrading conditions of the spoken sound) are used as illustration. Very clear interpretable solutions and core matrices. Contains illustrations of rotation of T2 core matrix to diagonality simultaneous for all frontal planes, and of joint plots of the components of two modes. The joint plots are related to the mixed mode matrices of Wainer et al. (1973)

  

• Kroonenberg, P. M., Dunn III, W. J., & Commandeur, J. J. F. (2003).
  Consensus molecular alignment based on generalized Procrustes analysis. Journal of Chemical Information and Computer Science,45, 69-97.
  One of the most serious problems in three-dimensional quantitative structure-activity relationship (3D-QSAR) studies is selection of an alignment rule for molecular super position of the compounds in the data set. In 3D-QSAR analyses of structure-activity data, a reference compound in a defined conformation is chosen, and all structures in the data set are aligned with the reference in a pairwise manner. In subsequent steps, conformation/alignment-dependent descriptors are computed for the compounds and compared to those of the reference. This approach gives much weight to the arbitrarily chosen reference molecule and can introduce a bias in the results. Here an alternative, and more general, approach to molecular alignment is presented that is based on Generalized Procrustes Analysis (GPA). The result is a consensus alignment that uses all molecules in the data set and avoids the bias introduced in the pairwise alignment strategy.

• Kroonenberg P.M., & Heiser, W.J. (1997).
  Parallel factor analysis with constraints on the configurations: An overview. In C.Hayashi, N. Ohsumi, K. Yajima, Y. Tanaka, H.-H. Bock, & Y. Baba (Eds.), Data science, classification, and related methods (pp. 587-597). Tokyo: Springer.
  The paper presents an overview of recent developments with respect to the use of constraints with the parallel factor analysis model. Constraints and the way they can be incorporated in the estimation process of the model are reviewed. Emphasis is placed on the relatively new triadic algorithm which provides a large number of new ways to use the Parafac model.

• Kroonenberg, P.M., & Kashima, Y. (1997).
  Rules in context. A three-mode principal component analysis of Mann et al.'s data on cross-cultural differences in respect for others. Journal of Cross-Cultural Psychology, 28, 463-480.
  The paper reports a secondary analysis of data from a study in which Australian and Japanese children's perceptions of interpersonal rules were compared. The per country/format matrices of Acts by Targets were double-centred and both separately and jointly analysed with three-mode principal component analysis using a 3x3x2 solution. Especially joint plot representations were found to be useful.

• Kroonenberg, P.M., Lammers, C.J., & Stoop, I. (1985).
  Three-mode principal component analysis of multivariate longitudinal organizational data. Sociological Methods & Research, 14, 99-136.
  The exploratory role three-mode principal component analysis can play in analyzing multivariate longitudinal organizational data is outlined by an exposition of the technique itself, and by its application to organizational data from Dutch hospitals. Relationships with some other techniques for such data are indicated.

• Kroonenberg, P.M., & Miyano, H. (1986).
  Tucker2 moderuniyoru tahenryô jikeiretu dêta no kaiseki - takaku seityô dêta [Analysis of multivariate longitudinal data by the Tucker2 model - growth curves data]. Bulletin of Industrial Products Research Institute, 105, 1-7.
  Data on the physical growth of Japanese girls between 6 and 14 years old were analysed to illustrate the application of three-mode principal component analysis, in particular the Tucker2 model, to growth curves. The individual differences in growth and growth speed were investigated using the deviation scores with respect to the average growth curves. The results were compared with a similar study of French girls, and with previous analyses of the data by C. Hayashi & F. Hayashi (1982). In addition, the problem of preprocessing data before a three-mode analysis is discussed.

• Kroonenberg, P.M. & Miyano, H. (1996c).
  Three-way data analysis and its recent developments. Japanese Psychological Review, 39, 386-407.
  Three-way data analysis methods are reviewed and explained from the viewpoint of Tucker models. A simple description of a three-way data set is followed by an introduction of typical tucker models, including the Tucker2, Tucker3, and Parafac models. A brief explanation of algorithms and preprocessing is given. The example of Kroonenberg & Miyano (1985) is discussed in some detail and several examples of recent,less common applications are presented (see also Kroonenberg, 1997a)

  

• Kroonenberg, P.M., Murakami, T., & Coebergh, J.W.W. (2002).
  Added value of three-way methods for the analysis of mortality trends illustrated with worldwide female cancer mortality (1968-1985). Statistical Methods in Medical Research, 11, 275-292.
  Trends in mortality rates are usually presented per tumour site or per country without an overall analysis of the complete data encompassing all three aspects (tumour sites, countries, trends). This paper presents a methodology for such an overall analysis using three-way methods applied to a data set on female mortality rates for 17 tumour sites of 43 countries for the years 1968-1985. Multivariate techniques like biplots and three-mode principal component analysis within an overall three-way analysis-of-variance framework were used. We confirmed the known patterns of comparatively high mortality for women due to cancer of the bladder, intestines, pancreas, rectum, breast, ovary, skin and leukaemia and the relatively low mortality rates for liver cancer in Western and Northern Europe, the USA, Australia and New Zealand. Also, the reverse pattern was observed for Middle and Southern Europe, Hong Kong, Singapore, and in Japan, and in some but not all Latin American countries. The relatively mortality due to cancer was high in the lungs, mouth, larynx and oesophagus in the British Isles, but was much less in other European countries. Mortality due to cancer of the thyroid, uterus, gall bladder and stomach was high in Middle European countries, as was the case in Japan, Chile and Costa Rica. Rates were low for Southern European countries, North America, Australia and New Zealand. Specific deviating patterns in the data were the more rapidly decreasing mortality rates for stomach cancer in Chile and Japan and the more rapidly increasing mortality rates for lung cancer in the USA, Scotland and Denmark. In conclusion, using three-way methods, it was feasible to analyse the cancer mortality data in their entirety. This enabled the simultaneous comparison of trends in relative mortality rates between all countries due to all tumour sites, as well as the identification of specific deviating trends for specific tumour sites in specific countries.

• Kroonenberg, P. M. & Oort, F. J. (2003).
  Three-mode analysis of multimode covariance matrices. British Journal of Mathematical & Statistical Psychology, 56, 305-335. Multimode covariance matrices, such as multitrait-multimethod matrices, contain the covariances of subject scores on variables for different occasions or conditions. This paper presents a comparison of three-mode component analysis and three-mode factor analysis applied to such covariance matrices. The differences and similarities between the non-stochastic and stochastic approaches are demonstrated by two examples, one of which has a longitudinal design. The empirical comparison is facilitated by deriving, as a heuristic device, a statistic based on the maximum likelihood function for three-mode factor analysis and its associated degrees of freedom for the three-mode component models. Furthermore, within the present context a case is made for interpreting the core array as second-order components.

• Kroonenberg, P.M., & Snyder Jr, C.W. (1989).
  Individual differences in assimilation resistance and affective responses in problem solving. Multivariate Behavioral Research, 24, 257-284.
  Data, comprising 6 judgment scales by 8 problem solving tasks by 32 thirteen- year-old boys, collected within the framework of Eckblad's (1981b) cognitive theory of affect are analyzed with three-mode principal component analysis. In general, this study illustrates the effectiveness of the three-mode principal component analysis (TUCKALS) method for the assessment of the nomothetic validity of a theoretical framework as it pertains to within and across person variation.

• Kroonenberg, P.M., & Ten Berge, J.M.F. (1987).
  Cross-validation of the WISC-R factorial structure using three-mode principal components analysis and perfect congruence analysis. Applied Psychological Measurement, 11, 195-210.
  By using three-mode principal components analysis and perfect congruence analysis in conjunction, the factorial structure of the 11 correlation matrices of the Wechsler Intelligence Scale for Children-Revised was analyzed within a single framework. This allows a unified description showing both the strong similarities between the standardization samples and some small differences related to age. Furthermore, claims about comparability between the WISC-R factorial structure, the structures of other independently conducted studies, and those of several translations of the WISC-R were evaluated. Again the overall similarity was striking, albeit most studies showed lower explained variances. Some age effects seemed to be present here as well. The contribution of three-mode principal components analysis was found to lie primarily in the simultaneous analysis of the standardization samples, while perfect congruence analysis allowed the evaluation of the strengths and the correlations of the common WISC-R components in all studies without encountering rotation problems.

• Kroonenberg, P.M., & Ten Berge, J.M.F. (1989).
  Three-mode principal component analysis and perfect congruence analysis for sets of covariance matrices. British Journal of Mathematical and Statistical Psychology, 42, 63-80.
  In this paper three-mode principal component analysis and perfect congruence analysis for weights applied to sets of covariance matrices are explained and detailed, and the relationships between the two techniques are explored. It is shown that given several assumptions are made for three-mode principal component analysis close links between the two techniques exist. The methods are illustrated with data pertaining to a theory of self-concept.

• Kroonenberg, P.M., Ten Berge, J.M.F., Brouwer, P., & Kiers, H.A.L. (1989).
  Gram-Schmidt versus Bauer-Rutishauser in alternating least-squares algorithms for three-mode principal component analysis. Computational Statistics Quarterly, 5, 81-87.
  The effect of replacing a Bauer-Rutishauser step using an eigendecomposition by a Gram-Schmidt orthogonalization step in an algorithm for three-mode principal component analysis was explored both theoretically and empirically. The results showed that the latter procedure has a slight to moderate advantage over the former one.

• Kroonenberg, P.M., & Van der Voort, T.H.A. (1987).
  Multiplicatieve decompositie van interacties bij oordelen over de werkelijkheidswaarde van televisiefilms [Multiplicative decomposition of interactions for judgements of realism of television films]. Kwantitatieve Methoden, 8, 117-144.
  Interactions of three-way factorial designs can be clarified by using multiplicative decompositions, such as the singular value decomposition and three-mode principal component analysis. This procedure is illustrated for a three-way analysis-of-variance design with a data set on the reality perception of television films by children.

• Kroonenberg, P.M., & Van IJzendoorn, M.H. (1987).
  Exploring children's behavior in the Strange Situation. In L.W.C. Tavecchio, & M.H. van IJzendoorn (Eds.), Attachment in social networks: Contributions to the Bowlby-Ainsworth attachment theory (pp. 379 -425). Amsterdam: North Holland.
  Using data from six different countries but disregarding nationality, an analysis was made of the behaviour of children and of subgroups of children in the Strange Situation. Employing three-mode principal component analysis, trends in behaviour were studied both for the Mother episodes, and for the Strange episodes separately, and for most episodes jointly. With continuous components, compact descriptions could be given of the behaviour, both in terms of idealised individuals and as members of the subgroups of Ainsworth's classification system. The rather complex patterns of avoidance towards the mother were studied and commented upon. Details are presented on the development of these components over the episodes. It was also shown that the components succeed to a reasonable degree to separate the subgroups.

• Kruskal, J.B. (1976).
  More factors than subjects, tests and treatments: An indeterminacy theorem for canonical decomposition and individual scaling. Psychometrika, 41, 281-293.
  Some methods that analyze three-way arrays of data (including INDSCAL and CANDECOMP/PARAFAC) provide solutions that are not subject to arbitrary rotation. This property is studied in this paper by means of the "triple product" [A, B, C] of three matrices. The question is how well the triple product determines the three factors. The answer: up to permutation of columns and multiplication of columns by scalars - under certain conditions. In this paper we greatly expand the conditions under which the result is known to hold. A surprising fact is that the nonrotability characteristic can hold even when the number of factors extracted is greater than every dimension of the three-way array, namely, the number of subjects, the number of tests, and the number of treatments.

• Kruskal, J.B. (1977).
  Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and Its Applications, 18, 95-138. (Corrections, 17-1-1984; available from author or The Three-mode Company)
  A triad is a multiplicative array. Analogous to the rank and the row rank of a matrix, we define rank (X) to be the minimum number of triads, x_ijk = a_Ib_jc_k, whose sum is X, and dim₁(X) to be the dimensionality of the space of matrices generated by the 1-slabs of X. We prove several lower bounds on rank, generalising a matrix theorem of Frobenius. We prove several sufficient conditions for the factors of a triple product, Sigma_ra_irb_jrc_kr, to be essentially unique. The results have applications to arithmetic complexity theory and to statistical models used in three-way multidimensional scaling.

• Kruskal, J.B. (1981).
  Multilinear models for data analysis. Behaviormetrika, 10, 1-20.
  This paper is about structural models which are bilinear, trilinear, or multilinear of higher order, such as the PARAFAC model, used to analyze two-way arrays, three-way arrays or many-way arrays of higher order. We will also discuss the geometrical meaning which many of these models have, using primarily the geometrical concepts of inner product and distance. Finally, we will present as an application the analysis of Harshman, Ladefoged & Goldstein (1977) tongue shape data, to illustrate these ideas.

• Kruskal, J.B. (1983a).
  Multilinear methods. Proceedings of Symposia in Applied Mathematics, 28, 75-104.
  This paper is about structural models which are bilinear, trilinear, or multilinear of higher order, used to analyze 2-way arrays, 3-way arrays or many-way arrays of higher order. Many of these models have geometrical content, which is explained briefly. The bilinear models touched on include factor analysis, principal component analysis, and multidimensional scaling. Trilinear models touched on include INDSCAL and PARARAFAC. An application of the latter is presented. Also, the new methods of preprocessing are described in an appendix, because no other description is available yet in the published literature.

  

• Kruskal, J.B. (1983b).
  Statement of some current results about three-way arrays. Informal notes.
  The concept of rank extends naturally to three-dimensional matrices, and the extension has applications to complexity theory and statistics. Calculating rank is difficult even for tiny matrices. New results are presented up to 3 × 3 × 3. Max 2 × 2 × 2 rank is 3; surprisingly, both rank 2 and rank 3 occur with positive measure. Their geometrical arrangement is described.

• Kruskal, J.B. (1985).
  Rank of N-way arrays and the geometry of 2×2×2 arrays. (Technical Memorandum), Murray Hill, NJ: AT&T Bell Laboratories.
  The concept of rank, which is fundamental to the theory of matrices, can be extended to many-way arrays of all orders. This extension is so useful and natural, particularly for 3-way arrays, that it has been introduced on several separate occasions for different purposes. The smallest 3-way arrays which are not effectively the same as matrices are 2 × 2 × 2. The maximum rank of such arrays is three. The rank 2 × 2 × 2 arrays is explored in considerable detail. The maximum rank of a 2 × J × J array is [3J/2]. More generally, if J is smaller than or equal to K, the maximum rank of a 2 × J × K array is J + min(J, [K/2]). This bound is sharp.

• Kruskal, J.B. (1989).
  Rank, decomposition, and uniqueness for 3-way and N-way arrays. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 7-18). Amsterdam: Elsevier.
  Decomposition of a matrix underlies both the bilinear methods (factor analysis, principal components analysis, and correspondence analysis) and the fundamental concept of matrix rank. The decomposition of a 3-way array as a sum or linear combination of outer product matrices underlies PARAFAC, and can be used to define rank of 3-way arrays. The many differences between 3-way arrays and 2-way arrays with respect to decomposition and rank are discussed. For 2-way arrays, rotational uniqueness of decompositions holds only in trivial cases, but for 3- way arrays, it holds for many decompositions of interest, including most PARAFAC solutions. Many people consider the rotational uniqueness of PARAFAC solutions to be a major advantage of this model. This paper also introduces the dimensionality vector of N-way arrays, which is closely connected to the number of factors used in 3-mode factor analysis.

• Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1985).
  Several mathematical relationships between PARAFAC-CANDECOMP and 3-mode factor analysis. Paper presented at the Annual Meeting of the Classification Society, St. John, Newfoundland, Canada. [Published as Kruskal, Harshman & Lundy, 1989].
  Cf. Kruskal, Harshman, & Lundy, 1989.

• Kruskal, J.B., Harshman, R.A., & Lundy, M.E. (1989).
  How 3-MFA data can cause degenerate PARAFAC solutions, among other relationships. In R. Coppi & S. Bolasco (Eds.), Multiway data analysis (pp. 115-122). Amsterdam: Elsevier.
  This paper discusses relationships among three models: (1) 3-mode factor analysis, (2) PARAFAC-CANDECOMP, and (3) CANDELINC. The most interesting relationship is that data satisfying model (1) can cause degenerate solutions when analyzed with model (2), as described by Theorem 1 and its corollary. Another interesting relationship connecting all three models at once is described by Theorem 2 and its corollaries.

• Krzanowski, W.J. (1979).
  Between-groups comparison of principal components. Journal of the American Statistical Association, 74, 703-707.
  A method is given for comparing principal component analyses conducted on the same variables in two different groups of individuals, and an extension to the case of more than two groups outlined. The technique leads to a latent root and vector problem, which has also arisen in the comparison of factor patterns in seperate factor analyses. Emphasis in the present article is on the underlying geometry and interpretation of the results. An illustrative example is provided.

• Krzanowski, W.J. (1984).
  Principal Component Analysis in the Presence of Group Structure Applied Statistics , 33, 164-168.
  A nested series of hypotheses on dispersion structure is identified when observations are grouped in a multivariate sample. A simple method of estimation is suggested for one of these hypotheses, and results using this method are compared with those previously obtained by maximum likelihood methods. Using these hypotheses, an analogy may be drawn between comparison of principal components between groups and comparison of regressions between groups.

• Krzanowski, W.J. (1990).
  Between-group analysis with heterogeneous covariance matrices: The common principal component model. Journal of Classification, 7, 81-98.
  This paper proposes two methods for between-group analysis when the common principal component model replaces the equal dispersion matrix assumption. One method is by extension of the two-stage approach to canonical variate analysis using the sequential principal component analysis as described by Campbell and Atchey (1981). The second method is by definition of a distance function between populations satislying the common principal component model, followed by metric scaling of the resulting between-populations distance matrix. The two methods are compared with each other and with ordinary canonical variate analysis on the previously introduced data.

       

• Kubista, M., Ismail, I. H., Forootan, A., & Sjogren, B. (2004).
  Determination of protolytic constants by trilinear fluorescence spectroscopy. Journal of Fluorescene,14, 139-144.
  Protolytic equilibria often have profound effects on chemical activity, since protolytic species usually behave quite differently. It is therefore important to characterize the protolytic properties of important chemicals. Here we present a new approach to study protolytic equilibria of fluorescent species that is extremely accurate and relies on minimum assumptions. We show that by measuring 2-dimensional excitation/ emission scans of samples at different pH, the 3-dimensional experimental data set, I (lambda(ex), lambda(em), C(pH)), can be unambiguously decomposed into the spectral responses of the protolytic species present as well as their concentration. The approach is demonstrated on the protolytic equilibrium of fluorescein. Although the fluorescein monoanion cannot be obtained in pure form, the spectra and concentrations of both fluorescein species, as well as the protolytic constant, are determined with excellent accuracy. The proposed method is general and can be applied not only for studies of protolytic equilibria, but on any chemical equilibria and chemical reactions involving fluorescent species.

• Kumar, A. & Dillon, W. R. (1992).
  An integrative look at the use of additive and multiplicative covariance structure models in the analysis of MTMM data. Journal of Marketing Research,29, 51-64.
  First-order confirmatory factor analytic models have had widespread use in the analysis of multitrait-multimethod (MTMM) data. In contrast to the usual first-order confirmatory factor analytic model for the analysis of MTMM data, other covariance structure models have recently been proposed and advocated. Two such models are Wothke's covariance component analysis model andBrowne's direct product model. The authors provide a conceptual and analytic discussion of those alternative procedures and compare them with the conventional first-order confirmatory factor analytic model. They consider the relationship between method factors and trait factors assumed under each model specification. General remarks about the nature of method factors and the likely reasons for lack of fit and ill-defined solutions frequently encountered with use of first-order factor models are presented. The authors also attempt to integrate the various approaches to modeling MTMM data and in so doing provide some perspective on selection of a particular covariance structure model for use in applied research.

• Kunert, J., & Qannari, E. M. (1999).
  A simple alternative to generalized procrustes analysis: Application to sensory profiling data. Journal of Sensory Studies,14, 197-208.
  A statistical method for analyzing sensory profiling data obtained by means of fixed vocabulary or free choice profiling is discussed the most interesting feature of this method is that it involves only simple statistical treatment and can therefore be performed using standard software packages. The outcomes of this method are compared to those of generalized procrustes analysis on the basis of two data sets obtained, respectively, by means of fixed vocabulary and free choice profiling. a significance test is also discussed in order to assess whether the overall configuration of the products is meaningful. This significance test is based upon a simulation study involving the permutation procedure.

• Kuze, T., Goto, M., Ninomiya, K., Asano, K., Miyazawa, S., Munekata, H., Ohno, H., & Uchiyama, I. (1985).
  A longitudinal study on development of adolescents' social attitudes. Japanese Psychological Research, 27, 195-205.
  The purpose of this study is to identify the factor structure of social attitudes among contemporary Japanese adolescents, and to examine the process of change in social attitudes through adolescence. Longitudinal data covering six years through junior high and high school years were obtained from 70 boys and 70 girls. Subjects were asked to respond to the 39-item social attitude questionnaire once a year. In order to explore the factor structure, the Quasi Three-Mode Principal Component Analysis (Murakami, 1983b) was employed on the items by time by sex data array. All factors were found to have basic stability.

   

• Kvaal, K., Wold, J. P., Indahl, U. G., Baardseth, P., & Naes, T. (1998).
  Multivariate feature extraction from textural images of bread. Chemomectrics and Intelligent Laboratory Systems, 42, 141-158.
  In order to compute the classical texture measures there is often a need to perform extensive calculations on the images and do a preprocessing in a specialised manner. Some of these texture measures are constructed to estimate specific information. Other texture measures seem to be more global in nature. The techniques presented in this paper define algorithms applied on the raw image without extensive preprocessing. We want to show that mathematical transformations of images on a vectorised form will easily enable the use of multivariate techniques and possibly model several features hidden in the images at the same time. In this paper we will compare five different methods of extracting features from textural images in food by multivariate modelling of the sensory porosity of wheat baguettes. The sample images are recorded from factorial designed baking experiments on wheat baguettes. The multivariate feature extraction methods to be treated are the angle measure technique (AMT), the singular value decomposition (SVD), the autocorrelation and autocovariance functions (ACF) and the so-called size and distance distribution (SDD) method. The methods will be tested on equal basis and the modelling of sensory porosity from extracted features is done using principal component regression (PCR) and partial least square regression (PLS). The difference between the behaviour of the methods will be discussed. The results show that all the methods are suited to extract sensory porosity but the AMT method prove to be the best in this case.

         

|Top | Algemene en Gezinspedagogiek - Datatheorie | Centre for Child and Family Studies | Education and Child Studies | The Three-Mode Company | Three-Mode bibliography |