Some clarifications of the TUCKALS2 algorithm applied to the IDIOSCAL problem

Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/PARAFAC to fit the INDSCAL model.     

A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n–1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.     

An efficient algorithm for PARAFAC of three-way data with large numbers of observation units

The CANDECOMP algorithm for the PARAFAC analysis ofn×m×p three-way arrays is adapted to handle arrays in whichn>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.The authors are obliged to Jos ten Berge for his comments on an earlier version of this paper. The research of Henk A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set ofn×n matrices that have a rank less thann has zero volume. Kruskal pointed out that a 2×2×2 array has rank three or less, and that the subsets of those 2×2×2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2×n×n arrays. Incidentally, it is shown that twon ×n matrices can be diagonalized simultaneously with positive probability.The author is obliged to Joe Kruskal and Henk Kiers for commenting on an earlier draft, and to Tom Wansbeek for raising stimulating questions.     

Uniqueness of N-way N-mode hierarchical classes models

This paper presents two uniqueness theorems for the family of hierarchical classes models, a collection of order preserving Boolean decomposition models for binary N-way N-mode data. The theorems are compared with uniqueness results for the closely related family of N-way N-mode principal component models. It is concluded that the two-way two-mode PCA and N-way N-mode TuckerN models suffer more from a lack of identifiability than their hierarchical classes analogues, whereas the uniqueness conditions for N-way N-mode PARAFAC/CANDECOMP models are less restrictive than the ones derived for their N-way N-mode hierarchical classes counterparts.     

The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J. The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.     

Effects of conditionality on INDSCAL and ALSCAL weights

The role of conditionality in the INDSCAL and ALSCAL procedures is explained. The effects of conditionality on subject weights produced by these procedures is illustrated via a single set of simulated data. Results emphasize the need for caution in interpreting subject weights provided by these techniques.     

Conflict detection,dual processes,and logical intuitions: Some clarifications

Recent studies on conflict detection during thinking suggest that reasoners are sensitive to possible conflict between their heuristic judgement and elementary logical or probabilistic principles. I have argued that this conflict sensitivity calls for the postulation of logical intuitions and has implications for the way we conceive the interaction between System-1 and System-2 in dual process theories. In this paper I clarify potential misconceptions about this work, discuss the link with other approaches, and sketch directions for further research.     

Confidence regions for INDSCAL using the jackknife and bootstrap techniques

Bootstrap and jackknife techniques are used to estimate ellipsoidal confidence regions of group stimulus points derived from INDSCAL. The validity of these estimates is assessed through Monte Carlo analysis. Asymptotic estimates of confidence regions based on a MULTISCALE solution are also evaluated. Our findings suggest that the bootstrap and jackknife techniques may be used to provide statements regarding the accuracy of the relative locations of points in space. Our findings also suggest that MULTISCALE asymptotic estimates of confidence regions based on small samples provide an optimistic view of the actual statistical reliability of the solution. The authors wish to thank Geert DeSoete, Richard A. Harshman, William Heiser, Jon Kettenring, Joseph B. Kruskal, Jacqueline Meulman, James O. Ramsay, John W. Tukey, Paul A. Tukey, and Mike Wish. Sharon L. Weinberg is a consultant at AT&T Bell Laboratories, Murray Hill, New Jersey 07974.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

Some models of balance applied to children's perceptions of international relations

Data obtained from four national samples of 6–11 year old children are presented, showing preference for four countries, perceived friendship of these countries with the subject's own country and perceived relationships among the target-countries. Two quantitative analogues of Heider's balance theory are used to examine the relations between these sets of data, one employing a multiplicative and the other a distance approach. Both models were found to fit the data adequately, further research being needed to discriminate between them. An explanation in terms of ‘ethnocentric correspondence’ is given to explain why prediction is noticeably better for British and Austrian than for Belgian and Greek samples.     

A latent class approach to fitting the weighted Euclidean model,clascal

A weighted Euclidean distance model for analyzing three-way proximity data is proposed that incorporates a latent class approach. In this latent class weighted Euclidean model, the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class. This model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions, and drastically reduces the number of parameters in the traditional INDSCAL model. The probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities. The maximum likelihood function is optimized by means of an EM algorithm; a modified Fisher scoring method is used to update the parameters in the M-step. A model selection strategy is proposed and illustrated on both real and artificial data.The second author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

A multidimensional scaling model for the size-weight illusion

The kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker's Three-mode Multidimensional Scaling (TMMDS) model or Carroll's Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.This paper is based on the first author's doctoral dissertation at the Department of Psychology, University of Illinois at Urbana-Champaign. The aid of Professor Ledyard R Tucker is gratefully acknowledged.     

Modern application of evolutionary theory to psychology: key concepts and clarifications

Darwinian selection has become the centerpiece of biology, and in the past few decades many psychologists and anthropologists have recognized the value of using an evolutionary perspective to guide their work. With a focus on evolved psychological mechanisms and associated information processing features, evolutionary psychology has risen as a compelling and fruitful approach to the study of human psychology and behavior. In this article we review the instrument of evolution: natural selection, the products of evolution, and the impact of evolutionary thinking on modern psychological science. We conclude that as prejudicial barriers are overcome, as more evolutionary psychological work is conducted, and as hypothesized psychological mechanisms are substantiated in other disciplines, evolutionary psychology will emerge as the metatheory for psychology.     

Some foundations and directions for applied developmental psychology

Applied developmental psychology (ADP) is an “umbrella” discipline which includes a range of psychological sciences and professions committed to use of the data base and methods of developmental psychology to address problems in education, health, human services and public policy. ADP emerges now as a manifestation of increasing recognition of the limits of developmental psychology's traditional models and paradigms, increasing recognition of our appropriate role in social problem-solving and public policy formation and an employment market which reflects these awarenesses. Consideration of clinical psychology as an ADP provides examples of synergies and distinctions helpful in defining ADP. The “linkage” concept inherent in a “scientist-practitioner” model is emphasized as a key in this definition and as a basis for designing appropriate doctoral training programs in ADP.