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Some Clarifications of the TUCKALS2 Algorithm Applied to the IDIOSCAL Problem

Published online by Cambridge University Press:  01 January 2025

Jos M. F. ten Berge,
Paul A. Bekker and
Henk A. L. Kiers
Jos M. F. ten Berge*
Affiliation:
University of Groningen
Paul A. Bekker
Affiliation:
University of Groningen
Henk A. L. Kiers
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Jos M. F. ten Berge, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, THE NETHERLANDS.
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Abstract

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.

Keywords

IDIOSCALTUCKALS2CANDECOMPPARAFAC

Information

Type
Original Paper
Information
Psychometrika , Volume 59 , Issue 2 , June 1994, pp. 193 - 201
Copyright
Copyright © 1994 The Psychometric Society

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References

Bekker, P. A. (1988). The positive semidefiniteness of partitioned matrices. Linear Algebra and its Applications, 111, 261–278.CrossRefGoogle Scholar
Carroll, J. D. & Chang, J. J. (1972, March). IDIOSCAL: A generalization of INDSCAL allowing IDIOsyncratic reference systems as well as an analytic approximation to INDSCAL. Paper presented at the Spring Meeting of the Psychometric Society, Princeton, NJ.Google Scholar
Carroll, J. D., Wish, M. (1974). Models and methods for three-way multidimensional scaling. In Krantz, D. H., Atkinson, R. C., Luce, R. D., Suppes, P. (Eds.), Contemporary developments in mathematical psychology (pp. 57–105). San Francisco: Freeman.Google Scholar
de Leeuw, J., Heiser, W. J. (1982). Theory of multidimensional scaling. In Krishnaiah, P. R., Kanai, L. N. (Eds.), Handbook of Statistics (pp. 285–316). Amsterdam: North-Holland.Google Scholar
Kiers, H. A. L. (1989). An alternating least-squares algorithm for fitting the two-way and three-way DEDICOM and the IDIOSCAL model. Psychometrika, 54, 515–521.CrossRefGoogle Scholar
Kiers, H. A. L., Cléroux, R. & ten Berge, J. M. F. (in press). Generalized canonical analysis based on optimizing matrix correlations and a relation with IDIOSCAL. Computational Statistics and Data Analysis.Google Scholar
Kroonenberg, P. M. (1983). Three-mode principal components analysis, Leiden: DSWO.Google Scholar
Kroonenberg, P. M., de Leeuw, J. (1980). Principal components analysis of three-way data by means of alternating least squares. Psychometrika, 45, 69–97.CrossRefGoogle Scholar
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 Components Analysis. Computational Statistics Quarterly, 2, 81–87.Google Scholar
Penrose, R. (1956). On the best approximate solutions of linear matrix equations. Proceedings of the Cambridge Philosophical Society, 52, 17–19.CrossRefGoogle Scholar
ten Berge, J. M. F. (1983). A generalization of Kristof's theorem on the trace of certain matrix products. Psychometrika, 48, 487–494.CrossRefGoogle Scholar
ten Berge, J. M. F., Kiers, H. A. L. (1991). Some clarifications of the CANDECOMP algorithm applied to INDSCAL. Psychometrika, 56, 317–326.CrossRefGoogle Scholar