Abstract
The Maxbet method is an alternative to the method of generalized canonical correlation analysis and of Procrustes analysis. Contrary to these methods, it does not maximize the inner products (covariances) between linear composites, but also takes their sums of squares (variances) into account. It is well-known that the Maxbet algorithm, which has been proven to converge monotonically, may converge to local maxima. The present paper discusses an eigenvalue criterion which is sufficient, but not necessary for global optimality. However, in two special cases, the eigenvalue criterion is shown to be necessary and sufficient for global optimality. The first case is when there are only two data sets involved; the second case is when the inner products between all variables involved are positive, regardless of the number of data sets.
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Chu, M.T., & Watterson, J.L. (1993). On a multivariate eigenvalue problem, Part I: Algebraic theory and a power method.SIAM Journal on Scientific Computing, 14(5), 1089–1106.
Frobenius, G. (1912). Über Matrizen aus nicht negativen Elementen [On matrices of nonnegative elements].Sitzungsberichte Preussische Akademie der Wissenschaft [Proceedings of the Prussian Academy of Science] (pp. 456–477). Berlin.
Horst, P. (1961). Relations amongm sets of measures.Psychometrika, 26, 129–149.
Hotelling, H. (1936). Relations between two sets of variates.Biometrika, 28, 321–377.
Kettenring, J.R. (1971). Canonical analysis of several sets of variables.Biometrika, 58, 433–451.
Perron, O. (1907). Grundlagen für eine Theorie des Jacobischen Kettenbruchalogrithmus. [Foundations for a theory of the Jacobian continued fraction algorithm.]Mathematische Annalen, 64, 11–76.
ten Berge, J.M.F. (1988). Generalized approaches to the MAXBET problem and the MAXDIFF problem, with applications to canonical correlations.Psychometrika, 53, 487–494.
Tucker, R.L. (1951).A method for synthesis of factor analysis studies (Personnel Research Section Rep. No. 984). Washington DC: Department of the Army.
van de Geer, J.P. (1984). Linear relations amongk sets of variables.Psychometrika, 49, 79–94.
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The authors are obliged to Henk Kiers for critical comments on a previous draft.
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Hanafi, M., ten Berge, J.M.F. Global optimality of the successive Maxbet algorithm. Psychometrika 68, 97–103 (2003). https://doi.org/10.1007/BF02296655
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DOI: https://doi.org/10.1007/BF02296655