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Linear Relations among k Sets of Variables

Published online by Cambridge University Press:  01 January 2025

John P. Van de Geer
John P. Van de Geer*
Affiliation:
Department of Data Theory, University of Leiden, Netherlands
*
Requests for reprints should be sent to John P. Van de Geer, Department of Data Theory, University of Leiden, Middelstegracht 4, 2312 TW Leiden, The Netherlands.
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Abstract

A family of solutions for linear relations among k sets of variables is proposed. It is shown how these solutions apply for k = 2, and how they can be generalized from there to k ≥ 3.

The family of solutions depends on three independent choices: (i) to what extent a solution may be influenced by differences in variances of components within each set; (ii) to what extent the sets may be differentially weighted with respect to their contribution to the solution—including orthogonality constraints; (iii) whether or not individual sets of variables may be replaced by an orthogonal and unit normalized basis.

Solutions are compared with respect to their optimality properties. For each solution the appropriate stationary equations are given. For one example it is shown how the determinantal equation of the stationary equations can be interpreted.

Keywords

canonical correlationlinear relations between setsmultiple regressionredundancy analysisridge regression
Type
Original Paper
Information
Psychometrika , Volume 49 , Issue 1 , March 1984 , pp. 79 - 94
Copyright
Copyright © 1984 The Psychometric Society

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References

Reference Notes

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Ten Berge, J. M. F. (1977). Optimizing factorial invariance. Groningen; Ph.D. Thesis, University of Groningen.Google Scholar

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