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Least-Squares Approximation of an Improper Correlation Matrix by a Proper One

Published online by Cambridge University Press:  01 January 2025

Dirk L. Knol  and
Jos M. F. ten Berge
Dirk L. Knol*
Affiliation:
University of Twente
Jos M. F. ten Berge*
Affiliation:
University of Groningen
*
Requests for reprints should be sent to Dirk Knol, University of Twente, Department of Education, PO Box 217, 7500 AE Enschede, THE NETHERLANDS or to
Jos ten Berge, University of Groningen, Vakgroep Psychologic, Grote Markt 31-32, 9712 HV Groningen, THE NETHERLANDS.
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Abstract

An algorithm is presented for the best least-squares fitting correlation matrix approximating a given missing value or improper correlation matrix. The proposed algorithm is based upon a solution for Mosier's oblique Procrustes rotation problem offered by ten Berge and Nevels. A necessary and sufficient condition is given for a solution to yield the unique global minimum of the least-squares function. Empirical verification of the condition indicates that the occurrence of non-optimal solutions with the proposed algorithm is very unlikely. A possible drawback of the optimal solution is that it is a singular matrix of necessity. In cases where singularity is undesirable, one may impose the additional nonsingularity constraint that the smallest eigenvalue of the solution be δ, where δ is an arbitrary small positive constant. Finally, it may be desirable to weight the squared errors of estimation differentially. A generalized solution is derived which satisfies the additional nonsingularity constraint and also allows for weighting. The generalized solution can readily be obtained from the standard “unweighted singular” solution by transforming the observed improper correlation matrix in a suitable way.

Keywords

missing value correlationtetrachoric correlationindefinite correlation matrixconstrained least-squares approximation
Type
Original Paper
Information
Psychometrika , Volume 54 , Issue 1 , March 1989 , pp. 53 - 61
Copyright
Copyright © 1989 The Psychometric Society

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