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Least-squares approximation of an improper correlation matrix by a proper one
Authors:Dirk L Knol  Jos M F ten Berge
Institution:(1) Department of Education, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands;(2) Vakgroep Psychologie, University of Groningen, Grote Markt 31-32, 9712 HV Groningen, The Netherlands
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 delta, where delta 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 ldquounweighted singularrdquo solution by transforming the observed improper correlation matrix in a suitable way.
Keywords:missing value correlation  tetrachoric correlation  indefinite correlation matrix  constrained least-squares approximation
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