A generalized solution of the orthogonal procrustes problem

A solutionT of the least-squares problemAT=B +E, givenA andB so that trace (EE)= minimum andTT=I is presented. It is compared with a less general solution of the same problem which was given by Green [5]. The present solution, in contrast to Green's, is applicable to matricesA andB which are of less than full column rank. Some technical suggestions for the numerical computation ofT and an illustrative example are given.This paper is based on parts of a thesis submitted to the Graduate College of the University of Illinois in partial fulfillment of the requirements for a Ph.D. degree in Psychology.The work reported here was carried out while the author was employed by the Statistical Service Unit Research, U. of Illinois. It is a pleasure to express my appreciation to Prof. K. W. Dickman, director of this unit, for his continuous support and encouragement in this and other work. I also gratefully acknowledge my debt to Prof. L. Humphreys for suggesting the problem and to Prof. L. R. Tucker, who derived (1.7) and (1.8) in summation notation, suggested an iterative solution (not reported here) and who provided generous help and direction at all stages of the project.     

On oblique procrustes rotation

The equations involved in the rotation of an arbitrary factor matrix to a least squares fit to a specified factor structure have been known for many years. These equations, in general, cannot be solved by purely algebraic means, and an approximate solution has previously been used in practical applications.In this paper an effective iterative method for obtaining the exact solution is developed. By algebraic manipulation the set of equations is expressed in the form of one polynomial equation in one unknown. Newton's method is suggested for solving this equation. Practical applications of the procedure indicate that convergence within small tolerance limits is generally attained after few iterations.Part of this research was carried out at the National Institute for Personnel Research (South Africa). It was completed while the author was a Visiting Research Psychologist at Educational Testing Service.     

On meredith's solution for weighted procrustes rotation

Meredith developed a criterion for weighted procrustes rotation. The solution was given using Lagrange multipliers. We show that the solution can be obtained more simply, and that our approach enables us to transform the orthogonalization problem, also considered by Meredith, to that of minimizing a quadratic form on the unit sphere.     

On browne's solution for oblique procrustes rotation

Browne [1967] has given a method of solving the problem (originally stated by Mosier, [1939]) of finding a least squares fit to a specified factor structure. The problem is one of minimizing the sum of squared residuals of —FT with Diag (T'T)=I. Browne's solution involves the eigenvectors and values ofF'F and leads to an iterative solution.This paper gives a form of the solution which does not involve solution of an eigenvalue problem but does require an iteration similar to Browne's. It suggests the possible existence of a singularity, and a simple modification of Browne's computational procedure is proposed which deals with this case. A better starting value for the iteration is also proposed for which convergence is guaranteed using the ordinary Newton iteration.Part of this work was presented at the April 1969 meetings of the Psychometric Society. The anthor is indebted to Dr. Ledyard Tucker for some helpful discussions. This work was supported in part by a PHS Research Grant No. MH-10006 from the National Institute of Mental Health, and Grant No. GM-12868 from the Institute of General Medical Sciences, Public Health Service.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

A weighted procrustes criterion

The Procrustes criterion is a common measure for the distance between two matricesX andY, and can be interpreted as the sum of squares of the Euclidean distances between their respective column vectors. Often a weighted Procrustes criterion, using, for example, a weighted sum of the squared distances between the column vectors, is called for. This paper describes and analyzes the performance of an algorithm for rotating a matrixX such that the column-weighted Procrustes distance toY is minimized. The problem of rotatingX intoY such that an aggregate measure of Tucker's coefficient of congruence is maximized is also discussed.We wish to thank Richard A. Harshman and C. F. Jeff Wu for valuable discussions in the early stages of this work. We would also like to thank Jos ten Berge, John Gower, and the Editor, Associate Editor, and referees whose comments and suggestions greatly improved this paper.     

On weighted procrustes and hyperplane fitting in factor analytic rotation

A weighted collinearity criterion for Procrustean rotation is developed, and it is shown that special cases with respect to the choice of weights and a vector norm are forms of hyperplane fitting, classical oblique Procrustes, etc.; a family of Procrustean transformation procedures is thereby generated. Numerical illustrations utilizing the Holzinger-Swineford data are presented.     

Orthogonal procrustes rotation maximizing congruence

Procedures for assessing the invariance of factors found in data sets using different subjects and the same variables are often using the least squares criterion, which appears to be too restrictive for comparing factors.Tucker's coefficient of congruence, on the other hand, is more closely related to the human interpretation of factorial invariance than the least squares criterion. A method maximizing simultaneously the sum of coefficients of congruence between two matrices of factor loadings, using orthogonal rotation of one matrix is presented. As shown in examples, the sum of coefficients of congruence obtained using the presented rotation procedure is slightly higher than the sum of coefficients of congruence using Orthogonal Procrustes Rotation based on the least squares criterion.The author is obliged to Lewis R. Goldberg for critically reviewing the first draft of this paper.     

A general approach to procrustes pattern rotation

A method is introduced for oblique rotation to a pattern target matrix specified in advance. The target matrix may have all or only some of its elements specified. Values are estimated by means of a general procedure for minimization with equality constraints. Results are shown using data from Harman and Browne.The author is indebted to K. G. Jöreskog and M. W. Browne for many helpful suggestions and comments throughout the course of this study. Thanks are also due P. H. Schönemann and H. Weiner.     

Standard errors for procrustes solutions

The asymptotic standard errors for the procrustes solutions are derived for orthogonal rotation, direct oblique rotation and indirect oblique rotation. The standard errors for the first two rotations are obtained using the augmented information matrices. For the indirect oblique solution, the standard errors of rotated parameters are derived from the information matrix of unrotated loadings using the chain rule for information matrices. For all three types of rotation, the standard errors of rotated parameters are presented for unstandardized and standardized manifest variables. Numerical examples show the similarity of theoretical and simulated values.     

Orthogonal procrustes rotation for two or more matrices

Necessary and sufficient conditions for rotating matrices to maximal agreement in the least-squares sense are discussed. A theorem by Fischer and Roppert, which solves the case of two matrices, is given a more straightforward proof. A sufficient condition for a best least-squares fit for more than two matrices is formulated and shown to be not necessary. In addition, necessary conditions suggested by Kristof and Wingersky are shown to be not sufficient. A rotation procedure that is an alternative to the one by Kristof and Wingersky is presented. Upper bounds are derived for determining the extent to which the procedure falls short of attaining the best least-squares fit. The problem of scaling matrices to maximal agreement is discussed. Modifications of Gower's method of generalized Procrustes analysis are suggested.     

A general solution to Mosier's oblique procrustes problem

Browne provided a method for finding a solution to the normal equations derived by Mosier for rotating a factor matrix to a best least squares fit with a specified structure. Cramer showed that Browne's solution is not always valid, and proposed a modified algorithm. Both Browne and Cramer assumed the factor matrix to be of full rank. In this paper a general solution is derived, which takes care of rank deficient factor matrices as well. A new algorithm is offered.     

A general solution of the weighted orthonormal procrustes problem

A general solution for weighted orthonormal Procrustes problem is offered in terms of the least squares criterion. For the two-demensional case. this solution always gives the global minimum; for the general case, an algorithm is proposed that must converge, although not necessarily to the global minimum. In general, the algorithm yields a solution for the problem of how to fit one matrix to another under the condition that the dimensions of the latter matrix first are allowed to be transformed orthonormally and then weighted differentially, which is the task encountered in fitting analogues of the IDIOSCAL and INDSCAL models to a set of configurations.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper.     

Univocal or orthogonal estimators of orthogonal factors

The defects of the least-squares or multiple-regression equation approach to estimating orthogonal factors are discussed and transformations of the beta weights are derived which remove these defects with minimum loss in correlations between estimators and true factor scores.The opinions expressed in this paper are those of the author and do not necessarily reflect official Department of the Army policy.     

On two problems of Harvey Friedman

The paper considers certain properties of intermediate and moda propositional logics.The first part contains a proof of the theorem stating that each intermediate logic is closed under the Kreisel-Putnam rule xyz/(xy)(xz).The second part includes a proof of the theorem ensuring existence of a greatest structurally complete intermediate logic having the disjunction property. This theorem confirms H. Friedman's conjecture 41 (cf. [2], problem 41).In the third part the reader will find a criterion which allows us to obtain sets satisfying the conditions of Friedman's problem 42, on the basis of intermediate logics satisfying the conditions of problem 41.Finally, the fourth part contains a proof of a criterion which allows us to obtain modal logics endowed with Hallden's property on the basis of structurally complete intermediate logics having the disjunction property.Dedicated to Professor Roman SuszkoThe author would like to thank professors J. Perzanowski and A. Wroski for valuable suggestions.