Convergence properties of an iterative procedure of ipsatizing and standardizing a data matrix,with applications to parafac/candecomp preprocessing

Centering a matrix row-wise and rescaling it column-wise to a unit sum of squares requires an iterative procedure. It is shown that this procedure converges to a stable solution. This solution need not be centered row-wise if the limiting point of the interations is a matrix of rank one. The results of the present paper bear directly on several types of preprocessing methods in Parafac/Candecomp.     

Fitting the off-diagonal dedicom model in the least-squares sense by a generalization of the harman and jones minres procedure of factor analysis

Harshman's DEDICOM model providesa framework for analyzing square but asymmetric materices of directional relationships amongn objects or persons in terms of a small number of components. One version of DEDICOM ignores the diagonal entries of the matrices. A straight-forward computational solution for this model is offered in the present paper. The solution can be interpreted as a generalized Minres procedure suitable for handing asymmetric matrices.     

Personality in proportion: a bipolar proportional scale for personality assessments and its consequences for trait structure

Trait structures resulting from personality assessments on Likert scales are affected by the additive and multiplicative transformations implied in interval scaling and correlational analysis. The effect comes into view on selecting a plausible alternative scale. To this end, we propose a bipolar bounded scale ranging from -1 to +1 representing an underlying process in which the assessor would review and discount positive and negative behavioral instances of a trait. As an appropriate index of likeness between variables X and Y, we propose LXY = SigmaXY/N, the average of the raw scores cross products. Using this index, we carried out a raw scores principal component analysis on data consisting of 133 participants who had each been rated by 5 assessors including self on 914 items. Contrary to the Big-Five structure that was found in these data on standard analysis, the results showed a relatively large first principal component F1 and 2 very small ones, F2 and F3. The sizes LFF=SigmaF2/N, the averages of the squared component scores, were modest to small. It thus appears that the scale, bipolar proportional versus standard, has a profound impact on the size and structure of personality assessments. The dissimilarity remains on analyzing self-ratings rather than averaged (over the 5 assessors) ratings.     

Retrieving the correlation matrix from a truncated PCA solution: The inverse principal component problem

Whenr Principal Components are available fork variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unlessr =k. In the present paper it is shown that, whenr is at or above the Ledermann bound,r principal components are enough to perfectly reconstruct the correlation matrix, albeit in a way more involved than taking the loading matrix times its transpose. In certain cases just below the Ledermann bound, recovery of the correlation matrix is still possible when the set of all eigenvalues of the correlation matrix is available as additional information.     

A case of extreme simplicity of the core matrix in three-mode principal components analysis

In three-mode Principal Components Analysis, theP ×Q ×R core matrixG can be transformed to simple structure before it is interpreted. It is well-known that, whenP=QR,G can be transformed to the identity matrix, which implies that all elements become equal to values specified a priori. In the present paper it is shown that, whenP=QR − 1,G can be transformed to have nearly all elements equal to values spectified a priori. A cllsed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation ofG are discussed. Constructive comments from anonymous reviewers are gratefully acknowledged.     

A series of lower bounds to the reliability of a test

Two well-known lower bounds to the reliability in classical test theory, Guttman's ₂ and Cronbach's coefficient alpha, are shown to be terms of an infinite series of lower bounds. All terms of this series are equal to the reliability if and only if the test is composed of items which are essentially tau-equivalent. Some practical examples, comparing the first 7 terms of the series, are offered. It appears that the second term (₂) is generally worth-while computing as an improvement of the first term (alpha) whereas going beyond the second term is not worth the computational effort. Possibly an exception should be made for very short tests having widely spread absolute values of covariances between items. The relationship of the series and previous work on lower bound estimates for the reliability is briefly discussed.The authors are obliged to Henk Camstra for providing a computer program that was used in this study.     

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 numerical approach to the approximate and the exact minimum rank of a covariance matrix

A concept of approximate minimum rank for a covariance matrix is defined, which contains the (exact) minimum rank as a special case. A computational procedure to evaluate the approximate minimum rank is offered. The procedure yields those proper communalities for which the unexplained common variance, ignored in low-rank factor analysis, is minimized. The procedure also permits a numerical determination of the exact minimum rank of a covariance matrix, within limits of computational accuracy. A set of 180 covariance matrices with known or bounded minimum rank was analyzed. The procedure was successful throughout in recovering the desired rank.The authors are obliged to Paul Bekker for stimulating and helpful comments.     

The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J. The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.