Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal. Part of this research was supported by (1) the Flemish Government: (a) Research Council K.U. Leuven: GOA-MEFISTO-666, GOA-Ambiorics, (b) F.W.O. project G.0240.99, (c) F.W.O. Research Communities ICCoS and ANMMM, (d) Tournesol project T2004.13; and (2) the Belgian Federal Science Policy Office: IUAP P5/22. Lieven De Lathauwer holds a permanent research position with the French Centre National de la Recherche Scientifique (C.N.R.S.). He also holds an honorary research position with the K.U. Leuven, Leuven, Belgium.     

Statistical inference of minimum rank factor analysis

For any given number of factors, Minimum Rank Factor Analysis yields optimal communalities for an observed covariance matrix in the sense that the unexplained common variance with that number of factors is minimized, subject to the constraint that both the diagonal matrix of unique variances and the observed covariance matrix minus that diagonal matrix are positive semidefinite. As a result, it becomes possible to distinguish the explained common variance from the total common variance. The percentage of explained common variance is similar in meaning to the percentage of explained observed variance in Principal Component Analysis, but typically the former is much closer to 100 than the latter. So far, no statistical theory of MRFA has been developed. The present paper is a first start. It yields closed-form expressions for the asymptotic bias of the explained common variance, or, more precisely, of the unexplained common variance, under the assumption of multivariate normality. Also, the asymptotic variance of this bias is derived, and also the asymptotic covariance matrix of the unique variances that define a MRFA solution. The presented asymptotic statistical inference is based on a recently developed perturbation theory of semidefinite programming. A numerical example is also offered to demonstrate the accuracy of the expressions.This work was supported, in part, by grant DMS-0073770 from the National Science Foundation.     

A generalization of Verhelst's solution for a constrained regression problem in alscal and related MDS-algorithms

Verhelst derived a solution for a constrained regression problem which occurs in the interval measurement application of ALSCAL and related MDS-algorithms. In the present paper it is shown that Verhelst's solution is based on an implicit nonsingularity assumption. A general solution, which contains Verhelst's solution as a special case, is derived by a simple completing-the-squares type approach instead of partial differentiation with a Lagrange multiplier. In addition, this approach permits the identification of a small interval which uniquely contains the optimal value of a parameter needed to solve the special case where Verhelst's solution is valid.The author is obliged to Dirk Knol and Klaas Nevels for helpful comments.     

Computational aspects of the greatest lower bound to the reliability and constrained minimum trace factor analysis

In the last decade several algorithms for computing the greatest lower bound to reliability or the constrained minimum-trace communality solution in factor analysis have been developed. In this paper convergence properties of these methods are examined. Instead of using Lagrange multipliers a new theorem is applied that gives a sufficient condition for a symmetric matrix to be Gramian. Whereas computational pitfalls for two methods suggested by Woodhouse and Jackson can be constructed it is shown that a slightly modified version of one method suggested by Bentler and Woodward can safely be applied to any set of data. A uniqueness proof for the solution desired is offered.The authors are obliged to Charles Lewis and Dirk Knol for helpful comments, and to Frank Brokken and Henk Camstra for developing computer programs.     

A general solution for a class of weakly constrained linear regression problems

This paper contains a globally optimal solution for a class of functions composed of a linear regression function and a penalty function for the sum of squared regression weights. Global optimality is obtained from inequalities rather than from partial derivatives of a Lagrangian function. Applications arise in multidimensional scaling of symmetric or rectangular matrices of squared distances, in Procrustes analysis, and in ridge regression analysis. The similarity of existing solutions for these applications is explained by considering them as special cases of the general class of functions addressed.The author is obliged to Henk Kiers and Willem Heiser for helpful comments.     

Correlation coefficients for more than one scale type: An alternative to the Janson and Vegelius approach

Janson and Vegelius have recently suggested a family of correlations for variables of mixed scale types, including nominal scales. The resulting correlations areE-coefficients, which means that they are unity if the variables involved are identical up to permissible transformations, and that they can be considered as inner products in a Euclidian space. Some of the coefficients of the correlation family suggested by Janson and Vegelius are generalized squared product-moment correlations and some are not. In the present paper, a family of correlations for variables of mixed scale types is advocated all members of which are generalized squared product-moment correlations. Some practical advantages of the latter family are explained. The authors are obliged to Klaas Nevels and Henk Kiers for helpful comments.     

On the equivalence of two oblique congruence rotation methods,and orthogonal approximations

Tucker's method of oblique congruence rotation is shown to be equivalent to a procedure by Meredith. This implies that Monte Carlo studies on congruence by Nesselroade, Baltes and Labouvie and by Korth and Tucker are highly comparable. The problem of rotating two matrices orthogonally to maximal congruence has not yet been solved. An approximate solution to this problem can be derived from Tucker's method. Even better results can be obtained from a Procrustes rotation followed by rotation to simple structure.     

A generalization of Takane's algorithm for dedicom

An algorithm is described for fitting the DEDICOM model for the analysis of asymmetric data matrices. This algorithm generalizes an algorithm suggested by Takane in that it uses a damping parameter in the iterative process. Takane's algorithm does not always converge monotonically. Based on the generalized algorithm, a modification of Takane's algorithm is suggested such that this modified algorithm converges monotonically. It is suggested to choose as starting configurations for the algorithm those configurations that yield closed-form solutions in some special cases. Finally, a sufficient condition is described for monotonic convergence of Takane's original algorithm.Financial Support by the Netherlands organization for scientific research (NWO) is gratefully acknowledged. The authors are obliged to Richard Harshman.     

On the relationship between Fortier's simultaneous linear prediction and van den Wollenberg's redundancy analysis

The purpose of this note is to show that Van den Wollenberg's method of redundancy analysis is a special case of a simultaneous linear prediction method offered by Fortier.     

Attachment styles and demographic factors as predictors of sociocultural and psychological adjustment of Eastern European immigrants in the Netherlands

The present study examined the relationship between adult attachment styles and psychological and sociocultural adjustment of Polish, Russian, and Hungarian immigrants (N = 631) to Dutch society. In addition, it also examined the relationship between demographic factors and adjustment and compared the predictive value of attachment styles and demographic factors for immigrants' adjustment. The Attachment Style Questionnaire was used to assess respondents' attachment. Psychological adjustment was measured with the Psychological Health Scale and the Satisfaction With Life Scale. Sociocultural adjustment was measured with the Social Support List - Interactions scale. Two scales for measuring identification and contact with the native and with the Dutch culture were developed and used as indicators of cultural adjustment. We found relations between attachment styles and psychological and sociocultural adjustment. Secure attachment was positively related (p<.01) to psychological and sociocultural adjustment, fearful attachment was negatively (p<.01) associated with psychological adjustment, and more negatively with identification with the Dutch culture than with identification with the native culture. Preoccupied attachment was negatively related (p<.01) to psychological adjustment and to identification with the Dutch culture. Dismissing attachment was weakly negatively related (p<.01) to sociocultural adjustment. Correlation patterns across the three immigrants' samples indicate that dismissing individuals remain relatively indifferent towards their native and the Dutch culture. Regarding demographic factors we found that education and age at immigration were positively associated with psychological and sociocultural adjustment, and length of residence appeared to be positively related to sociocultural adjustment. In general, demographic factors showed a stronger association with sociocultural than with psychological adjustment. Regression analysis revealed that attachment styles were better predictors of immigrants' psychological and sociocultural adjustment than demographic factors-education, age at immigration, and length of residence. The results indicate that immigrant studies would benefit from taking an attachment perspective.