Multi‐set factor analysis by means of Parafac2

We consider multi‐set data consisting of observations, k = 1,…, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J × J covariance matrices , k = 1,…, K, where the common part is modelled by Parafac2 and the unique variances , k = 1,…, K, are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k, and a common factor correlation matrix. We estimate the unique variances by minimum rank factor analysis on for each k. The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k. Also, the Parafac2 part is rotationally unique under mild conditions.     

Three-Mode Factor Analysis by Means of Candecomp/Parafac

A three-mode covariance matrix contains covariances of N observations (e.g., subject scores) on J variables for K different occasions or conditions. We model such an JK×JK covariance matrix as the sum of a (common) covariance matrix having Candecomp/Parafac form, and a diagonal matrix of unique variances. The Candecomp/Parafac form is a generalization of the two-mode case under the assumption of parallel factors. We estimate the unique variances by Minimum Rank Factor Analysis. The factors can be chosen oblique or orthogonal. Our approach yields a model that is easy to estimate and easy to interpret. Moreover, the unique variances, the factor covariance matrix, and the communalities are guaranteed to be proper, a percentage of explained common variance can be obtained for each variable-condition combination, and the estimated model is rotationally unique under mild conditions. We apply our model to several datasets in the literature, and demonstrate our estimation procedure in a simulation study.     

Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher

The Candecomp/Parafac (CP) model decomposes a three-way array into a prespecified number R of rank-1 arrays and a residual array, in which the sum of squares of the residual array is minimized. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate solutions, in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large. We consider the real-valued CP model in which p × p × 2 arrays of rank p + 1 or higher are decomposed into p rank-1 arrays and a residual array. It is shown that the CP objective function does not have a minimum in these cases, but an infimum. Moreover, any sequence of CP approximations, of which the objective value approaches the infimum, will become degenerate. This result extends Ten Berge, Kiers, & De Leeuw (1988), who consider a particular 2 × 2 × 2 array of rank 3. Request for reprints should be sent to Alwin Stegeman, Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. The author is obliged to Jos ten Berge and Henk Kiers for helpful comments. Also, the author would like to thank the Associate Editor and the anonymous reviewers for many suggestions on how to improve the contents and the presentation of the paper.     

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.     

On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.     

Simultaneous Component Analysis by Means of Tucker3

A new model for simultaneous component analysis (SCA) is introduced that contains the existing SCA models with common loading matrix as special cases. The new SCA-T3 model is a multi-set generalization of the Tucker3 model for component analysis of three-way data. For each mode (observational units, variables, sets) a different number of components can be chosen and the obtained solution can be rotated without loss of fit to facilitate interpretation. SCA-T3 can be fitted on centered multi-set data and also on the corresponding covariance matrices. For this purpose, alternating least squares algorithms are derived. SCA-T3 is evaluated in a simulation study, and its practical merits are demonstrated for several benchmark datasets.     

Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p×q×3, with a two-valued typical rank. These are the 5×3×3 and 8×4×3 arrays, and the 3×3×4 and 3×3×5 arrays with symmetric 3×3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m,m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p×p×2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur. The author is supported by the Dutch Organisation for Scientific Research (NWO), VENI grant 451-04-102.     

Modelling non‐normal data: The relationship between the skew‐normal factor model and the quadratic factor model

Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew‐normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew‐normal factor are equivalent to those under a quadratic model up to third‐order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew‐normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.