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 kinds of individual differences in perceptions permitted by the weighted euclidean model for multidimensional scaling (e.g., INDSCAL) are much more restricted than those allowed by Tucker’s Three-mode Multidimensional Scaling (TMMDS) model or Carroll’s Idiosyncratic Scaling (IDIOSCAL) model. Although, in some situations the more general models would seem desirable, investigators have been reluctant to use them because they are subject to transformational indeterminacies which complicate interpretation. In this article, we show how these indeterminacies can be removed by constructing specific models of the phenomenon under investigation. As an example of this approach, a model of the size-weight illusion is developed and applied to data from two experiments, with highly meaningful results. The same data are also analyzed using INDSCAL. Of the two solutions, only the one obtained by using the size-weight model allows examination of individual differences in the strength of the illusion; INDSCAL can not represent such differences. In this sample, however, individual differences in illusion strength turn out to be minor. Hence the INDSCAL solution, while less informative than the size-weight solution, is nonetheless easily interpretable.

The use of Candecomp to fit scalar products in the context of INDSCAL is based on the assumption that the symmetry of the data matrices involved causes the component matrices to be equal when Candecomp converges. Ten Berge and Kiers gave examples where this assumption is violated for Gramian data matrices. These examples are believed to be local minima. It is now shown that, in the single-component case, the assumption can only be violated at saddle points. Chances of Candecomp converging to a saddle point are small but still nonzero.

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.

A weighted Euclidean distance model for analyzing three-way dissimilarity data (stimuli by stimuli by subjects) for heterogeneous subjects is proposed. First, it is shown that INDSCAL may fail to identify a common space representative of the observed data structure in presence of heterogeneity. A new model that removes the rotational invariance of the classical multidimensional scaling problem and specifies K common homogeneous spaces is proposed. The model, called mixture INDSCAL in K classes, or briefly K-INDSCAL, still includes individual saliencies. However, the large number of parameters in K-INDSCAL may produce instability of the estimates and therefore a parsimonious model will also be discussed. The parameters of the model are estimated in a least-squares fitting context and an efficient coordinate descent algorithm is given. The usefulness of K-INDSCAL is demonstrated by both artificial and real data analyses.

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.

Several methods have been developed for the analysis of a mixture of qualitative and quantitative variables, and one, called PCAMIX, includes ordinary principal component analysis (PCA) and multiple correspondence analysis (MCA) as special cases. The present paper proposes several techniques for simple structure rotation of a PCAMIX solution based on the rotation of component scores and indicates how these can be viewed as generalizations of the simple structure methods for PCA. In addition, a recently developed technique for the analysis of mixtures of qualitative and quantitative variables, called INDOMIX, is shown to construct component scores (without rotational freedom) maximizing the quartimax criterion over all possible sets of component scores. A numerical example is used to illustrate the implication that when used for qualitative variables, INDOMIX provides axes that discriminate between the observation units better than do those generated from MCA.

Very general multilinear models, called CANDELINC, and a practical least-squares fitting procedure, also called CANDELINC, are described for data consisting of a many-way array. The models incorporate the possibility of general linear constraints, which turn out to have substantial practical value in some applications, by permitting better prediction and understanding. Description of the model, and proof of a theorem which greatly simplifies the least-squares fitting process, is given first for the case involving two-way data and a bilinear model. Model and proof are then extended to the case of N-way data and an N-linear model for general N. The case N = 3 covers many significant applications. Two applications are described: one of two-way CANDELINC, and the other of CANDELINC used as a constrained version of INDSCAL. Possible additional applications are discussed.

Carroll and Chang have claimed that CANDECOMP applied to symmetric matrices yields equivalent coordinate matrices, as needed for INDSCAL. Although this claim has appeared to be valid for all practical purposes, it has gone without a rigorous mathematical footing. The purpose of the present paper is to clarify CANDECOMP in this respect. It is shown that equivalent coordinate matrices are not granted at global minima when the symmetric matrices are not Gramian, or when these matrices are Gramian but the solution not globally optimal.

A new procedure is discussed which fits either the weighted or simple Euclidian model to data that may (a) be defined at either the nominal, ordinal, interval or ratio levels of measurement; (b) have missing observations; (c) be symmetric or asymmetric; (d) be conditional or unconditional; (e) be replicated or unreplicated; and (f) be continuous or discrete. Various special cases of the procedure include the most commonly used individual differences multidimensional scaling models, the familiar nonmetric multidimensional scaling model, and several other previously undiscussed variants.

The procedure optimizes the fit of the model directly to the data (not to scalar products determined from the data) by an alternating least squares procedure which is convergent, very quick, and relatively free from local minimum problems.

The procedure is evaluated via both Monte Carlo and empirical data. It is found to be robust in the face of measurement error, capable of recovering the true underlying configuration in the Monte Carlo situation, and capable of obtaining structures equivalent to those obtained by other less general procedures in the empirical situation.

A weighted Euclidean distance model for analyzing three-way proximity data is proposed that incorporates a latent class approach. In this latent class weighted Euclidean model, the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class. This model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions, and drastically reduces the number of parameters in the traditional INDSCAL model. The probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities. The maximum likelihood function is optimized by means of an EM algorithm; a modified Fisher scoring method is used to update the parameters in the M-step. A model selection strategy is proposed and illustrated on both real and artificial data.

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of order n × n as a linear combination of 1/2n(n + 1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetric n × n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n + 1) fixed binary matrices of rank one. The decomposition implies that an n × n × p array consisting of p symmetric n × n slabs has maximal rank 1/2n(n + 1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n + 1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n − 1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large as p, the number of matrices analyzed.