A modification of the TUCKALS3 algorithm is proposed that handles three-way arrays of order I × J × K for any I. When I is much larger than JK, the modified algorithm needs less work space to store the data during the iterative part of the algorithm than does the original algorithm. Because of this and the additional feature that execution speed is higher, the modified algorithm is highly suitable for use on personal computers.

Kruskal, Harshman and Lundy have contrived a special 2 × 2 × 2 array to examine formal properties of degenerate Candecomp/Parafac solutions. It is shown that for this array the Candecomp/Parafac loss has an infimum of 1. In addition, the array will be used to challenge the tradition of fitting Indscal and related models by means of the Candecomp/Parafac process.

Millsap and Meredith (1988) have developed a generalization of principal components analysis for the simultaneous analysis of a number of variables observed in several populations or on several occasions. The algorithm they provide has some disadvantages. The present paper offers two alternating least squares algorithms for their method, suitable for small and large data sets, respectively. Lower and upper bounds are given for the loss function to be minimized in the Millsap and Meredith method. These can serve to indicate whether or not a global optimum for the simultaneous components analysis problem has been attained.

A loglinear IRT model is proposed that relates polytomously scored item responses to a multidimensional latent space. The analyst may specify a response function for each response, indicating which latent abilities are necessary to arrive at that response. Each item may have a different number of response categories, so that free response items are more easily analyzed. Conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear IRT models.

Harshman's DEDICOM model providesa framework for analyzing square but asymmetric materices of directional relationships among n 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.

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.

In three-mode Principal Components Analysis, the P ×Q ×R core matrix G can be transformed to simple structure before it is interpreted. It is well-known that, when P = 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, when P = QR − 1, G can be transformed to have nearly all elements equal to values specified a priori. A closed-form solution for this transformation is offered. Theoretical and practical implications of this simple structure transformation of G are discussed.

Kroonenberg and de Leeuw have suggested fitting the IDIOSCAL model by the TUCKALS2 algorithm for three-way components analysis. In theory, this is problematic because TUCKALS2 produces two possibly different coordinate matrices, that are useless for IDIOSCAL unless they are equal. Kroonenberg has claimed that, when IDIOSCAL is fitted by TUCKALS2, the resulting coordinate matrices will be identical. In the present paper, this claim is proven valid when the data matrices are semidefinite. However, counterexamples for indefinite matrices are also constructed, by examining the global minimum in the case where the data matrices have the same eigenvectors. Similar counterexamples have been considered by ten Berge and Kiers in the related context of CANDECOMP/PARAFAC to fit the INDSCAL model.

When r Principal Components are available for k variables, the correlation matrix is approximated in the least squares sense by the loading matrix times its transpose. The approximation is generally not perfect unless r =k. In the present paper it is shown that, when r 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.

Whereas the unique axes properties of PARAFAC1 have been examined extensively, little is known about uniqueness properties for the PARAFAC2 model for covariance matrices. This paper is concerned with uniqueness in the rank two case of PARAFAC2. For this case, Harshman and Lundy have recently shown, subject to mild assumptions, that PARAFAC2 is unique when five (covariance) matrices are analyzed. In the present paper, this result is sharpened. PARAFAC2 is shown to be usually unique with four matrices. With three matrices it is not unique unless a certain additional assumption is introduced. If, for instance, the diagonal matrices of weights are constrained to be non-negative, three matrices are enough to have uniqueness in the rank two case of PARAFAC2.

A procedure is described for minimizing a class of matrix trace functions. The procedure is a refinement of an earlier procedure for minimizing the class of matrix trace functions using majorization. It contains a recently proposed algorithm by Koschat and Swayne for weighted Procrustes rotation as a special case. A number of trial analyses demonstrate that the refined majorization procedure is more efficient than the earlier majorization-based procedure.

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.

Multitrait-Multimethod (MTMM) matrices are often analyzed by means of confirmatory factor analysis (CFA). However, fitting MTMM models often leads to improper solutions, or non-convergence. In an attempt to overcome these problems, various alternative CFA models have been proposed, but with none of these the problem of finding improper solutions was solved completely. In the present paper, an approach is proposed where improper solutions are ruled out altogether and convergence is guaranteed. The approach is based on constrained variants of components analysis (CA). Besides the fact that these methods do not give improper solutions, they have the advantage that they provide component scores which can later on be used to relate the components to external variables. The new methods are illustrated by means of simulated data, as well as empirical data sets.

Procedures for oblique rotation of factors or principal components typically focus on rotating the pattern matrix such that it becomes optimally simple. An important oblique rotation method that does so is Harris and Kaiser's (1964) independent cluster (HKIC) rotation. In principal components analysis, a case can be made for interpreting the components on the basis of the component weights rather than on the basis of the pattern, so it seems desirable to rotate the components such that the weights rather than the pattern become optimally simple. In the present paper, it is shown that HKIC rotates the components such that both the pattern and the weights matrix become optimally simple. In addition, it is shown that the pattern resulting from HKIC rotation is columnwise proportional to the associated weights matrix, which implies that the interpretation of the components does not depend on whether it is based on the pattern or on the component weights matrix. It is also shown that the latter result only holds for HKIC rotation and slight modifications of it.

Three-Mode Factor Analysis (3MFA) and PARAFAC are methods to describe three-way data. Both methods employ models with components for the three modes of a three-way array; the 3MFA model also uses a three-way core array for linking all components to each other. The use of the core array makes the 3MFA model more general than the PARAFAC model (thus allowing a better fit), but also more complicated. Moreover, in the 3MFA model the components are not uniquely determined, and it seems hard to choose among all possible solutions. A particularly interesting feature of the PARAFAC model is that it does give unique components. The present paper introduces a class of 3MFA models in between 3MFA and PARAFAC that share the good properties of the 3MFA model and the PARAFAC model: They fit (almost) as well as the 3MFA model, they are relatively simple and they have the same uniqueness properties as the PARAFAC model.

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.

Bailey and Gower examined the least squares approximation C to a symmetric matrix B, when the squared discrepancies for diagonal elements receive specific nonunit weights. They focussed on mathematical properties of the optimal C, in constrained and unconstrained cases, rather than on how to obtain C for any given B. In the present paper a computational solution is given for the case where C is constrained to be positive semidefinite and of a fixed rank r or less. The solution is based on weakly constrained linear regression analysis.

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.

Breastfeeding represents a strong selective factor for shaping the infant gut microbiota. Besides providing nutritional requirements for the infant, human milk is a key source of oligosaccharides, human milk oligosaccharides (HMOs), and diverse microbes in early life. This study aimed to evaluate the influence of human milk microbiota and oligosaccharides on the composition of infant faecal microbiota at one, three, and nine months postpartum. We profiled milk microbiota, HMOs, and infant faecal microbiota from 23 mother–infant pairs at these time points. The predominant genera in milk samples were Streptococcus, Staphylococcus, and an unclassified Enterobacteriaceae genus-level taxon (Enterobacteriaceae uncl.), whereas the infant faecal microbiota was predominated by Bifidobacterium, Bacteroides, and Enterobacteriaceae uncl. Mother–infant dyads frequently shared bacterial amplicon sequence variants (ASVs) belonging to the genera Bifidobacterium, Streptococcus, Enterobacteriaceae uncl., Veillonella, Bacteroides, and Haemophilus. The individual HMO concentrations in the milk showed either no change or decreased over the lactation period, except for 3-fucosyllactose (3-FL), which increased. Neither maternal secretor status nor HMO concentrations were significantly associated with microbiota composition at the different ages or the bacterial ASVs of maternal milk and infant faeces. This study suggests an age-dependent role of milk microbes in shaping the gut microbiota, while variations in HMO concentrations show limited influence.