Journal of Chemometrics

Candecomp/Parafac is a generalized method of Principal Component Analysis for three-way data, yielding unique components under mild conditions. Harshman has given sufficient conditions for uniqueness which are easy to prove, and general enough to handle most practical applications. However, these conditions are less powerful than more complicated conditions proposed by Kruskal. This paper shows how to relax Harshman's conditions to become as powerful as Kruskal's in the case where two of the three component matrices have full column rank.

A generalization of the PARAFAC model is developed that improves its properties when applied to multi-way problems involving linearly dependent factors. This model is called PARALIND (PARAllel profiles with LINear Dependences). To avoid problems from linearly dependent factors, any set of components that in theory should be rank deficient are re-expressed in PARALIND as a product of two matrices, one that explicitly represents their dependency relationships and another, with fewer columns, that captures their patterns of variation.

This paper presents a tensor-based high-precision method for the estimation of the parameters of a signal, modelled as a finite sum of complex damped exponentials, whose poles may be close. We show that the multilinear approach fully reveals the underlying structure of the signal model. In this framework, the use of unsymmetric tensor dimensionality reduction makes it possible to circumvent the problem of ill-conditioning. The method outperforms competing matrix techniques.

Estimating the adequate number of components is an important yet difficult problem in multi-way modeling. We demonstrate how a Bayesian framework for model selection based on automatic relevance determination (ARD) can be adapted to the Tucker and CandeComp/PARAFAC (CP) models such that the model order can be estimated at the cost of fitting an ordinary Tucker/CP model. The approach is demonstrated to successfully estimate the number of components of both real and synthetic data.

DEDICOM is a model for the analysis of square asymmetric tables. The speed of convergence in Takane's original algorithm for DEDICOM was substantially improved by incorporating the minimal polynomial extraporation (MPE) method. The efficiency of the new algorithm was demonstrated by numerical examples.

Warped factor analysis (WFA), presented here, generalizes shifted factor analysis (SFA) so as to fit factor variation not only in position shifts and systematic weighting but also in more flexible shape changes of sequential factors. A quasi alternating least squares procedure is developed for WFA, based on warping of segmented sequential factors. As two-mode SFA was shown to yield an essentially unique solution, two-mode WFA produces essentially unique components provided that the data contain sufficient sources of shape variation.

An overview is given of the literature on the non-existence of optimal solutions and the occurrence of ‘degenerate’ solutions in the best fitting decomposition of a three-way array into outer product arrays (i.e. the Candecomp/Parafac (CP) model). Using the simultaneous generalized Schur decomposition (SGSD) is a known remedy in the case of I× J× 2 arrays. The SGSD method is evaluated on ill-conditioned data of more general sizes.

Various aspects of tensor decompositions are addressed: existence, uniqueness and computation. The state of the art is surveyed, by making the difference between conjectures and proved results. Some numerical algorithms are described in details, and their numerical complexity is evaluated. The slowness of numerical algorithms is often due to a form of ill-conditioning of the tensor to be decomposed. In particular, Richard Harshman called ‘bottleneck’ the fact that two or more factors in a mode are almost collinear.

In this paper in silico flux distributions are generated from a genome-scale metabolic network of a bacterium (Lactococcus lactis MG1363). Multiple environmental conditions are used and correlation matrices for each experimental condition between all metabolic reactions are calculated. To identify variant and invariant reactions in the network across the different conditions multi-way analysis (PARAFAC and PCA) is applied. The discussion of the results of both methods leads to the question whether latent variable models are suitable for analyzing this type of data.

A new method for resolving peaks across large sections or even entire GC × GC-TOFMS chromatograms using PARAFAC in an automated fashion is presented. This method builds upon previously developed automated PARAFAC methods for small subsections of GC × GC-TOFMS data. The new method is demonstrated on three different GC × GC-TOFMS data sets. The results of these analyses show that the presented method can resolve peaks (including overlapping peaks) and provide useful quantitative and identification information.

We study the decomposition of a nonnegative tensor into a minimal sum of outer product of nonnegative vectors and the associated parsimonious naïve Bayes probabilistic model. We show that the corresponding approximation problem, which is central to nonnegative PARAFAC, will always have optimal solutions. The result holds for any choice of norms and, under a mild assumption, even Bregman divergences.

In the Parafac algorithm it is possible to constrain one or more component matrices to have uncorrelated columns. In applications where one mode has very many entries, the algorithm is computationally impractical. Two alternative algorithms are offered to handle such situations efficiently. By means of a simulation study, it is demonstrated that these algorithms are much more efficient than the original algorithm.