A celebration of multiway analysis

The interesting impact of psychometrics in analytical chemistry will be described using calibration as an example. Calibration is the task of quantifying the amount of analytes in a sample and is an everyday task in analytical chemistry. The multiway methods developed in psychometrics, however, have provided completely new and revolutionary tools for solving this task. Multiway methods have also be applied to monitoring of batch chemical processes, which are sensibly modeled by three-way methods were the modes are variables, batch time and batch number. In monitoring applications, models are developed on good batches, then new batches are compared with the models.

In the past decades, tremendous progress has been made in the theory and application of multiway methods. We give a short survey of developments in the understanding of basic algebraic aspects and in the numerical computation of tensor decompositions. We focus on applications in signal processing, paying special attention to techniques for signal separation and factor analysis

The transformational indeterminacy of Tucker3 solutions has given rise to the study of transformations to simplicity of the component matrices and the core array. Early experiences with Simplimax have shown that extreme simplicity (a vast majority of zero's) in the core can be attained in a number of cases. This led to the search for closed-form solutions for the transformations involved. Ten Berge will present an overview of such transformations.
In the first extensive study on Tucker models, Pieter Kroonenberg (1984) carefully and quite quite extensively addressed the issues of algorithms and their quality, as well as possibilities for rotation (or more generally transformation) of the solutions. His seminal work has led to a host of further developments on algorithms and techniques for rotation to improve interpretation.
Kiers will present an overview of a selection of such developments. Three-way models and multi-set models have been fruitfully illustrated using multisubject, multivariate longitudinal data. When facing such data in empirical practice, selecting a proper model can be rather cumbersome. The appearance of the data offers little direction, because multi-set data can be disguised as multiway data and vice versa.
Timmerman will present an overview of multiway and multiset models and discuss how to track a suitable modeling approach in empirical practice.

We illustrate the use of three-way methods in psychology with a brief outline of four applications: (1) a PARAFAC analysis of real-valued three-way three-mode (family by scale by judgment condition) data on behaviors of parents towards their children (and vice versa) from developmental psychology (Kroonenberg, Harshman & Murakami, 2009), (2) a Boolean PARAFAC (INDCLAS) analysis of binary three-way three-mode (person by frustrating situation by hostile behavior) data from personality psychology (Vansteelandt & Van Mechelen, 1998), (3) a constrained linked-mode three-mode/two-mode partitioning (CLASSI) analysis of a binary three-way three-mode (person by situation by appraisal) data block coupled with a binary two-way two-mode (person by situation) data block on experienced anger from the psychology of emotions (Ceulemans, Kuppens, & Van Mechelen, 2012), and (4) an INDSCAL analysis of real-valued three-way two-mode (person by affective state by affective state) data on correlations between affective states across voxels (with regard to neural activity) from neuropsychology (Baucom et al., 2012). For each application, we briefly summarize the research context and research question, the type of data, the type of model, and the gist of the modeling output. We conclude with a short discussion of threats and opportunities for the application of three-way methods in psychology.

Having introduced Pieter Kroonenberg to genotype × environment × attribute data from plant breeding trials more than twenty-five years ago, I reflect on his contribution to three-way three-mode analytical methods in agriculture. Not only has Pieter provided theoretical developments, he has analysed and interpreted data from plant improvement programs on cotton, maize, groundnut, common bean, adzuki bean and wheat. In this address, I comment on some common themes and my perception of how the focus has changed over time. I conclude with extracts from a paper which aimed to develop a detailed understanding of the heritable variation in the wheat genome and to directly translate this knowledge into gains in wheat breeding. This was achieved via a Wheat Phenome Atlas (a collection of diagrammatic representations of chromosome regions that affect trait inheritance) and three-way principal component analysis of the genotype marker-trait association profiles.

Metabolomics is the new kid on the block in functional genomics and relies heavily on advanced instrumental techniques such as GC-MS, NMR and LC-MS. It can be used to probe metabolism in cellular organisms, to analyze metabolites in body-fluid samples, plant extracts and food to name a few examples. The purpose of these measurements is dictated by the biological question underlying the study. Multiway analysis has proven to be a very useful tool in several stages of analyzing metabolomics data. An overview of these applications will be given and some examples will be worked out in more detail.

There were some potential demands for methods of multiway data analysis in 70's and 80's of the previous century in Japan. Models such as TUCKER3, PARAFAC, and INDSCAL were appreciated with considerable enthusiasm by groups of research workers, and a lot of application works were done in several laboratories. We will demonstrate some of them with graphical representations.
Some researchers have made a few innovations in the methodology of multiway data analysis, for example, improvements of the algorithms, proposals of ways of transformations of output to facilitate interpretations. Also in the context of multidimensional scaling, a couple of the scaling methods for analysing asymmetric similarity data have been developed. We will explain some of them briefly.
Finally, we will introduce some unique works for multiway categorical data analysis in Japan. Although they were not known well even in Japan, and have not yet been in practical use, they seem to give some useful intuitions of complex interactions in a higher order cross classification table.
Prof. Pieter Kroonenberg has visited and stayed in Japan many times, and has been interested in most of works that we will demonstrate in this presentation. He has been interacting with many Japanese psychometricians in a positive way, collaborating with some of them.

Multiway data analysis has been a source of inspiration for us not only because it obviously is valuable for the analysis of multiway data, but even more so because multiway thinking can be a fruitful strategy for solving certain problems in two-way data analysis. We give some examples of this strategy from our own work, which is often concerned with multiple dissimilarity- or attribute relations among individuals or between categories and individuals. In an attempt to predict what lies ahead for multiway data analysis, we will point out some recent work in the areas of relational learning and preference learning that is becoming increasingly important, for example, in social network modeling, the semantic web, and comorbidity networks of psychiatric symptoms. Here we see the borrowing strength strategy in action for the prediction of links between individuals and categories, with new alternatives of well-known three-way models.