Title: MULTILINEAR MODELS FOR THREE-WAY DATA REDUCTION

Abstract: Multilinear models were born in psychometrics as possible extensions of Principal Component Analysis, and/or Factor Analysis, to the case of three-way data.

In this mini-course, we review them following least squares and model based approaches. We illustrate their use to perform Principal Component Analysis, Factor Analysis, Multidimensional Scaling and Cluster analysis on three-way data.

Keywords: Principal Component Analysis, Factor Analysis, Cluster Analysis, Three-way Data, Data Reduction
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Three-way data refer to situations where the same two-way data structure is observed under different conditions. For example, when the same variables are measured on the same subjects in different occasions. This type of data, also named matrix data, is found in almost all fields of application such as psychometrics, chemistry, economics, biology, etc. Various methods have been proposed in literature to analyze three-way data (see for example [4] and [2]).

In this context, we review a particular class of models named multilinear because parameters can be partitioned such that the model is linear with respect to a subset of parameters given the others. They have been initially proposed to extend Principal Component Analysis, that is bilinear, from two-way to three-way data. Nowadays, they are largely used to perform even Factor Analysis, Multidimensional Scaling and Cluster Analysis in several fields of application on three-way data.

A tentative syllabus is the following:

• Tucker3 and tensorial bases;

• CANDECOMP-PARAFAC and the tensorial rank;

• Component approach: least squares estimation;

• Structural approach: model based estimation [1];

• Simultaneous clustering and dimensionality reduction [5], [3].

References

[1] P. Giordani, R. Rocci, and G. Bove. Factor uniqueness of the structural parafac model. Psychometrika, 85:555 – 574, 2020.

[2] P.M. Kroonenberg. Applied multiway data analysis. Wiley, 2008.

[3] R. Rocci, M. Vichi, and M. Ranalli. Mixture models for simultaneous classification and reduction of three-way data. Submitted, 2023.

[4] A.K. Smilde, R. Bro, and P. Geladi. Multi-way analysis: applications in the chemical sciences. John Wiley & Sons, 2005.

[5] M. Vichi, R. Rocci, and H.A.L. Kiers. Simultaneous component and clustering models for three-way data: Within and between approaches. Journal of Classification, 24:71–98, 2007.