The Analysis of Three Way Data Matrices: A Method Based on Relation Measures Between Units.

The analysis of data matrices X = { xijk } by means of factorial techniques is considered in this paper. The most relevant contributions to this subject are due to Carroll (1968), Tucker (1972) and Kroonenberg (1983), within the framework of the "Anglo-saxon school". On the other side, in the French school of "Analyse des donnees" , the works of Saporta (1975), Escoufier (19S0) and Escofier, Pages (1984) can be mentioned. Our own contribution shows that dealing with this type of data matrices implies using a "strategy of analysis" and leads to the generalization of the notion of "dimensions" of X. To this purpose the neutral term mode is utilized. This is justified by a detailed examination of the different ways statistical information is usually manipulated in order to set up the data matrices to be eventually analysed. After having provided a technical definition of the notions of "strategy", "point of view" and "common component", the different types of three-mode data matrices are illustrated. These can be classified into three different categories: compact, non compact and sparse matrices, according to the characteristics of the various two-way section of X. For each type, the structures which can be defined in the different spaces (where the elements belonging to each mode can be represented) are studied in detail, according to a given “strategy”. The results of this analysis can be summarized as follows. In the compact matrices the symmetry of analysis allows for a complete study of the three possible strategies. In particular it is shown that by means of the "inter-structure" analysis of the "trajectories" of the same element according to another mode a global analysis of X is achieved, even if we limit ourselves to only one strategy. In the non compact matrices the choice of suitable metrics for each two-way section, within a given strategy, allows for the adoption of a common space, Rj , where the inter-structure analysis of the single shapes and of the "trajectories" is made possible either by deriving "common factors" or by means of the " Escoufier’s operators" . In the sparse matrices the difficulty of componing the different two-way sections gives rise to new methodological probelms for a three-way analysis of this type of data. In fact, the only techniques available for this case cannot be included within the "strategical approach", since they are capable to analyse only two-way "marginal tables"