Representation, synthesis, variability and data preprocessing of a three-way data set

Abstract In the first section of this paper we describe the structures (vectors and matrices) on which a three-way data set X can be organized, and the information we can point out when using these structures. Many three-way analyses are based on pooled representations of X , that are systematically studied. The information given by a three-way data set can be synthesized according to the structures utilized to represent X . In the second section we define one-way, two-way and three-way syntheses of X . Also the variability of a three-way data set is evaluated, in section three, according three different levels: one-way or fiber variability, two-way or slab variability and three-way variability. The syntheses and the variability indices of X can be used for data preprocessing of X , which is here discussed in section four. Furthermore we discuss, in Section 5, the Principal Matrices Analysis on the base of three-way variability indices.

[1]  Y. Escoufier.  LE TRAITEMENT DES VARIABLES VECTORIELLES , 1973 .

[2]  Jérôme Pagès,et al.  L'analyse factorielle multiple , 1984 .

[3]  P. Robert,et al.  A Unifying Tool for Linear Multivariate Statistical Methods: The RV‐Coefficient , 1976 .

[4]  R. A. Harshman,et al.  Data preprocessing and the extended PARAFAC model , 1984 .

[5]  Henk A. L. Kiers,et al.  Alternating least squares algorithms for simultaneous components analysis with equal component weight matrices in two or more populations , 1989 .

[6]  B. Flury.  Common Principal Components in k Groups , 1984 .

[7]  R. Coppi.  Analysis of Three-Way Data Matrices Based on Pairwise Relation Measures , 1986 .

[8]  Henk A. L. Kiers,et al.  Simultaneous Components Analysis , 1992 .

[9]  J. Levin,et al.  Simultaneous factor analysis of several Gramian matrices , 1966, Psychometrika.

[10]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[11]  J. Barra,et al.  Recent Developments in Statistics , 1978 .

[12]  L. Tucker.  Relations between multidimensional scaling and three-mode factor analysis , 1972 .

[13]  Roger E. Millsap,et al.  Component analysis in cross-sectional and longitudinal data , 1988 .

[14]  J. Kruskal.  MULTILINEAR MODELS FOR DATA ANALYSIS , 1981 .

[15]  B. Escofier,et al.  Multiple factor analysis results of a three-year utilization , 1989 .

[16]  J. Gower,et al.  Metric and Euclidean properties of dissimilarity coefficients , 1986 .

[17]  Alfredo Rizzi.  On the synthesis of three way data matrices , 1989 .

[18]  D. M. Titterington,et al.  Recent Developments in Statistical Inference and Data Analysis , 1982 .

[19]  H. Hotelling.  Relations Between Two Sets of Variates , 1936 .