La connessione e la correlazione tra due matrici dei dati componenti una matrice a tre indici

Abstract

In this work, starting from the statistical knowledge about connection and correlation between couples of variables, the author carries out a systematic study and a critical examination on these two relations among couples of data matrices belonging to a three-way data matrix. Analogously to linear measures among variables, the measures of correlation between two matrices have to indicate the "direction" of dependence of the whole variables set of a data matrix with that of the other data matrix. In other words measures of correlation between couples of matrices have to evaluate whether all variables of a matrix are more frequently concordant or discordant with all variables of the other matrix. After defining the mathematical notation for the three-way data matrix, two different ways to measure the correlation between couples of matrices are examined and also conditions necessary to define it are given. The covariance and correlation between couples of matrices are matrices are introduces . These indices have two main requirements they become well-known measures of correlation among couples of variables when matrices degenerate into vectors; they measure the correlation among couples of matrices as a function of the measures of correlation among couples of variables, these last in the couple each one belonging to a matrix. The statistical meaning about connection between a couple of matrices can be explained according to the well-known meaning of this relation among variables. Two matrices are not connected with each other if any variable of one matrix is not connected with any variable of the other matrix. Also the measures of connection between couples of data matrices have two main properties they become well-known measures of connection between couples of variables when matrices degenerate into vectors; they measure the connection between two matrices as a function of measures of connection between couples of variables, these last each one belonging to a matrix.