Multivariate Analysis of a Sequence of Paired Data Matrices: Succesive and Simultaneous Approaches

Abstract

The relationship between two data matrices has been studied in the interbattery factor analysis. When two data matrices are partitioned in rows, the relationship between two data matrices has been studied in the STATICO method. The main advantage of this method is the optimality of the compromise of co-structures. It is well known that the weighting coefficients of the compromise may be contrary sign in some cases and make it uninterpretable. Thus, many multivariate data analysis methods have been developed, particularly those designed to tackle the fundamental issue: the description of the relationships between two data matrices. This can be studied by successive modeling approaches as well as by a simultaneous modeling approach. These methods are based on co-inertia and can be reduced to finding the maximum, minimum, or other critical values of a ratio of quadratic forms. However, all these methods are successive. In this paper, we propose two algorithms. The first one called sDO-CCSWA (successive Double-Common Component and Specific Weight Analysis) maximizes the sum of squared covariances, by first finding the best pair-component solution, and repeating that process in the respective residual spaces. The sDO-CCSWA is a new monotonically convergent algorithm obtained by searching for a fixed point of the stationary equations. The second approach is a simultaneous algorithm (DO-CCSWA) which maximizes the sum of squared covariances.