Regression of a data matrix on descriptors of both its rows and of its columns via latent variables: L-PLSR

Abstract

A new approach is described, for extracting and visualising structures in a data matrix Y in light of additional information BOTH about the ROWS in Y, given in matrix X, AND about the COLUMNS in Y, given in matrix Z. The three matrices Z– Y– X may be envisioned as an “L-shape”; X( I× K) and Z( J× L) share no matrix size dimension, but are connected via Y( I× J). A few linear combinations (components) are extracted from X and from Z, and their interactions are used for bi-linear modelling of Y, as well as for bi-linear modelling of X and Z themselves. The components are defined by singular value decomposition (SVD) of X′ YZ. Two versions of the L-PLSR are described—using one single SVD for all components, or component-wise SVDs after deflation. The method is applied to the analysis of consumer liking data Y of six products assessed by 125 persons, in light of 10 other product descriptors X and 15 other person descriptors Z. Its performance is also checked on artificial data.