Iterative deflation algorithm, eigenvalue equations, and PLS2

Abstract

PLS2 is probably the most used algorithm to perform projection to latent structures regression in the case of multivariate response. However, several criticisms pointed to the theoretical limits of its original formulation, highlighting the need of a more robust foundation within the theory of regression analysis. The iterative deflation algorithm is here introduced as a starting point to obtain a family of regression methods, which includes PLS2, principal component regression (PCR), and elastic component regression (ECR), where different eigenvalue equations are used to calculate the weight vectors. Within this framework, an original portrait of PLS2 is drawn. The main mathematical properties useful to understand what PLS2 is and how PLS2 behaves are derived. A new regression method called iterative deflation algorithm‐based regression (IDAR) is introduced to describe the limit behaviour of PLS2, PCR, and ECR. The post‐transformation method is presented as a general property of the iterative deflation algorithm. Two data sets, one simulated and the other experimental, are investigated to illustrate the main properties of PLS2.