A unified framework for the study of the PLS estimator's properties

TL;DR

This paper introduces a unified framework linking the Partial Least Squares (PLS) estimator to discrete orthogonal polynomials, simplifying analysis and providing new insights into its properties.

Contribution

It presents a novel approach connecting PLS to residual polynomials, enabling simplified proofs and deeper understanding of PLS properties.

Findings

Unified framework for PLS analysis

Simplified proofs of classical results

New insights into PLS behavior

Abstract

In this paper we propose a new approach to study the properties of the Partial Least Squares (PLS) estimator. This approach relies on the link between PLS and discrete orthogonal polynomials. Indeed many important PLS objects can be expressed in terms of some specific discrete orthogonal polynomials, called the residual polynomials. Based on the explicit analytical expression we have stated for these polynomials in terms of signal and noise, we provide a new framework for the study of PLS. Furthermore, we show that this new approach allows to simplify and retreive independent proofs of many classical results (proved earlier by different authors using various approaches and tools). This general and unifying approach also sheds new light on PLS and helps to gain insight on its properties.