Nonlinear Partial Least Squares An Overview

Nonlinear Partial Least Squares An Overview

Roman Rosipal (Medical University of Vienna, Austria and Pacific Development and Technology, LLC, USA)
DOI: 10.4018/978-1-61520-911-8.ch009
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Abstract

In many areas of research and industrial situations, including many data analytic problems in chemistry, a strong nonlinear relation between different sets of data may exist. While linear models may be a good simple approximation to these problems, when nonlinearity is severe they often perform unacceptably. The nonlinear partial least squares (PLS) method was developed in the area of chemical data analysis. A specific feature of PLS is that relations between sets of observed variables are modeled by means of latent variables usually not directly observed and measured. Since its introduction, two methodologically different concepts of fitting existing nonlinear relationships initiated development of a series of different nonlinear PLS models. General principles of the two concepts and representative models are reviewed in this chapter. The aim of the chapter is two-fold i) to clearly summarize achieved results and thus ii) to motivate development of new computationally efficient nonlinear PLS models with better performance and good interpretability.
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Background

The concept of nonlinear PLS modeling was introduced by S. Wold, Kettaneh-Wold, and Skagerberg (1989). Already in this seminal work, the authors distinguished and described two basic principles for modeling curved relationships between sets of observed data. The first principle, here denoted as Type I, is well-known and used in mathematical statistics and other research fields. The principle applies first a nonlinear transformation to observed variables. In the new representation a linear model is constructed. This principle can be easily applied to PLS, and indeed several different nonlinear PLS models were proposed and applied to real data sets. The first nonlinear PLS models in this category were constructed by using simple polynomial transformations of the observed data (Berglund & Wold, 1997, 1999). However, the proposed polynomial transformation approach possesses several computational and generalization limitations. To overcome these limitations, a computationally elegant kernel PLS method was proposed by Rosipal and Trejo (2001). The powerful concept of a kernel mapping function allows to construct highly flexible but still computationally simple nonlinear PLS models. However, in spite of the ability of kernel PLS to fit highly complex nonlinear data relationships, the model represents a ‘black-box’ with limited possibility to interpret the results with respect to the original data.

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