23 - Calculating the Solution for Regression Techniques: Part 3 — Partial Least Squares Regression Made Simple

Abstract

This chapter illustrates the straightforward calculations used for partial least squares (PLS) regression utilizing singular value decomposition. The PLS form of singular value decomposition (SVD) includes the use of the concentration vector c as well as the data matrix A. All mathematical operations are completed using MATLAB (Matrix Laboratory) software. The example of a simple data matrix denoted by A is used to illustrate the calculation procedure. A is used to represent a set of absorbances for three samples and three data channels, as a rows x columns matrix. For this example, each row represents a different sample spectrum and each column a different data channel, absorbance, or frequency. Both A and the concentration vector c are necessary for calculating the special case of PLS singular value decomposition (PLSSVD). The operation performed in PLSSVD is sometimes referred to as the PLS form of eigen analysis or factor analysis. If PLSSVD is performed on the A matrix and the c vector, the result is three matrices, termed the left singular values (LSV) matrix or the U matrix, the singular values matrix (SVM) or the S matrix, and the right singular values matrix (RSV) or the V matrix. The PLS Scores matrix and PLS Loadings matrix can then be evaluated.