22 - Calculating the Solution for Regression Techniques: Part 2 — Principal Component(s) Regression Made Simple

Abstract

This chapter identifies a simple data matrix denoted by A. A is used to represent a set of absorbances for three samples 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. The data necessary to calculate the singular value decomposition (SVD) for matrix A is thus obtained. The operation performed in SVD is sometimes referred to as eigenanalysis, principal components analysis, or factor analysis. If SVD is performed on the A matrix, 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. From this information, the scores matrix and loadings matrix can be found. The Loadings matrix is simply the fight singular values matrix or the V matrix; this matrix is referred to as the P matrix in principal components analysis terminology. The Scores matrix is calculated as the product of the data matrix A and the Loadings matrix V.