Orthogonal signal correction: algorithmic aspects and properties

Abstract

AbstractThe objective of this paper is to present a modified algorithm for the orthogonal signal correction (OSC) filter based on the approaches proposed by Wold, Antti, Lindgren and Öhman (Chemometrics Intell. Lab. Syst. 1998; 44: 175–185) and Fearn (Chemometrics Intell. Lab. Syst. 2000; 50: 47–52). An OSC filter consists of a trio of building blocks: weights, components and loadings, $\font\mathbfit=cmmib10\def\bfit#1{\hbox{\mathbfit #1}} {\bf \{w_{{\bfit j}}, t_{{\bfit j}}, p_{{\bfit j}}\}_{{\bfit j}=1}^{{\bfit A}}}$. The original OSC filter of Wold et al. was based on the framework of the non‐linear iterative partial least squares (NIPALS) algorithm. Adopting this approach enabled the mathematical justification for the selection of the loading vectors p j and components t j, but there was no theoretical foundation for the selection of w j. In contrast, the approach of Fearn described an objective function for the selection of the weight vectors w j, but in this case there is no theoretical justification for either p j or t j. Combining both approaches, within a NIPALS framework, enables a clear theoretical basis for the selection of all three building blocks to be established. A number of orthogonal and optimal properties of the NIPALS‐based OSC algorithm, although previously reported, are also theoretically proven. Finally, it is shown that the modified OSC algorithm is equivalent to Fearn's OSC but is interpretable as a consequence of it being presented from a NIPALS perspective. This enables the possible extension of OSC to dynamic and non‐linear systems. Copyright © 2002 John Wiley & Sons, Ltd.