Desensitizing models using covariance matrix transforms or counter‐balanced distortions

Abstract This paper presents a generalization of the Lagrange multiplier equation for a regression subject to constraints. It introduces two methods for desensitizing models to anticipated spectral artifacts such as baseline variations, wavelength shift, or trace contaminants. For models derived from a covariance matrix such as multiple linear regression (MLR) and principal components regression (PCR) models, the first method shows how a covariance matrix can be desensitized to an artifact spectrum, v , by adding σ 2 v  ⊗  v to it. For models not derived from a covariance matrix, such as partial least squares (PLS) or neural network (NN) models, the second method shows how distorted copies of the original spectra can be prepared in a counter‐balanced manner to achieve desensitization. Unlike earlier methods that added random distortions to spectra, these new methods never introduce any accidental correlations between the added distortions and the Y ‐block. The degree of desensitization is controlled by a parameter, σ, for each artifact from zero (no desensitization) to infinity (complete desensitization, which is the Lagrange multiplier limit). Unlike Lagrange multipliers, these methods permit partial desensitization so we can individually vary the degree of desensitization to each artifact, which is important when desensitization to one artifact inhibits desensitization to another. Copyright © 2005 John Wiley & Sons, Ltd.