Tikhonov regularization in standardized and general form for multivariate calibration with application towards removing unwanted spectral artifacts

Abstract

AbstractTikhonov regularization (TR) is an approach to form a multivariate calibration model for y = Xb. It includes a regulation operator matrix L that is usually set to the identity matrix I and in this situation, TR is said to operate in standard form and is the same as ridge regression (RR). Alternatively, TR can function in general form with L ≠ I where L is used to remove unwanted spectral artifacts. To simplify the computations for TR in general form, a standardization process can be used on X and y to transform the problem into TR in standard form and a RR algorithm can now be used. The calculated regression vector in standardized space must be back‐transformed to the general form which can now be applied to spectra that have not been standardized. The calibration model building methods of principal component regression (PCR), partial least squares (PLS) and others can also be implemented with the standardized X and y. Regardless of the calibration method, armed with y, X and L, a regression vector is sought that can correct for irrelevant spectral variation in predicting y. In this study, L is set to various derivative operators to obtain smoothed TR, PCR and PLS regression vectors in order to generate models robust to noise and/or temperature effects. Results of this smoothing process are examined for spectral data without excessive noise or other artifacts, spectral data with additional noise added and spectral data exhibiting temperature‐induced peak shifts. When the noise level is small, derivative operator smoothing was found to slightly degrade the root mean square error of validation (RMSEV) as well as the prediction variance indicator represented by the regression vector 2‐norm $\left\| {{\rm \hat b}} \right\|_2$ thereby deteriorating the model harmony (bias/variance tradeoff). The effective rank (ER) (parsimony) was found to decrease with smoothing and in doing so; a harmony/parsimony tradeoff is formed. For the temperature‐affected data and some of the noisy data, derivative operator smoothing decreases the RMSEV, but at a cost of greater values for $\left\| {{\rm \hat b}} \right\|_2$. The ER was found to increase and hence, the parsimony degraded. A simulated data set from a previous study that used TR in general form was reexamined. In the present study, the standardization process is used with L set to the spectral noise structure to eliminate undesirable spectral regions (wavelength selection) and TR, PCR and PLS are evaluated. There was a significant decrease in bias at a sacrifice to variance with wavelength selection and the parsimony essentially remains the same. This paper includes discussion on the utility of using TR to remove other undesired spectral patterns resulting from chemical, environmental and/or instrumental influences. The discussion also incorporates using TR as a method for calibration transfer. Copyright © 2006 John Wiley & Sons, Ltd.