Prediction of sample properties using spectroscopic data with multivariate calibration is often enhanced by wavelength selection. This paper reports on a built-in wavelength selection method in which the estimated regression vector contains zero to near-zero coefficients for undesirable wavelengths. The method is based on Tikhonov regularization with the model 1-norm (TR1) and is applied to simulated and near-infrared (NIR) spectral data. Models are also formed from wavelength subsets determined by the standard method of stepwise regression (SWR). Harmonious (bias/variance tradeoff) and parsimonious considerations are compared with and without wavelength selection for principal component regression (PCR), ridge regression (RR), partial least squares (PLS), and multiple linear regression (MLR). Results show that TR1 models generally contain large baseline regions of near-zero coefficients, thereby essentially achieving built-in wavelength selection. For example, wavelengths with spectral interferences and/or poor signal-to-noise ratios obtain near zero regression coefficients. Results often improve with TR1 models, compared to full wavelength PCR, RR, and PLS models. The SWR subset results are similar to those for the TR1 models using the NIR data and worse with the simulated spectral situations. In general, wavelength selection improves prediction accuracy at a sacrifice to a potential increase in variance and the parsimony remains nearly equivalent compared to full wavelength models. New insights gained from the reported studies provide useful guidelines on when to use full wavelengths or use wavelength selection methods. Specifically, when a small number of large wavelength effects (good sensitivity and selectivity) exist, subset selection by SWR (with caution) and TR1 do well. With a small to moderate number of large to moderate sized wavelength effects, TR1 is better. Lastly, when a large number of small effects are present, full wavelengths with the methods of PCR, RR, or PLS are best.

Tikhonov regularization (TR) is an approach to form a multivariate calibration model for y ¼ 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 6 ¼ 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-normb 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 forb 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.

For multivariate calibration with the relationship y ¼ Xb, it is often necessary to determine the degrees of freedom for parsimony consideration and for the error measure root mean square error of calibration (RMSEC). This paper shows that degrees of freedom can be estimated by an effective rank (ER) measure to estimate the model fitting degrees of freedom and the more parsimonious model has the smallest ER. This paper also shows that when such a measure is used on the X-axis, simultaneous graphing of model errors and other regression diagnostics is possible for ridge regression (RR), partial least squares (PLS) and principal component regression (PCR) and thus, a fair comparison between all potential models can be accomplished. The ER approach is general and applicable to other multivariate calibration methods. It is often noted that by selecting variables, more parsimonious models are obtained; typically by multiple linear regression (MLR). By using the ER, the more parsimonious model is graphically shown to not always be the MLR model. Additionally, a harmony measure is proposed that expresses the bias/variance tradeoff for a particular model. By plotting this new measure against the ER, the proper harmony/parsimony tradeoff can be graphically assessed for RR, PCR and PLS. Essentially, pluralistic criteria for fairly valuating and characterizing models are better than a dualistic or a single criterion approach which is the usual tactic. Results are presented using spectral, industrial and quantitative structure activity relationship (QSAR) data.

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