Wavelength Selection for Multivariate Calibration Using Tikhonov Regularization

Abstract

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.

Impact of standardization sample design on Tikhonov regularization variants for spectroscopic calibration maintenance and transfer

“…When E is rectangular, then such a sparse solution is not guaranteed. The special case of TR in Expression (6) with E ¼ I and y E ¼ 0 is more commonly known as the least absolute shrinkage and selection operator (LASSO) [45][46][47][48]. A version known as the adaptive LASSO results when E is a diagonal array and y E is the zero vector [45,49].…”

Multivariate calibration maintenance and transfer through robust fused LASSO

“…Section 4 will discuss the computation used to solve the nonlinear optimization. LASSO has been performed in multivariate calibration with wavelength selection in prior papers [6,7]. While LASSO does perform wavelength selection and yields a sparse model, it ignores the order of wavelength predictors.…”

Impartial graphical comparison of multivariate calibration methods and the harmony/parsimony tradeoff

“…Every second measurement is used in this study for a total of 350 wavelengths for the full wavelength investigations. The 19 wavelength subset evaluated was that determined in a previous study based on TR with the regression vector 1-norm [43]. The moisture content of the corn is used.…”

“…Using variables selected in previous work [43,50], Tables IV and V contain results for the Corn and CAI data sets. It should be noted that the CAI selected variables were those deemed best for an artificial network.…”