Overview of two‐norm (L₂) and one‐norm (L₁) Tikhonov regularization variants for full wavelength or sparse spectral multivariate calibration models or maintenance

Abstract

Building a multivariate calibration model is typically accomplished using partial least squares, principal component regression, or ridge regression, also derived as the standard form of Tikhonov regularization (TR). These approaches can be used in a full variable mode (full wavelengths for spectroscopic data) or with wavelength selection (bands and/or individual for sparse models). Calibration maintenance is an important aspect of multivariate calibration and describes the situation of maintaining acceptable predictions from a model over time. In terms of TR, this amounts to updating an existing model (determined under primary conditions) to handle new secondary conditions such as a new instrument, sample matrix, or environmental conditions. This paper overviews TR in its ability to form a primary calibration model or update a primary model to new secondary conditions while using full wavelengths or simultaneously selecting wavelengths to form respective sparse models. These objectives can be accomplished in two‐norm (L2) or one‐norm (L1) or combined formats. Also included is a TR design that minimizes the effect of the standardization set composition, i.e., reduces the effect of an outlier. Ongoing work that removes all reference samples from the TR framework is also described. Copyright © 2012 John Wiley & Sons, Ltd.

Least absolute shrinkage and selection operator and dimensionality reduction techniques in quantitative structure retention relationship modeling of retention in hydrophilic interaction liquid chromatography

“…analytical chemistry. Several variants of TR exist, and although they are not TR per se, they are still derived from the general principles of TR [24]. Regularized (penalized) regression methods include ridge, LASSO (least absolute shrinkage and selection operator), elastic net, bridge regression and their extensions [25].…”

Atmospheric particulate matter characterization by Fourier Transform Infrared spectroscopy: a review of statistical calibration strategies for carbonaceous aerosol quantification in US measurement networks

“…The contribution from PTFE can presumably be removed with the appropriate baseline correction technique (Section 3.3.2). Though not been tested 20 extensively across various filter types, successful prediction has been reported between two PTFE filter types (Weakley et al, 2018b (Ottaway et al, 2012;Kalivas, 2012;Wise and Roginski, 2015), though also requires evaluation. Changing atmospheric composition can be addressed by updating the calibration set with new samples which contain new analytes or different regimes in concentration.…”

“…Many algorithms in the domain of statistical learning, machine learning, and chemometrics have demonstrated utility in building calibration models with spectra measurements: neural networks (Long et al, 1990;Walczak and Massart, 2000), Gaussian process regression , support vector regression (Thissen et al, 2004;Balabin and Smirnov, 2011), principal components regression (Hasegawa, 2006), ridge regression (Hoerl and Kennard, 1970;Tikhonov and Arsenin, 1977;Kalivas, 2012), wavelet regression (Brown et al, 2001;Zhao et al, 2012), functional regression (Saeys et al, 2008), partial least squares 30 (Rosipal and Krämer, 2006); among others. There is no lack of algorithms for supervised learning with continuous response variables that can potentially be adapted for such an application (Hastie et al, 2009).…”

Sum of ranking differences (SRD) to ensemble multivariate calibration model merits for tuning parameter selection and comparing calibration methods

“…The minimization expression for the TR variant RR [24,[42][43][44] The L-curve for selecting tuning parameters [3,20,21,[24][25][26][27]29] can be formed by plotting mean RMSEC or RMSECV against a model variance or complexity measure. Models in the corner region of the L-curve represent acceptable compromises for the bias/variance tradeoff,…”

“…Only a limited set of combinations of model merits were evaluated with SRD in this study. Not studied in this paper was using other model merits such as Mallow's Cp criterion, AIC [42][43][44][45], etc. to build up the number of objects for SRD.…”