A General Computational Method for Converting Normal Spectra into Derivative Spectra

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.

“…Thus, selecting a tuning parameter pair is easier for this situation. Expression (22) could have been used to obtain other sparse models.…”

Section: Selecting Tuning Parameter Pairsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kalivas

2012

“…Besides using TR for basic calibration, removal of unwanted spectral artifacts, and calibration transfer, TR approaches have been used as a spectral smoothing process (smooth X) [37] and to convert raw measured spectra to derivative spectra [38]. A TR-type approach can also be used to select variables (wavelength) without knowledge of the spectral error structure as examined in this paper.…”

Section: Resultsmentioning

confidence: 99%

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

Stout

Kalivas

2006

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.

“…The method of TR is beginning to prove itself as versatile tool in multivariate calibration with application to a wide range of problems. Besides using TR in 2-norm for basic calibration [5,10] and TR in 1-norm for variable selection [2,4,[33][34][35][36][37][38][39][40][41][42][43][44], TR type approaches have been used to remove unwanted spectral artifacts [6,12,54] (with additional on-going investigations in our laboratory), calibration transfer [55,56] (with additional on-going investigations in our laboratory), smoothing X [57], smoothing the regression vector [6] and converting raw measured spectra to derivative spectra [58]. …”

Section: Discussionmentioning

confidence: 99%

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

Stout¹,

Baines²,

Kalivas³

2006

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.