Pareto Optimal Multivariate Calibration for Spectroscopic Data

Abstract

Multivariate calibration of spectra l data is considered with an emphasis on prediction. An abundance of methods are available to develop such calibration m odels. Using a harmonious approach with target vector optimization, the best calibration models are identi ed relative to the criteria used. Criteria utilized to determine the adequacy of models are minimization of the root mean square error of calibration (RMSEC) and the norm of the regressio n vector. Because of the simplicity of the optimization response surfaces, the method of sim plex was found to function much faster than generalized simulated annealing. Using a near infrared spectral example to demonstrate concepts, a family of models are established to be good, i.e., Pareto optimal models. For the data set investigated, it is found that the Pareto optimal models are essentially the same as models obtained by ridge regression and generalized ridge regression and are more harmonious than models obtained by principal component regressio n (PCR), partial least-squares (PLS), continuum regression, and cyclic subspace regression as the Pareto optimal models have smaller regressio n vector norms, RMSEC, and root mean square error of validation (RMSEV) values. The PLS models are found to be Pareto optimal relative to the PCR models. The paper presents an explanation of when the RR model will not be acceptable.

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

“…However, the small parsimony reduction is offset by the small degradation in harmony thereby forming a harmony/parsimony tradeoff much like the bias/variance tradeoff for harmony. As noted in previous work [35,36], the usual parsimony trend PCR, PLS, TR (in order of most to least) is observed for both L I and L 2 .…”

“…As has been pointed out in References [35] and [36], TR generates the more harmonious curve, but in the optimal model regions located at the bends in respective L-curves, the TR, PCR and PLS optimal models are essentially equivalent. Listed in Table I are evaluation values for model comparisons.…”

“…Displayed in Figure 2 are the harmonious curves for TR, PCR and PLS showing that the typical harmonious trend TR, PLS, PCR (in order from most to least Pareto) without smoothing [35,36] is maintained with smoothing. As has been pointed out in References [35] and [36], TR generates the more harmonious curve, but in the optimal model regions located at the bends in respective L-curves, the TR, PCR and PLS optimal models are essentially equivalent.…”

Effective rank for multivariate calibration methods

“…Optimal PCR, PLS and RR models were determined from plots of the variance measure kb bk 2 against the bias measures RMSEC, RMSEV and respective R 2 values [1,4,15,16]. The approach of using these plots for optimal model selection is sometimes termed L-curve owing to the characteristic Lshape of the plots [4,17].…”

“…Equally important to comparing models is the harmony. The more harmonious model will have a good balance of bias and variance [1].…”