Selectivity‐relaxed classical and inverse least squares calibration and selectivity measures with a unified selectivity coefficient

Abstract

Two popular calibration strategies are classical least squares (CLS) and inverse least squares (ILS). Underlying CLS is that the net analyte signal used for quantitation is orthogonal to signal from other components (interferents). The CLS orthogonality avoids analyte prediction bias from modeled interferents. Although this orthogonality condition ensures full analyte selectivity, it may increase the mean squared error of prediction. Under certain circumstances, it can be beneficial to relax the CLS orthogonality requisite allowing a small interferent bias if, in return, there is a mean squared error of prediction reduction. The bias magnitude introduced by an interferent for a relaxed model depends on analyte and interferent concentrations in conjunction with analyte and interferent model sensitivities. Presented in this paper is relaxed CLS (rCLS) allowing flexibility in the CLS orthogonality constraints. While ILS models do not inherently maintain orthogonality, also presented is relaxed ILS. From development of rCLS, presented is a significant expansion of the univariate selectivity coefficient definition broadly used in analytical chemistry.The defined selectivity coefficient is applicable to univariate and multivariate CLS and ILS calibrations. As with the univariate selectivity coefficient, the multivariate expression characterizes the bias introduced in a particular sample prediction because of interferent concentrations relative to model sensitivities. Specifically, it answers the question of when can a prediction be made for a sample even though the analyte selectivity is poor? Also introduced are new component-wise selectivity and sensitivity measures. Trends in several rCLS figures of merit are characterized for a near infrared data set.

Reliable Model Selection without Reference Values by Utilizing Model Diversity with Prediction Similarity

“…Another difficulty in selecting calibration or maintenance models is the large number of diverse models that can be formed with unique features and yet predict accurately. − This problem is compounded in model maintenance when two or more tuning parameters are involved. Thus, there are many tuning parameter values that can form applicable models in the effective predictive domain of the new samples.…”

Calibration Model Updating to Novel Sample and Measurement Conditions without Reference Values

“…Also important are the calibration sample density and respective error structures, number of primary samples, and the number of secondary samples used in the model updating algorithm. The inability to interpret weight values is much like the situation of the inability to interpret model regression vectors due to the many model regression vectors that can accurately predict the same samples. − …”

Diagnosing indirect relationships in multivariate calibration models

“…This may be less problematic if the indirect relationship found in the calibration data is conserved in a new sample to which the regression model is applied. Brown and Ridder 3 and Kalivas et al 7 even show how such an indirect relationship may support the model in providing a smaller prediction error. However, as soon as the indirect relationship in the calibration data is not representative for the new sample, which may happen due to several reasons including seasonal changes when dealing with biological samples, calibration validity may be compromised 3,7,8 .…”

“…Brown and Ridder 3 and Kalivas et al 7 even show how such an indirect relationship may support the model in providing a smaller prediction error. However, as soon as the indirect relationship in the calibration data is not representative for the new sample, which may happen due to several reasons including seasonal changes when dealing with biological samples, calibration validity may be compromised 3,7,8 . Model validity is easier compromised when a model is built on indirect relationships rather than direct relationships.…”

“…However, as soon as the indirect relationship in the calibration data is not representative for the new sample, which may happen due to several reasons including seasonal changes when dealing with biological samples, calibration validity may be compromised. 3,7,8 Model validity is easier compromised when a model is built on indirect relationships rather than direct relationships. Therefore, a tool to diagnose indirect relationships in multivariate regression modeling is useful.…”