Weighting schemes for updating regression models—a theoretical approach

While multivariate calibration has been successfully employed in the monitoring of chemical processes, difficulties arise in that sensors are inherently prone to drift and processes are susceptible to unmodeled upsets. Having detected an unmodeled source of variance within new samples, the usual remedy is to update the model with additional calibration samples that contain the new chemical interferent or instrumental variation. In the event that relatively few new calibration samples are available, these new samples can be assigned higher weights by incorporating two or more copies of each when constructing the updated model. While weighting has been suggested as a means of improving prediction estimates for samples containing a new source of variance, no theoretical explanation has been provided as to why weighting is advantageous and no criteria have been proposed in selecting weights for the new calibration samples. In this paper, the utility of sample weighting is explained theoretically using both model error and leverage arguments and a leverage-based criterion for selecting weights for the new calibration samples is presented. Employing both simulated and process spectral data, a close correspondence is demonstrated between weights selected using prediction error and leverage-based criteria. Additionally, paired simulation experiments show that the reduction in prediction error achieved by sample weighting increases as the level of noise in the responses increases, suggesting that this method will be of particular value when constructing calibration models using noisy instrumental responses.