Fusion strategies for selecting multiple tuning parameters for multivariate calibration and other penalty based processes: A model updating application for pharmaceutical analysis

• Multiple measures of model quality are simultaneously assessed.
• Multiple tuning parameter values (penalty weights) can be automatically selected.
• Selected tuning parameter values balance bias and variance.

New multivariate calibration methods and other processes are being developed that require selection of multiple tuning parameter (penalty) values to form the final model. With one or more tuning parameters, using only one measure of model quality to select final tuning parameter values is not sufficient. Optimization of several model quality measures is challenging. Thus, three fusion ranking methods are investigated for simultaneous assessment of multiple measures of model quality for selecting tuning parameter values. One is a supervised learning fusion rule named sum of ranking differences (SRD). The other two are non-supervised learning processes based on the sum and median operations. The effect of the number of models evaluated on the three fusion rules are also evaluated using three procedures. One procedure uses all models from all possible combinations of the tuning parameters. To reduce the number of models evaluated, an iterative process (only applicable to SRD) is applied and thresholding a model quality measure before applying the fusion rules is also used. A near infrared pharmaceutical data set requiring model updating is used to evaluate the three fusion rules. In this case, calibration of the primary conditions is for the active pharmaceutical ingredient (API) of tablets produced in a laboratory. The secondary conditions for calibration updating is for tablets produced in the full batch setting. Two model updating processes requiring selection of two unique tuning parameter values are studied. One is based on Tikhonov regularization (TR) and the other is a variation of partial least squares (PLS). The three fusion methods are shown to provide equivalent and acceptable results allowing automatic selection of the tuning parameter values. Best tuning parameter values are selected when model quality measures used with the fusion rules are for the small secondary sample set used to form the updated models. In this model updating situation, evaluation of all possible models, thresholding, and iterative SRD performed equivalently for the three fusion rules with TR and PLS performed worse. While the application is model updating, the fusion processes are applicable to other situations requiring selection of multiple tuning parameter values.

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