Role of the calibration process in reducing model predictive error

An equation is derived through which the variance of predictive error of a calibrated model can be calculated. This equation has two terms. The first term represents the contribution to predictive error variance that results from an inability of the calibration process to capture all of the parameterization detail necessary for the making of an accurate prediction. If a model is “uncalibrated,” with parameter values being supplied solely through “outside information,” this is the only term required. The second term represents the contribution to predictive error variance arising from measurement noise. In an overdetermined system, such as that which may be obtained through “parameter lumping” (e.g., through the introduction of a spatial zonation scheme), this is the only term required. It is shown, however, that parameter lumping is a form of “implicit regularization” and that ignoring the implied first term of the predictive error variance equation can potentially lead to underestimation of predictive error variance. A model's role as a predictor of environmental behavior can be enhanced if it is calibrated in such a way as to reduce the variance of those predictions which it is required to make. It is shown that in some circumstances this can be accomplished through “overfitting” against historical field data. It can also be accomplished by giving greater weight to those measurements which carry the greatest information content with respect to a required prediction. This suggests that a departure may be necessary from the custom of using a single “calibrated model” for the making of many different predictions. Instead, model calibration may need to be repeated many times so that in each case the calibration process is optimized for the making of a specific model prediction.