estimated discrepancy more closely predicts the model form error. Naturally, the fidelity of the estimated discrepancy bias to the “true” discrepancy will be increased if it is calibrated with more experimental data. 4 A FORECASTING METRIC FOR PREDICTIVE MODELING In Section 2, the discrepancy bias is defined as the difference between the calibrated model predictions at the settings where experiments are available. At these settings, the discrepancy bias can be obtained with high fidelity. However, simulation models are rarely used to make predictions at settings where experiments are readily available. Therefore, to ensure the success of a certification campaign dominated by forecasts of simulation models, the focus must go beyond the discrepancy bias and consider the errors associated with forecasting that is executing simulation models to predict at settings where experiments are not available. Herein, we introduce a new term, forecasting error, known as the difference between the calibrated and bias corrected model solutions at settings where experiments are unavailable. Knowing the true form of discrepancy function permits the full correction of bias errors. In reality however, the true form of discrepancy function will be unavailable and can only be estimated. Estimated discrepancy bias will inherently have inaccuracies, ultimately contributing to forecasting errors. However, because an increased number of experiments can better define the discrepancy function, the forecasting errors are reduced. Therefore, we use the term “forecasting error” as a forecasting metric to quantify the forecasting ability of a simulation model for a given set of experimental measurements. For this, when the simulation model is used in predictive forecasting, we consider an additional term to be added to Equation 1 to compensate for the inaccuracies in the model simulation caused by (1) the remaining model parameter uncertainties and (2) the remaining inaccuracies in the definition of the discrepancy function. ( ) ( , ) ( ) ( ) ( ) f f f f sim f obs x E x x y x y x + + + = ε θ δ (3) In Equation 3, xf represents the settings at which the model is executed to forecast. Yobs(x f) represents the truth observed at the settings of xf. The surrogate model y sim(x f,ș) is a statistical process that supplies the model prediction, while the discrepancy term į(xf), also a statistical process, supplies the corresponding bias error. This calibrated and bias corrected statistical process model, therefore, yields our best possible estimate. This model is expected to successfully confine the experiments at the tested 438
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