Crisp discharge forecasts and grey uncertainty bands using data-driven models

A data-driven artificial neural network (ANN) model and a data-driven evolutionary polynomial regression (EPR) model are here used to set up two real time crisp discharge forecasting models whose crisp parameters are estimated through the least square criterion on the basis of literature techniques. In order to represent the total uncertainty of each model in performing the forecast, their parameters are then considered as grey numbers: their estimation is made through a calibration procedure that imposes a constraint whereby the envelope of the corresponding intervals representing the outputs (grey discharges, calculated at different points in time) must include a preset percentage of observed values and, at the same time, be as narrow as possible. Comparison of the results obtained through the application of the two models to a real case study shows that the crisp forecasting models based on ANN and EPR provide similar accuracy for short forecasting lead times, whereas for long forecasting lead times the performance of the model based on EPR deteriorates with respect to that of the ANN model, particularly in the validation phase. As regards the uncertainty bands produced by the grey formulation of the two data driven models, it is shown that, in the ANN case, these bands are, on average, narrower than those obtained by using a standard technique such as the Box-Cox transformation of the errors, whereas, in the EPR case, these bands are, on average, larger. These results thus suggest that the performance of a grey data-driven model depends on its inner structure and that, for the specific models here considered, the ANN is to be preferred since it shows a better and more flexible behaviour for forecasting both the crisp discharge values and the grey uncertainty bands.