Fuzzy Neural Networks for water level and discharge forecasting with uncertainty

This paper proposes a new procedure for water level (or discharge) forecasting under uncertainty using artificial neural networks: uncertainty is expressed in the form of a fuzzy number. For this purpose, the parameters of the neural network, namely, the weights and biases, are represented by fuzzy numbers rather than crisp numbers. Through the application of the extension principle (Zadeh, 1965), the fuzzy number representative of the output variable is then calculated at each time step on the basis of a set of crisp inputs and fuzzy parameters of the neural network. The fuzzy parameters of the neural network are estimated from the modelling process, that is through a calibration procedure that imposes a constraint whereby for an assigned h-level the envelope of the corresponding intervals representing the outputs (forecasted levels or discharges, calculated at different points in time) must include a preset percentage of observed values. The application of the model to two specific cases and a comparison of the results with those provided by other data-driven models - Bayesian neural networks (Neal, 1992) and the Local Uncertainty Estimation Model (Shrestha and Solomatine, 2006) - show its effectiveness in estimating water levels or discharges under uncertainty. The fuzzy neural network enables us to define bands that are expected to contain given percentages of forecasted level/discharge values for each lead time selected; an analysis of the results obtained reveals that these bands (e.g. 99%, 95% and 90% uncertainty bands) generally have a slightly smaller width compared to the bands obtained using other data-driven models.