Bayesian neural network for rainfall-runoff modeling

In this paper, a Bayesian learning approach is introduced to train a multilayer feed-forward network for daily river flow and reservoir inflow simulation in a cold region river basin in Canada. In Bayesian approach, uncertainty about the relationship between inputs and outputs is initially taken care of by an assumed prior distribution of parameters ( weights and biases). This prior distribution is updated to posterior distribution using a likelihood function following Bayes' theorem while data are observed. This posterior distribution is called the objective function of a network in the Bayesian learning approach. The objective function is maximized using a suitable optimization technique. Once the network is trained, the predictive distribution of the network outputs is obtained by integrating over the posterior distribution of weights. In this study, Gaussian prior distribution and a Gaussian noise model are used in defining posterior distribution. The network has been optimized using a scaled conjugate gradient technique. Posterior distribution of weights is approximated to Gaussian during prediction. Prediction performance of the Bayesian neural network ( BNN) is compared with the results obtained from a standard artificial neural network ( ANN) model and a widely used conceptual rainfall-runoff model, namely, HBV-96. The BNN model outperformed the conceptual model and slightly outperformed the standard ANN model in simulating mean, peak, and low river flows and reservoir inflows. The significant contribution of the Bayesian method over the conventional ANN approach, among others, is the uncertainty estimation of the outputs in the form of confidence intervals which are particularly needed in practical water resources applications. Prediction confidence limits ( or intervals) indicate the extent to which one can rely on predictions for decision making. It is shown that the BNN can provide reliable streamflow and reservoir inflow forecasts without a loss in model prediction accuracy as compared to standard ANN and conceptual model HBV. Another significant advantage of BNN approach is that the overfitting and underfitting problems are automatically taken care of by the Bayesian learning algorithm, which conversely remain serious problems with conventional ANN learning algorithm.