Uncertainty Reduction in Streamflow Prediction Using Mixture Density Networks

Streamflow prediction with higher accuracy and less uncertainty has been a challenge for many decades. Some studies have advanced uncertainty estimation using deep learning (DL), involving state-of-the-art recurrent neural networks, Gaussian mixture models (GMMs), and Monte Carlo dropout. However, there is a growing need to accurately characterize and reduce uncertainty in hydrological predictions. This study aims to reduce uncertainty in streamflow prediction by introducing a custom loss function (CLF) modified to include a penalty term for variance, in addition to the negative log-likelihood (NLL) traditionally used in mixture density networks (MDNs). We also focused on filling the research gap of uncertainty reduction without compromising accuracy, coverage of observed values, and the sharpness of forecasted distributions. Three models were used: long short-term memory (LSTM) for deterministic prediction of streamflow, MDN with NLL, and MDN with CLF, to predict the mean streamflow from the probabilistic distribution (MDNs) generated by combining multiple aussian components. MDNs also aim to estimate 90% and 50% uncertainty bounds and analyze the percentage change in these bounds between the two versions of MDNs. The results show that MDN with CLF has slightly higher prediction accuracy compared with LSTM, but significantly reduces the uncertainty bounds (average: 16%; median: 18%) compared with MDN with NLL. Additionally, the reliability of the three models was evaluated, and MDN with CLF exhibits almost no under- or overestimation for high and extreme flows, higher sharpness and accuracy of forecasted distributions, and good coverage of the observed values within the uncertainty bounds.