Enhancing probabilistic hydrological predictions with mixture density Networks: Accounting for heteroscedasticity and Non-Gaussianity

Improving probabilistic streamflow prediction requires proper characterization of residual errors in hydrological models. Previous studies often adopt data transformation techniques to stabilize the variability of residual errors to make them more Gaussian. Here, we introduce a novel residual error model that directly captures the intrinsic properties of residual errors, thereby avoiding potential pitfalls such as misinterpretation and conflict assumptions during data transformation. The proposed model, based on a Mixture Density Network (MDN), combines a neural network with a mixture density model to generate time-varying mixture distributions conditioned on the magnitude of streamflow and other variables. The model was examined using both synthetic data as well as data from a real catchment in the source of the Yellow River. Two model selection criteria, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), were adopted to choose the optimal number of mixture components. Compared to the benchmark model, which assumes homoscedastic Gaussian residual errors, our proposed model yields uncertainty intervals over 20% narrower and provides more accurate uncertainty coverage, particularly enhancing the capture of low and high flows. Besides, it effectively characterized heteroscedasticity and non-Gaussianity in residual errors following model assumptions checking. Further, we discovered that the optimal number of mixture components was better identified using AIC rather than BIC as the criterion.