Reliable and Adaptive Probabilistic Forecasting for Event-Driven Water-Quality Time Series Using a Gated Hybrid–Mixture Density Network

Real-time, reliable forecasting of water quality (WQ) is a critical component of sustainable environmental management. A key challenge, however, is modeling time-varying uncertainty (heteroscedasticity), particularly during disruptive events like storms where predictability decreases dramatically. Standard probabilistic models often fail in these high-stakes scenarios, producing forecasts that are either too conservative during calm periods or dangerously overconfident during volatile events. This paper introduces the Gated Hybrid–Mixture Density Network (GH-MDN), an architecture explicitly designed for adaptive uncertainty estimation. Its core innovation is a dedicated gating network that learns to adaptively modulate the prediction interval width in response to a domain-relevant, event-precursor signal. We evaluate the GH-MDN on both synthetic and real-world WQ datasets using a rigorous cross-validation protocol. The results demonstrate that our gated model provides robust calibration and trustworthy adaptive coverage; specifically, it successfully widens prediction intervals to capture extreme events where standard benchmarks fail. We further show that common aggregate metrics such as CRPS can mask over-confident behavior during rare events, underscoring the need for evaluation approaches that prioritize calibration. This science-informed approach to modeling heteroscedasticity prioritizes reliable risk coverage over aggregate error minimization, marking a critical step towards the development of more trustworthy environmental forecasting systems.