Adaptive Horizon-Aware Model Selection and Routing for Multi-Step Water Quality Prediction: An Integrative Review
- 1Information Center of Ministry of Ecology and Environment, Beijing 100029, China
- 2School of Intelligence and Information Engineering, Heilongjiang University of Science and Technology, Harbin 150022, China
- 3School of Environment, Harbin Institute of Technology, Harbin 150001, China
Abstract
Multi-step water quality prediction comprises a set of related forecasting tasks whose information structure, uncertainty, and relative model competence vary with forecast horizon, monitoring context, and system state. Conventional model comparison commonly relies on performance averaged across lead times, which can obscure horizon-dependent error accumulation, model-ranking crossover, and regime-specific failure. This review examines adaptive horizon-aware model selection and routing from a mathematical and statistical perspective. Recursive, direct, multi-output, ensemble, mixture-of-experts, and adaptive routing strategies are compared according to their dependence structure, parameter sharing, loss coupling, and error-propagation mechanisms. Particular emphasis is placed on conditional risk as a unified representation of model competence across monitoring sections, temporal resolutions, target indicators, forecast horizons, and environmental states. Routing regret, competence margins, and selection uncertainty are further discussed to characterize the potential gain, stability, and reliability of conditional model allocation. The review also analyzes how hydrological memory, transport delay, cross-indicator dependence, external forcing, data scarcity, and concept drift reshape predictability across horizons. Statistical evaluation is considered through leakage-free temporal validation, rolling-origin assessment, dependence-aware model comparison, model-ranking uncertainty, probabilistic calibration, and selective prediction. This synthesis provides a mathematically structured basis for understanding when different forecasting mechanisms remain reliable and how horizon-dependent model competence can support more robust adaptive routing in multi-step water quality prediction.
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BibTeX
@article{yan2026adaptive,
title = {{Adaptive Horizon-Aware Model Selection and Routing for Multi-Step Water Quality Prediction: An Integrative Review}},
author = {Qinian Yan and Hongyu Hao and Fengxia Zhang and Chuanhao Xin and Lixin Li},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14213902},
url = {https://www.mdpi.com/2227-7390/14/21/3902},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Adaptive Horizon-Aware Model Selection and Routing for Multi-Step Water Quality Prediction: An Integrative Review
AU - Qinian Yan
AU - Hongyu Hao
AU - Fengxia Zhang
AU - Chuanhao Xin
AU - Lixin Li
PY - 2026
DA - 2026/10/26
JO - Mathematics
DO - 10.3390/math14213902
UR - https://www.mdpi.com/2227-7390/14/21/3902
AB - Multi-step water quality prediction comprises a set of related forecasting tasks whose information structure, uncertainty, and relative model competence vary with forecast horizon, monitoring context, and system state. Conventional model comparison commonly relies on performance averaged across lead times, which can obscure horizon-dependent error accumulation, model-ranking crossover, and regime-specific failure. This review examines adaptive horizon-aware model selection and routing from a mathematical and statistical perspective. Recursive, direct, multi-output, ensemble, mixture-of-experts, and adaptive routing strategies are compared according to their dependence structure, parameter sharing, loss coupling, and error-propagation mechanisms. Particular emphasis is placed on conditional risk as a unified representation of model competence across monitoring sections, temporal resolutions, target indicators, forecast horizons, and environmental states. Routing regret, competence margins, and selection uncertainty are further discussed to characterize the potential gain, stability, and reliability of conditional model allocation. The review also analyzes how hydrological memory, transport delay, cross-indicator dependence, external forcing, data scarcity, and concept drift reshape predictability across horizons. Statistical evaluation is considered through leakage-free temporal validation, rolling-origin assessment, dependence-aware model comparison, model-ranking uncertainty, probabilistic calibration, and selective prediction. This synthesis provides a mathematically structured basis for understanding when different forecasting mechanisms remain reliable and how horizon-dependent model competence can support more robust adaptive routing in multi-step water quality prediction.
SN - 2227-7390
ER -