Hu et al. (2026) Monte-Carlo-assisted endo-exo temporal transformer for high-confidence interval forecasting of daily runoff

Identification

• Journal: Stochastic Environmental Research and Risk Assessment
• Year: 2026
• Date: 2026-04-01
• Authors: Xiao-xue Hu, Dong-mei Xu, Wenchuan Wang, Jun Wang, Zong Li
• DOI: 10.1007/s00477-026-03206-1

Research Groups

College of Water Resources, North China University of Water Resources and Electric Power, Zhengzhou, China

Short Summary

This study introduces the Endo-Exo Temporal Transformer (ETT) model, which fuses endogenous and exogenous hydrological features with a Monte Carlo-assisted interval forecasting framework, significantly improving daily runoff prediction accuracy and uncertainty quantification across diverse watersheds.

Objective

• To develop a high-confidence interval forecasting model for daily runoff that addresses low prediction accuracy and challenges in feature identification under climate variability and human activities, by integrating endogenous-exogenous feature fusion, multi-scale temporal context capture, and Monte Carlo-assisted uncertainty quantification.

Study Configuration

• Spatial Scale: Four hydrological stations in the USA, representing diverse climate-morphological conditions: Cow Creek (Oregon, 167 square kilometers), Crystal River (Colorado, 432 square kilometers), Minam River (Oregon, 618 square kilometers), and Alsea River (Oregon, 857 square kilometers).
• Temporal Scale: Daily runoff data spanning from January 1, 2002, to December 31, 2009 (2,922 days). An optimal sliding window size of 12 days was determined.

Methodology and Data

• Models used:
  • Proposed: Endo-Exo Temporal Transformer (ETT) model, built upon a Transformer architecture.
  • Key components: Endogenous-Exogenous Feature Fusion Module (EFFM), Temporal Context Fusion Unit (TCFU), Long Short-Term Memory (LSTM) networks, Efficient Additive Attention (EAA) mechanism.
  • Uncertainty quantification: Monte Carlo (MC) random sampling method for constructing multi-confidence prediction intervals.
  • Benchmark models for comparison: TCFU-Transformer, TCFU, LSTM, EFFM-Transformer, LSTM-Transformer, EAA-Transformer, and Transformer.
• Data sources: CAMELS dataset (Catchment Attributes and Meteorology for Large-sample Studies).
  • Input features: Daily runoff, sunshine duration (Dayl), precipitation (Prcp), snow water equivalent (Swe), solar radiation (Srad), daily maximum temperature (Tmax), daily minimum temperature (Tmin), and atmospheric pressure (Vp).

Main Results

• The ETT model consistently achieved superior performance in daily runoff prediction across all four diverse hydrological stations compared to seven benchmark models.
• At the Crystal River station, the ETT model achieved a Nash–Sutcliffe Efficiency (NSE) of 0.997 and a Kling-Gupta Efficiency (KGE) of 0.979.
• Peak flow prediction errors (EP) were reduced by an average of 68.3% compared to the comparative models. For instance, at Cow Creek, ETT's average EP was 12.15%, significantly lower than the 22.80%-94.71% range of benchmark models.
• The Monte Carlo-assisted interval prediction framework demonstrated high reliability, with Prediction Interval Coverage Probability (PICP) ranging from 0.861 to 0.961 across 80% to 95% confidence levels.
• The Mean Prediction Interval Width (MPIW) was compressed by more than 65% compared to benchmark models, and the Mean Prediction Interval Center Deviation (MPICD) showed significant improvement, indicating enhanced reliability and optimized uncertainty quantification.
• Factor sensitivity analysis revealed that the dominant sensitive factors influencing runoff prediction vary significantly across different climatic regions (e.g., daily maximum temperature in wet regions, sunshine duration in semi-arid regions, snow water equivalent in high-altitude arid regions).
• Parameter sensitivity analysis indicated that num_heads = 8 and E-D layers = 2 generally provided the optimal balance between model performance and computational efficiency.
• Model interpretability, achieved through attention weight visualization and SHAP analysis, confirmed that the ETT model adaptively captures key temporal nodes and feature contributions that align with actual hydrological processes.

Contributions

• Proposed the Endogenous-Exogenous Feature Fusion Module (EFFM) for synchronous modeling of endogenous runoff fluctuations (high-frequency/low-frequency) and exogenous meteorological drivers, achieving dynamic feature fusion.
• Introduced the Efficient Additive Attention (EAA) mechanism to enhance the Transformer's response to local abrupt features (e.g., flood peaks), addressing the boundary blurring issue of traditional global attention.
• Designed an LSTM-EAA cascaded architecture that leverages LSTM's long-term memory for seasonal trends and EAA's local sensitivity for abrupt events like heavy rainfall and snowmelt.
• Developed a "Factor Elimination-Performance Degradation" experimental framework to quantify the region-specific contributions of seven types of driving factors (e.g., temperature, precipitation).
• Established a multi-confidence interval prediction framework using Monte Carlo random sampling, systematically evaluating Prediction Interval Coverage Probability (PICP), Mean Prediction Interval Width (MPIW), Percentage of Uncertainty Coverage Index (PUCI), and Mean Prediction Interval Center Deviation (MPICD) to bridge the gap between point and interval prediction.

Funding

• Key Special Projects of National Key Research and Development Program on “Major Natural Disasters and Public Safety” (No: 2024YFC3012300)
• Henan Province Centrally Guided Local Science and Technology Development Fund Projects for 2024 (No: Z20241471017)

Citation

@article{Hu2026MonteCarloassisted,
  author = {Hu, Xiao-xue and Xu, Dong-mei and Wang, Wenchuan and Wang, Jun and Li, Zong},
  title = {Monte-Carlo-assisted endo-exo temporal transformer for high-confidence interval forecasting of daily runoff},
  journal = {Stochastic Environmental Research and Risk Assessment},
  year = {2026},
  doi = {10.1007/s00477-026-03206-1},
  url = {https://doi.org/10.1007/s00477-026-03206-1}
}