An explainable hybrid framework for estimating daily reference evapotranspiration: Combining extreme gradient boosting with Nelder-Mead method

Accurate estimation of reference evapotranspiration (ETo) is essential for effective water resources management, irrigation system design, and various hydrological and agricultural applications. This study employed extreme gradient boosting (XGBoost) model, signal decomposition techniques, and XGBoost coupled with Nelder–Mead (NM) method to enhance ETo prediction across two meteorological stations in Iran. This study proposed a novel framework which lies in its comprehensive integration of advanced techniques to create a model that is both interpretable and highly accurate for ETo estimation. For this aim, sixty meteorological variables were categorized into solar and cloud-based, temperature-based, wind and humidity-based, and pressure-based groups to analyze their effect on ETo estimation. Feature selection methods, including the Gradient Boosting Machine, Kendall's Tau, and Relief Algorithm, were employed to identify the most influential predictors. Variables with normalized weights equal to or greater than 0.9 were selected for model input, resulting in the top 10% of variables being utilized. The XGBoost models were then developed using these selected inputs (level 1), a wavelet-based hybrid was developed according to the most effective features (level 2), and the XGBoost model was coupled by NM algorithm (level 3) for ETo estimation. Results of estimated ETo by levels 1 to 3 were compared with four common empirical approaches. The results indicated that Kendall's Tau-based feature selection provided the most accurate predictions in Shiraz, achieving an RMSE of 0.721 (mm/day) for solar and cloud-based variables during the test phase. Additionally, the application of wavelet analysis further refined the model inputs, which enhanced ETo estimation in most variable groups. Integrating the NM algorithm with XGBoost demonstrated significant improvements in ETo estimation, where it could improve RMSE to 0.091 and 0.155 (mm/day) for testing section in Fasa and Shiraz, respectively.