# Evapotranspiration Prediction Method Based on K-Means Clustering and QPSO-MKELM Model
## 1 Introduction  
With the growth of the global population, the shortage of freshwater resources is becoming increasingly severe. However, worldwide, agriculture accounts for 70% of total water usage; of that, approximately 60% is lost through evaporation. The prediction of evapotranspiration (ET\({}_{0}\)) is crucial for calculating agricultural irrigation requirements, as well as for the design, optimization, and management of irrigation systems and water resources [1, 2, 3, 4]. The widely recognized standard for calculating ET\({}_{0}\) is the FAO-56 Penman-Monteith equation [5], proposed by the Food and Agriculture Organization (FAO), which has been extensively used for calculating reference evapotranspiration (ET\({}_{0}\)) values [6, 7, 8, 9]. However, the FAO-56 Penman-Monteith equation requires numerous meteorological variables, many of which are often incomplete or missing, particularly in developing countries. This limitation restricts the direct application of FAO-56 [10, 11]. As a result, researchers have developed and calibrated various simplified models and empirical formulas for estimating evapotranspiration, such as the Makkin, Priestley-Taylor, and Hargreavesmodels. However, evaluations in practical applications have shown that these models often exhibit significant deviations in prediction accuracy [12; 13; 14], and many coefficients require region-specific adjustments [15]. Therefore, there is a need to identify methods that can more accurately predict ET\({}_{0}\) under conditions with limited meteorological data. In recent years, with the rapid development of machine learning, it has been widely applied to predicting evapotranspiration. Many common machine learning algorithms, such as Artificial Neural Network (ANN) [16], Random Forest (RF) [6; 17], Support Vector Machine (SVM) [17; 18; 19], Adaptive Neuro-Fuzzy Inference Systems (ANFIS) [16; 18], Multivariate Adaptive Regression Splines (MARS) [18], and MSP [17], have been used to model and predict ET\({}_{0}\), yielding promising results. Among these, the Extreme Learning Machine (ELM) model has garnered significant attention in academia and industry due to its fast and efficient computation, excellent generalization ability, and strong adaptability [20]. [PERSON] et al. (2015) were the first to apply the ELM model to predict evapotranspiration. They used the ELM model and the Feedforward Backpropagation Neural Network (FFBP) to predict ET\({}_{0}\) at three sites across Iraq. Their study found that the ELM model exhibited excellent predictive capability for daily ET\({}_{0}\) outperformed the FFBP model in prediction accuracy and efficiency, even when incomplete meteorological data were used. They strongly recommended the ELM model for evapotranspiration prediction [21]. Since then, many researchers have attempted to use the ELM model to predict evapotranspiration in different regions and under various input conditions, all finding that it delivers superior performance in ET\({}_{0}\) prediction. For example, [PERSON] et al. (2016) used ELM, ANN, Genetic Programming (GP), and SVM to model and predict ET\({}_{0}\) for Pusa in Bihar, India. They found that ELM achieved significantly higher prediction accuracy than the other three models with much shorter computation times [22]. Similarly, [PERSON] et al. (2018) used SVM, ELM, and four tree-based ensemble models to model and predict daily evapotranspiration for different climatic regions in China. Their results showed that SVM and ELM had higher prediction accuracy and stability than the other models, with the ELM model slightly outperforming the SVM model [17]. In machine learning, a model's performance is closely related to the setting of its parameters. Optimization algorithms can continuously refine the parameters by searching for the optimal solution, significantly improving the predictive accuracy of machine learning models. Therefore, optimization algorithms are widely used in conjunction with machine learning models, and their advantages are continuously validated in practice [23; 24; 25].