# Evapotranspiration Prediction Method Based on K-Means Clustering and QPSO-MKELM Model
## 1 Introduction
The ELM model does not require iterations compared to other machine learning models. The connection weights between the input layer and the hidden layer, as well as the thresholds of the hidden layer, are randomly generated, which gives ELM a breakneck computational speed. However, this also means that ELM may not consistently achieve optimal performance and leaves room for further optimization. As a result, many optimization algorithms have been applied to enhance ELM. In the field of evapotranspiration prediction, Particle Swarm Optimization (PSO) has been repeatedly shown to be one of the most effective methods for optimizing ELM, making it a popular and widely used model in evapotranspiration prediction. For example, [PERSON] et al. (2020) optimized the ELM model using PSO for evapotranspiration modeling and prediction in northwestern China's temperate continental climate region. They compared the performance of the PSO-ELM model with the original ELM, ANN, RF, and six empirical models regarding prediction accuracy. The results indicated that PSO enhanced the performance of ELM, outperforming other machine learning and empirical models [12]. Similarly, [PERSON] et al. (2021) used Genetic Algorithm (GA), PSO, and Artificial Bee Colony (ABC) algorithms to optimize ELM for ET\({}_{0}\) prediction at 12 representative stations across five different climatic regions in China. Their results showed that all optimization algorithms significantly improved the prediction accuracy of ELM, with the PSO-ELM model performing the best [13]. Additionally, [PERSON] et al. (2023) optimized the ELM model using GA, PSO, and Salp Swarm Algorithm (SSA) to predict ET\({}_{0}\) at 23 stations in China. Their findings demonstrated that the PSO-ELM model had the highest prediction accuracy and most excellent applicability [26]. These studies highlight the significant performance improvements achieved by using PSO to optimize ELM for ET\({}_{0}\) prediction, establishing PSO-ELM as a practical and superior approach. However, the traditional PSO algorithm has some drawbacks, such as the tendency to get stuck in local optima in complex multimodal problems, the lack of randomness in particle position updates, and the rapid decrease in population diversity as iterations progress, which negatively impacts its global search capability. To address these limitations, the Quantum Particle Swarm Optimization (QPSO) algorithm, which incorporates quantum theory, has been proposed to enhance the diversity of the population, improve global search ability, and balance global and local searches, thus reducing the likelihood of getting trapped in local optima [27]. QPSO has been applied in many fields and has improved performance [28; 29; 30]. In recent years, the improved Kernel Extreme Learning Machine (KELM) model has been proposed. It introduces a kernel function into the ELM model, which maps the data to a higher-dimensional space, thereby enhancing the ELM model's ability to handle nonlinear data. Since the factors involved in evapotranspiration (ET\({}_{0}\)) prediction are complexly interrelated and highly nonlinear, this approach has also been applied to ET\({}_{0}\) prediction and has yielded promising results [31; 32]. However, using a single kernel function limits its adaptability to handle complex and diverse data types. To overcome this, multi-kernel functions can be introduced, which, by optimizing the weightings of different kernels, further enhance the model's ability and adaptability to handle data with different features [33; 34; 35; 36]. In the KELM model, computing the kernel matrix becomes very time-consuming when the sample size is large. By applying clustering algorithms to group-related samples and building models for each group, the computational load of the kernel matrix can be significantly reduced, resulting in lower computational costs [32]. Therefore, the main objectives of this study are as follows: (1) To propose an improved QPSO-MKELM model based on the PSO-ELM model, then introduce the K-means-QPSO-MKELM model.