# Evapotranspiration Prediction Method Based on K-Means Clustering and QPSO-MKELM Model
## 3 Results and Analysis  
### Model Input Selection  
Table 1 provides the basic statistical summary of meteorological factors in the dataset for the study area. First, the average temperature (T), humidity (Rh), 2 m average wind speed (u), sunshine duration (n), daily precipitation (Dr), and ET\({}_{0}\) were normalized. Then,  
The correlation coefficients of temperature and sunshine duration with ET\({}_{0}\) reached as high as 0.8660 and 0.9298 (Table 3), which are highly correlated and much higher than the other features. Their normalized variances are 0.0454 and 0.0551 (Table 2), respectively, which are also the highest among all features. Therefore, they are initially considered to contain the most information about ET\({}_{0}\) and are the two most important meteorological factors for predicting daily ET\({}_{0}\). The correlation coefficients of humidity, wind speed, and precipitation with ET\({}_{0}\) are 0.5895, 0.5705, and 0.5536 (Table 3), respectively, relatively close. Their normalized variances are 0.0268, 0.015, and 0.0035 (Table 2). The variance of the humidity feature is greater than that of wind speed and much larger than that of precipitation. If the correlation between features is not considered, humidity and wind speed would be the third and fourth most important features. The heatmap of feature correlations shows that the correlation coefficient between humidity and precipitation is as high as 0.8496 (Figure 2), the only feature pair with a correlation coefficient above 0.8, indicating a strong correlation and significant information redundancy. Therefore, precipitation was excluded from the input. It is concluded that humidity and wind speed are the third and fourth most important factors for predicting evapotranspiration in the current region, although their importance is far less than that of temperature and sunshine duration. Evaptranspiration exhibits a uniform variation trend throughout the year [46,47], and this is also true for the region in this study. As shown in Figure 3, evapotranspiration in the study area changes uniformly and shows strong periodicity. Moreover, the day of the year can be converted directly without additional sensor measurements.Therefore, the day of the year was also considered as an input to observe its impact on the machine learning model's prediction accuracy. Since the day of the year is periodic, the last day of the year and the first day of the following year are theoretically closest, yet numerically, they are farthest apart. To address this, trigonometric functions, sin(day) and cos(day), were used to transform the day of the year for input. Based on this, the study proposed and developed eight input models for evaluation (Table 4). \begin{table}
\begin{tabular}{c c} \hline \hline
**Input Model Number** & **Input Features** \\ \hline
1 & T, rh, u, n \\
2 & T, rh, n \\
3 & T, u, n \\
4 & T, n \\
5 & T, rh, u, n day\_sin, day\_cos \\
6 & T, rh, n day\_sin, day\_cos \\
7 & T, u, n day\_sin, day\_cos \\
8 & T, n day\_sin, day\_cos \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Correspondence between input models and features. Figure 3: Monthly average of daily evapotranspiration over 44 years in the study area. From Figure 4 and Table 5, it can be seen that for models with the same number of clusters, the training time trends under different feature inputs are generally similar. The training time for the QPSO-MKELM model under different feature inputs ranged from 18,364 \(s\) to 20,933 \(s\), significantly higher than the training time of other models. The average training times for the Kmeans-QPSO-MKELM-10, Kmeans-QPSO-MKELM-20, Kmeans-QPSO-MKELM-30, and Kmeans-QPSO-MKELM-40 models were 10%, 4.19%, 2.39%, and 2.29% of the QPSO-MKELM model's training time, respectively. As the number of clusters increases, the training time first decreases rapidly and then decreases more slowly.