## 4 Results
### Model Performance at Different Observations  
Since each meteorological station produces observation data containing multiple meteorological elements, and the units of these meteorological elements are not consistent, this section used two dimensionless indicators, the determination coefficient (R2) and the symmetric mean absolute percentage error (SMAPE), to evaluate the imputation ability of different machine-learning methods for meteorological datasets with missing values. The closer the R2 is to 1, the more accurately the algorithm fits the data, and the closer the SMAPE is to 0, the smaller and more precise the prediction error of the algorithm is. Their definitions are shown in:  
\[R^{2}=1-\frac{\sum_{l=1}^{m}(y_{l}-\hat{y_{l}})^{2}}{\sum_{l=1}^{m}(y_{l}- \overline{y})^{2}} \tag{2}\]  
\[SMAPE=\frac{100\%}{m}{\sum_{l=1}^{m}\frac{|\hat{y_{l}}-y_{l}|}{(|\hat{y_{l}}| +|y_{l}|)/2}} \tag{3}\] \begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \hline
**Method** & \multicolumn{1}{c}{**Short Description**} \\ \hline Linear regression & Multivariate linear regression (MLR), ridge regression (Ridge), lasso regression (Lasso), and ElasticNet regression (ENet). These methods are used to establish the relationship between independent and dependent variables by fitting the data through minimizing the sum of squared residuals. The difference between these four methods is that they improve the generalization ability of the model by adding different types of regularization terms. \\ \hline Probability-based & Bayesian ridge regression (BR) and automatic relevance determination regression (ARD) are based on Bayesian linear regression. ARD is a linear regression model that is solved using Bayesian inference in statistics, assuming that the prior distribution of the regression coefficients is an elliptical Gaussian distribution parallel to the coordinate axis, whereas BR assumes that the prior distribution of the regression coefficients is a spherical normal distribution. \\ \hline Instance-based & K-nearest neighbor (KNN). For a given test sample, based on the distance metric, find the K-closest training samples in the training set, and then make predictions based on the information of these K “neighbors”. \\ \hline Tree-based & Decision tree regression (DTR), random forest (RF), adaptive boosting algorithm (AdaBoost), gradient boosting decision tree (GBDT), extremely randomized tree (ERT), bootstrap aggregating (Bagging). DTR is a simple regression algorithm, whereas Bagging, RF, AdaBoost, GBDT, and ERT are ensemble learning algorithms based on decision trees. They improve prediction performance by using different methods to construct and combine decision trees. \\ \hline Kernel-based & Support vector regression (SVR) uses kernel functions to transform data into higher-dimensional space to simulate nonlinearity. \\ \hline Neural network-based & Perceptron, multilayer perceptron neural networks (MLP), recurrent neural networks (RNN), long short-term memory networks (LSTM), bidirectional LSTM networks (BiLSTM), and temporal convolutional networks (TCN) are based on the perceptron, but with differing structures and functions. For example, MLP is a feedforward neural network, and RNN, LSTM, and BiLSTM can all handle sequence data. On the other hand, TCN uses convolution to handle sequence data. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical machine-learning regression method. where \(m\) represents the sample size, \(y_{l}\) represents the true value, \(\widehat{y_{l}}\) represents the model output value, and \(\overline{y}\) represents the mean value of attribute \(y\). Table 2 displays the average results of 20 machine-learning methods used to impute missing values in 124 meteorological observation datasets under different missing value rates within the MMDIF. According to the results recorded in the table, the data missing rate is the primary factor that impacts the accuracy of imputation.