## 4 Results
### Model Performance at Different Observations
This may be related to its special structure, which can capture the time series characteristics in the data, and this may enable it to cope with missing data [44]. The results of the six linear regression methods (MLR, Ridge, Lasso, Enet, BR, and ARD) are very close on R\({}^{2}\) and SMAPE, and there is no obvious difference in their performance. They have a weak ability to impute missing data, because these algorithms are all based on linear assumption models, whereas GMOD have nonlinear or complex patterns, which causes the models to not fit or predict the data well. We noticed that ERT's R\({}^{2}\) was the lowest at all missing rates, suggesting that ERT is not suitable for imputing missing values in GMOD. This may lead to a decrease in the correlation and distribution between the imputed data and the original data. As the missing rate increases, the prediction accuracy of all algorithms decreases, but the degree of decrease varies. This paper used Equations (4) and (5) to calculate the decrease in the imputation accuracy of each algorithm when the missing rate rose from 10% to 80%, where \(R^{2}_{10\%}\) and \(R^{2}_{80\%}\), \(SMAPE_{10\%}\) and \(SMAPE_{80\%}\) represent the R\({}^{2}\) and SMAPE obtained using one of the methods when the missing rate is 10% and 80%, respectively. The results are shown in Figure 4.  
\[R^{2}\;decline\;range=\left(R^{2}_{10\%}-R^{2}_{80\%}\right)/R^{2}_{10\%} \tag{4}\]  
\[SMAPE\;decline\;range=\left(SMAPE_{80\%}-SMAPE_{10\%}\right)/SMAPE_{10\%} \tag{5}\]  
In general, there was a certain positive correlation between the decrease in R\({}^{2}\) and the decrease in SMAPE, that is, the larger the decrease in R\({}^{2}\), the larger the decrease in SMAPE. However, Figure 4 shows that neural network-based algorithms (such as Perceptron, BiLSTM, MLP, LSTM, RNN, etc.) have smaller R\({}^{2}\) decreases and larger SMAPE decreases than other types of algorithms, such as linear regression, probability-based algorithms, and tree-based methods. Such a difference suggests that neural network algorithms perform better on the R\({}^{2}\) metric but worse on the SMAPE metric. A possible explanation for this could be that neural network algorithms can capture the complex features and nonlinear relationships in the data more effectively [45], which improves the R\({}^{2}\) value, but they also tend to overfit the noise or outliers in the data, which lowers the SMAPE value. Figure 3: Distribution of the imputation results of RF. As shown previously, RF is less accurate than TCN, except for at 60% and 80% missing rates, but more accurate at 10%, 20% and 40%. However, Figure 4 reveals that the R\({}^{2}\) drops by 19.9%, and the SMAPE rises by 86.2% for RF, whereas TCN only sees a 11.1% R\({}^{2}\) decline and a 36.2% SMAPE increase as the missing rate grows from 10% to 80%. This means TCN is more robust for missing rate changes, which is crucial in practice, as we cannot control or predict a missing rate. A sensitive algorithm would require frequent parameter tuning or replacement, increasing our work and cost. An insensitive algorithm can reliably handle any missing data without affecting the imputation quality. In addition to RF, five other tree-based methods (DTR, AdaBoost, GBDT, ERT, Bagging) saw the R\({}^{2}\) drop by 21.4%-39.2% and the SMAPE rise by 58.4%-93.2%, with ERT exhibiting the worst performance for both metrics, showing its low stability and accuracy for missing data imputation. The kernel-based (SVR) and instance-based (KNN) methods have similar imputation accuracy declines as the missing rate increases. Their R\({}^{2}\) falls by around 24% and their SMAPE climbs by around 70% when the missing rate goes from 10% to 80%. They interpolated well with increasingly low missing data but performed poorly on increasingly high missing data.