## 4 Results and Discussion
### Implementation of AQI Forecasting Model  
This section describes a simulation-like AQI forecasting using stacking ensemble and AdaBoost (the two best methods from the analyses in Section 4.2) as backend techniques. Each prediction is accompanied by a prediction interval (PI) within a 95% confidence level, which describes a given tolerance for the prediction value such that there is 95% chance that the actual observation could fall within this range. The prediction interval is calculated using the formula below [21]:  
\[PI=z_{\alpha/2}\times\sigma(3)\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Without Imputation**} & \multicolumn{3}{c}{**With Imputation**} \\ \cline{2-7}  & **RMSE** & **MAE** & \(R^{2}\) & **RMSE** & **MAE** & \(R^{2}\) \\ \hline SVM-Polynomial & 7.072 & 5.556 & 0.974 & 7.275 & 5.821 & 0.973 \\ SVM-RBF & 9.119 & 5.542 & 0.957 & 8.324 & 4.702 & 0.964 \\ SVM-Linear & 9.529 & 7.400 & 0.953 & 8.485 & 6.621 & 0.963 \\ Random Forest & 2.971 & 1.869 & 0.995 & 2.979 & 1.868 & 0.995 \\ AdaBoost-Square & 3.020 & 1.771 & 0.995 & 2.996 & 1.766 & 0.995 \\ AdaBoost-Linear & 2.995 & 1.767 & 0.995 & 2.983 & 1.760 & 0.995 \\ AdaBoost-Exponential & 3.020 & 1.771 & 0.995 & 2.996 & 1.766 & 0.995 \\ ANN & 3.966 & 2.544 & 0.992 & 3.821 & 2.585 & 0.992 \\ Stacking Ensemble & 2.925 & 1.823 & 0.996 & 2.921 & 1.814 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of ML Algorithms for Fengshan F1-AQI Prediction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Without Imputation**} & \multicolumn{3}{c}{**With Imputation**} \\ \cline{2-7}  & **RMSE** & **MAE** & \(R^{2}\) & **RMSE** & **MAE** & \(R^{2}\) \\ \hline SVM-Polynomial & 25.658 & 19.806 & 0.659 & 25.512 & 19.606 & 0.663 \\ SVM-RBF & 25.810 & 20.392 & 0.655 & 25.665 & 20.248 & 0.659 \\ SVM-Linear & 36.292 & 26.859 & 0.318 & 29.598 & 23.028 & 0.546 \\ Random Forest & 16.634 & 12.111 & 0.857 & 16.606 & 12.100 & 0.857 \\ AdaBoost-Square & 16.440 & 11.399 & 0.860 & 16.498 & 11.391 & 0.859 \\ AdaBoost-Linear & 16.367 & 11.364 & 0.861 & 16.339 & 11.367 & 0.862 \\ AdaBoost-Exponential & 16.387 & 11.373 & 0.861 & 16.398 & 11.367 & 0.861 \\ ANN & 19.112 & 14.285 & 0.811 & 18.975 & 14.182 & 0.814 \\ Stacking Ensemble & 16.302 & 11.517 & 0.862 & 16.279 & 11.527 & 0.863 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Results of ML Algorithms for Fengshan F8-AQI Prediction. where \(\sigma\) represents the standard deviation of the residual errors defined as [22]:  
\[\sigma=\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}{(y_{i}-\hat{y}_{i})^{2}}} \tag{4}\]  
Prediction intervals that reflect the uncertainty of a model's output should be adjusted dynamically as new observations are received every hour, thus ensuring that the prediction interval is always current. The one-month samples (December 2018) from the Zhongli dataset are used to obtain the standard deviation. As shown in Figure 8a, the higher the prediction time step, the wider the tolerance needed to represent the estimation. AdaBoost and stacking ensemble outperform the other techniques tested in the previous section, obtaining similar predictions and prediction intervals. The predictions here are all based on authentic data, where the best models in each prediction category are reused. Figure 8b shows another forecast constructed during winter, providing an example of poor air quality cases captured in the prediction of F1-AQI, F8-AQI, and F24-AQI using AdaBoost and stacking ensemble. Figure 9 provides an illustration on how the information will be provided and visualized given a sample of upcoming data for the monitoring and forecasting of the air quality. Noted that as shown by the graph, the higher the time step of prediction the wider the tolerance needed to escort the estimation. AdaBoost and stacking are two methods that outperform other techniques tested in the previous section. Their predictions are close to each other and so are the prediction intervals.