## 4 Results
### Analysis of Short-Term Prediction Results
It is clear that as the number of epochs increases beyond 150, the generalization capacity does not. In addition, the optimization of all models seems to be a bit slow, and overfitting seems likely when the number of epochs exceeds 150. The more epochs there are, the more computational resources are used. On the other hand, as epochs increase, the training performance of the model may improve, but this can also lead to overfitting problems. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & **RMSE** & **MAE** & **R\({}^{2}\)** & **IA** & **MAPE** \\ \hline LSTM & 34.39 & 23.03 & 0.796 & 95.30\% & 0.707 \\ GRU & 32.62 & 22.25 & 0.824 & 95.93\% & 0.683 \\ CNN-LSTM & 31.69 & 21.81 & 0.832 & 96.15\% & 0.672 \\ DAQFF & 30.12 & 21.02 & 0.849 & 96.47\% & 0.669 \\ FPHFA & 28.15 & 19.19 & 0.877 & 97.04\% & 0.561 \\ \hline \hline \end{tabular} Note: window size = 24, epochs = 100, and average of model performance evaluation indicators (RMSE, MAE, R\({}^{2}\), and IA) over the next 1-12 h.  
\end{table} 
Table 2: The model performance evaluation indicators of models in short-term PM\({}_{2.5}\) concentration prediction. Figure 6: ACF and Partial Autocorrelation Function (PACF) plots for Actual Value (first row), ACF and PACF plots obtained with LSTM (second row), GRU (third row), CNN-LSTM (fourth row), DAQFF (fifth row), FPHFA (sixth row). To further investigate the short-term prediction performance of models, we investigate the PM\({}_{2.5}\) concentration forecasting ability of FPHFA and other deep learning models throughout the course of a month (744 observations in total). Figure 9 shows the comparison of the actual PM\({}_{2.5}\) value with the value predicted 12 h ahead by the models LSTM, GRU, CNN-LSTM, DAQFF, and the proposed FPHFA model in the experiment with the Beijing dataset. As shown in Figure 9, compared to other deep learning models, the CNN-LSTM model has a lower match between the actual value and forecasted value. Beijing had the highest PM\({}_{2.5}\) concentration in December, and the CNN-LSTM model may not be sensitive to such high values of PM\({}_{2.5}\) concentration. It is obvious that the FPHFA outperforms LSTM, GRU, CNN-LSTM, and DAQFF in the task of 12-h forward prediction, especially as regards the time periods between the peaks and valleys of PM\({}_{2.5}\) data. In addition, as shown in Figure 9, the prediction results of the FPHFA are highly similar to the observed results, while it also has a good fit at the points of sudden change in PM\({}_{2.5}\) concentration. In summary, compared to the short-term PM\({}_{2.5}\) concentration prediction under different experiment conditions, the hybrid deep learning models' forecasting performance is generally not poor, and FPHFA continues to have the greatest performance. Given how easy it is to anticipate time series in the near run, high prediction performance can frequently be obtained by simply following the trend of the preceding hours. Figure 8: Window size is 56, prediction length is 12 h, and batch size is 128. Effect of number of epochs on RMSE and IA of models in the Beijing dataset. (**a**) RMSE, (**b**) IA. Figure 7: Prediction step is 12 h, batch size is 128, and number of epochs is 40. Effect of window size on RMSE and IA of the models in the Beijing dataset. (**a**) RMSE, (**b**) IA.