## 4 Results
### Analysis of Long-Term Prediction Result
In the spring and summer forecast, the forecast error is low, but the consistency between the predicted data and the real data is not high. In autumn and winter, larger forecast errors correspond to a higher agreement. Considering the seasonal characteristics of PM2.5 concentration, we found that the prediction error may be related to PM2.5 concentration and dispersion in different seasons. In summer, PM2.5 concentration is usually low, and the prediction error is small, but the consistency between the predicted value and the real value is not high. In winter, the concentration of PM2.5 is high, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Models** & **RMSE** & **MAE** & **R2** & **IA** & **MAPE** \\ \hline LSTM & 33.95 & 24.00 & 0.804 & 95.53\% & 0.831 \\ GRU & 33.01 & 23.33 & 0.814 & 95.80\% & 0.829 \\ CNN-LSTM & 31.32 & 22.14 & 0.830 & 96.17\% & 0.746 \\ DAQFF & 29.15 & 20.57 & 0.864 & 96.83\% & 0.691 \\ FPHFA & 22.12 & 15.27 & 0.932 & 98.30\% & 0.438 \\ \hline \hline \end{tabular} Note: window size = 48, epochs = 100, and average of model performance evaluation indicators (RMSE, MAE, R2, and IA) over the next 13-24 h.  
\end{table} 
Table 3: The model performance evaluation indicators of FPHFA in long-term PM2.5 concentration prediction in comparison to other comparison models. dispersion is large, and the uncertainty of the variation of pollutant concentration is also high, which leads to the difficulty of pollutant prediction. High PM\({}_{2.5}\) concentration limits the prediction ability of the deep learning model, leading to the largest prediction error in winter among all seasons [41]. This is also the reason why the PM\({}_{2.5}\) concentration values in December 2016 and January 2017 were selected as the prediction criteria in this study. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Models**} & \multicolumn{4}{c}{**Spring**} & \multicolumn{4}{c}{**Summer**} \\ \cline{2-9}  & **RMSE** & **MAE** & **R\({}^{2}\)** & **IA** & **RMSE** & **MAE** & **R\({}^{2}\)** & **IA** \\ \hline LSTM & 24.58 & 17.53 & 0.856 & 96.63\% & 21.78 & 16.19 & 0.668 & 92.72\% \\ GRU & 22.45 & 15.76 & 0.873 & 97.12\% & 19.87 & 14.79 & 0.716 & 93.90\% \\ CNN-LSTM & 21.04 & 14.34 & 0.896 & 97.55\% & 17.84 & 13.23 & 0.791 & 95.32\% \\ DAQFF & 20.20 & 14.10 & 0.905 & 97.76\% & 19.05 & 14.44 & 0.771 & 94.73\% \\ FPHFA & 15.87 & 10.80 & 0.949 & 98.71\% & 13.18 & 9.78 & 0.917 & 97.84\% \\ \hline \hline \end{tabular} Note: window size = 56, epochs = 100, and average of model performance evaluation indicators (RMSE, MAE, R\({}^{2}\), and IA) over the next 24 h.  
\end{table} 
Table 4: The model performance evaluation indicators of models in spring and summer. Figure 10: Window size is 56, number of epochs is 150, and batch size is 128. RMSE, MAE, R\({}^{2}\), and IA of models at different prediction sizes in the Beijing dataset. **(a)** RMSE. **(b)** MAE. **(c)** R\({}^{2}\). **(d)** IA. To evaluate the long-term prediction performance of FPHFA and other deep learning models on the Beijing dataset, we investigate the PM\({}_{2.5}\) concentration forecasting capability of FPHFA and other deep learning models under different prediction sizes (24 h and 96 h) throughout the course of a month (744 observations in total). Comparing the actual and predicted PM\({}_{2.5}\) values for several models (LSTM, GRU, CNN-LSTM, DAQFF, and FPHFA) at various time steps is shown in Figure 11 (24 h and 96 h). Figure 11 shows that the long-term predictive performance of FPHFA is superior to both classical deep learning models and other hybrid deep learning models at different time steps, especially with regard to the peak and valley periods of the test data. In addition, for prediction tasks, including sudden changes in pollutant concentration, FPHFA outperforms comparative models. Moreover, the FPHFA model consistently has the greatest prediction performance for long-term PM\({}_{2.5}\) concentration predictions at any time step.