## 4 Results
### Analysis of Short-Term Prediction Results  
LSTM, GRU, CNN-LSTM, and DAQFF (The model is proposed in the Deep Air Quality Forecasting Using Hybrid Deep Learning Framework text) are used as excellent models for processing time series data, and we use them as comparison models with the same model parameters set as FPHFA. Table 2 presents the short-term prediction quantitative results of the short-term prediction of PM\({}_{2.5}\) from the Beijing dataset, which gives a comparison of LSTM, GRU, CNN-LSTM, DAQFF, and the FPHFA model in terms of RMSE, MAE, R\({}^{2}\), and IA. From Table 2, we can see that the FPHFA model performs better than other deep learning models in the task of short-term PM\({}_{2.5}\) concentration prediction for the Beijing dataset. In the Beijing dataset, FPHFA improves R\({}^{2}\) to 0.877, IA to 97.04%, and MAPE to 0.561 while reducing RMSE to 28.15 and MAE to 19.19 compared to the other comparison models, which represents an obvious improvement in the accuracy of the prediction. Additionally, the classic deep learning models' model performance evaluation indicators are comparable but inferior to those of the hybrid deep learning models. The implication is that hybrid deep learning models are  
Figure 5: Spatiotemporal distribution features of PM\({}_{2.5}\) concentration. superior to traditional deep learning models for short-term PM\({}_{2.5}\) concentration prediction. Moreover, our model performs the best among hybrid deep learning models. Compared with the DAQFF model, FPHFA improved R\({}^{2}\) by 0.018 and IA by 0.57% while reducing RMSE by 1.97 and MAE by 1.83. This is because FPHFA can learn local trend features through the unique multi-channel 1D CNNs, and long-term dependence of PM\({}_{2.5}\) concentration can be obtained by Bi LSTM. Moreover, the additional most important attention mechanism effectively focuses on information that is more significant for prediction at particular moments, thus improving the final prediction results. In order to clearly describe the prediction results of the model, examples of the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are plotted in Figure 6. We use the predicted results of the model to make an example diagram. Compared with other models, the prediction results of our model also have a significant correlation. This shows that our model is superior to other models. In addition, the selection of window size (representing the input size of the model's historical observations) also has an impact on short-term prediction performance. We investigate the influence of the window size for the deep learning model in the Beijing dataset. As shown in Figure 7, when the window size is larger than 32, the FPHFA model outperforms the other models in prediction performance. This is due to the fact that when the window is too tiny, the historical data available for learning is insufficient, and the capacity to forecast is hampered by data from nearby sites, leading to inaccurate PM\({}_{2.5}\) concentration predictions by the FPHFA model for the target station. The FPHFA's performance evaluation indicators optimize as the window size increases, and the RMSE of FPHFA reaches its minimum value (and IA its maximum) when the window size is about 56. Beyond this point, the model performance evaluation indicators remain constant or slightly increase, which may be a sign that the prediction models are overfitting. Next, we study the influence of the number of epochs on prediction performance by the different models. Figure 8 shows the model performance evaluation indicators (RMSE and IA) curves of the FPHFA relative to different epochs and provides comparisons with other prediction models. It is evident that FPHFA consistently outperforms the other deep learning models at almost any number of epochs. Moreover, the RMSE of FPHFA reaches its minimum (and the IA its maximum) when the number of epochs is about 150. This is followed by progressively unstable model performance as the epoch size keeps expanding.