## 4 Conclusions  
In this paper, we propose the TF-BiLSTM-attention model for PM2.5 concentration prediction at 12 sites in Beijing. In order to capture the frequency information in the PM2.5 concentration series, firstly, the DCT is used to transform the time domain series into the frequency domain series. Secondly, the original time domain series is united with the frequency domain series, allowing the model to effectively utilize both the time and frequency domain information. The input features at different times of the time series have different effects on the predicted values, while the basic BiLSTM network assigns equal weight values to all the input features, ignoring the degree of influence of the input features on the predicted values. For this reason, we add the attention mechanism after the BiLSTM network to give higher weights to the factors with higher influence. At the same time, taking advantage of the high spatial dependence of PM2.5 concentration data from different sites, a total of 12 variables, including the PM2.5 concentration at its own site and that of the remaining 11 sites in the study area, are used as input variables for the multivariate model. The results demonstrate superior prediction performance of the model incorporating frequency domain information compared to the model solely utilizing time domain information. Furthermore, the hybrid model augmented with the attention mechanism outperforms the hybrid model without this augmentation. Our proposed TF-BiLSTM-attention model outperforms all the basic and hybrid models in the experiment. The improved model will be able to capture key information and trends in the time series data more accurately. This will help to improve the accurate prediction of pollutant concentrations, meteorological conditions, and other factors, thus improving the overall prediction accuracy. Nevertheless, using DCT and adding the attention mechanism to the base BiLSTM model results in a model with higher complexity. It takes longer time to train a complex model, and the longer computation time may affect the real-time availability of air quality predictions, making it more difficult to update and provide information in a timely manner. Therefore, in future applications, we will strive to find a suitable balance between model complexity and computational efficiency. In addition, this study is limited by the lack of spatial feature information related to PM2.5 to explain the generalization of the model to different regions. For this reason, a large amount of real and valid air quality related data will be collected in the future, and the model proposed in this paper will be applied to several datasets from different regions for validation. **Author Contributions:** Conceptualization, X.T. and Y.P. ; methodology, X.T. and N.W. ; software, X.T. and N.W. ; formal analysis, Y.P. ; data curation, X.T. ; writing--original draft preparation, X.T. and N.W. ; writing--review and editing, X.T. and N.W. ; funding acquisition, Y.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China (No. 62267005), Chinese Guangxi Natural Science Foundation (No. 2023 GXNSFAA026493), Guangxi Collaborative Innovation Center of Multi-source Information Integration and Intelligent Processing; Innovation Project of Guangxi Graduate Education (No. YCSW2023437). **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Publicly available datasets were analyzed in this study. The datasets can be found here: [[https://doi.org/10.24432/C5](https://doi.org/10.24432/C5) RK5G]([https://doi.org/10.24432/C5](https://doi.org/10.24432/C5) RK5G) (accessed on 29 October 2023). **Acknowledgments:** I wish to express my sincere gratitude for the reviewers, for their insightful and valuable comments on the manuscript that helped to improve its quality to a great degree.