## 5 Conclusions  
This paper proposed a correlation filtered spatial-temporal long short-term memory (CFST-LSTM) model for PM2.5 concentrations prediction. For a target station, not only the historical data of itself, but also the data of its surrounding stations are used. A spatial-temporal filter (STF) layer was designed  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Models** & **CFST-LSTM** & **O-LSTM** & **F-LSTM** \\ \hline Average R Square & 0.9155 & 0.8899 & 0.8613 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Average R Square value of different LSTM models on all sites.  
Figure 10: The spatial interpolation of \(R_{\Delta}^{2}\) in California State.  
to automatically remove the station data with low correlation with the target station. Hourly PM2.5 concentrations data of 30 stations in California were collected to validate the model effectiveness. Prediction performance of the CFST-LSTM model was compared with other traditional machine learning models and commonly seen neural networks. Results show that:  
* The proposed CFST-LSTM model outperforms other commonly seen machine learning/deep learning models with a better fitting degree and higher prediction accuracy. Its R\({}^{2}\) can reach 0.9583.
* Compared with ordinary LSTM, our method not only considers the influence from nearby stations but also filters out less related time series inputs, and this helps increase 2.88% R\({}^{2}\) performance in our tests on 30 sites. On the other hand, if only simply adding the time series inputs from other stations, the model performance will drop 3.32% due to a higher level of noise.
* According to the experiment on the R\({}^{2}\) improvements over the sites in California, the proposed method exhibited a higher improvement over ordinary LSTM in areas with denser sites, but lower improvement in sparser districts. This reflects that our method performs better at places with denser spatial inputs.
* Parameter optimization of the newly designed STF layer is quite important to the proposed method. The experiment showed that the difference in R\({}^{2}\) between proper and improper parameters can reach around 5.39% of the overall performance.  
The main contribution of this study is that we proposed an improved neural network model for spatial-temporal time series predictions. The model is modified based on deep learning techniques. Besides the prediction of PM2.5 concentrations, the model is expected to be applicable to other types of spatial-temporal time series problems, such as the prediction of weather, wind power, and other types of air pollutants. Of course, further studies need to be conducted for verifications.  
Due to the data availability, only the historical PM2.5 concentration data are collected and tested in this paper. Other possible influential factors of PM2.5, such as meteorological characteristics and traffic emissions, are not considered. Further studies could be conducted to explore the feasibility of implementing the proposed method on multivariate inputs.  
Conceptualization, Y.D. and C.Z.; Data curation, Y.D. and J.M.; Formal analysis, Y.D.; Investigation, C.Z.; Methodology, Y.D., Z.L. and J.M.; Project administration, C.Z.; Software, Y.D., Z.L., C.Z. and J.M.; Supervision, C.Z.; Validation, Y.D. and C.Z.; Visualization, Y.D. and J.M.; Writing--original draft, Y.D. and Z.L. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.