## 3 Data and Methods
### Methods
#### 3.3.2 CFST-LSTM  
However, when predicting air pollutant concentrations, the ordinary LSTM model neither takes into account the spatial correlation among the monitoring stations nor considers the strength of the spatial-temporal correlations. Consequently, its performance is limited when the air pollution in one area is largely influenced by other areas. To overcome the problem, this study proposed a correlation filtered spatial-temporal long short-term memory (CFST-LSTM) model, which can analyze the lagged  
Figure 3: The basic structure of the long short-term memory (LSTM) memory block. spatial-temporal correlation of the input data and select the time series data that meet the pre-set threshold. The architecture of the CFST-LSTM model is presented in Figure 4. It consists of five layers, including input layer, STF layer, LSTM layer, fully connected (FC) layer, and output layer. Working process of the CFST-LSTM model is shown as follows. Firstly, given a dataset X, STF layer calculates the correlation coefficient matrix R for the time series of the target stations and the lagged time series of all the other stations (including the target station itself). The formulation of R is shown in Equation (8). \[\text{R}=\text{Corr}\big{(}\text{S}_{\text{target\_site}},\text{S}_{\text{ other\_sites}}\big{)} \tag{8}\]  
where \(\text{Corr}(\cdot)\) denotes the function for calculating the correlation coefficient matrix, \(\text{S}_{\text{target\_site}}\) represents the time series of the target station, \(\text{S}_{\text{other\_sites}}\) represents the lagged time series matrix of other sites, which can be formulated as Equation (9). \[\text{S}_{\text{other\_sites}}=\left[ \begin{array}{ccc}\text{S}_{\text{site \_1}}^{\text{t-1}}&\dots&\text{S}_{\text{site\_1}}^{\text{t-r}}\\ \dots&\dots&\dots\\ \text{S}_{\text{site\_n}}^{\text{t-1}}&\dots&\text{S}_{\text{site\_n}}^{\text {t-r}}\end{array} \right] \tag{9}\]  
where \(n\) represents the number of sites, \(r\) represents the largest time lag, \(t\) represents the length of time series, each \(\text{S}_{\text{site\_i}}^{\text{t-j}}\) represents the lagged j-moment time series of the number \(i\)th station for the target station, \(1\leq\text{i}\leq\text{n},\ 1\leq\text{j}\leq\text{r}\). For \(\text{Corr}(\cdot)\), each \(\text{R}_{\text{(ij)}}=\rho\big{(}\text{S}_{\text{target\_site}},\text{S}_{ \text{site\_i}}^{\text{t-j}}\big{)}\). \(\rho(\cdot)\) is the Pearson correlation function. Its calculation is shown in Equation (10). \[\rho(\text{X},\text{Y})=\frac{\text{Cov}(\text{X},\text{Y})}{\sigma_{\text{X}} \sigma_{\text{Y}}} \tag{10}\]  
Figure 4: The architecture of the correlation filtered spatial-temporal long short-term memory (CFST-LSTM) model. where X represents the original input of the model, Y represents the output, \(\text{Cov}(\text{X},\text{Y})\) represents the covariance of X and Y, \(\sigma_{\text{X}}\) and \(\sigma_{\text{Y}}\) represent the standard deviation of X and Y, respectively, \(\rho\in(-1,1)\). The formulation of X can be expressed as Equation (11). \[\text{X}=\left[ \begin{array}{ccc}\text{site\_1^{t-1}}&\ldots&\text{site\_1^ {t-r}}\\ \ldots&\ldots&\ldots\\ \text{site\_n^{t-1}}&\ldots&\text{site\_n^{t-r}}\end{array} \right] \tag{11}\]  
where site\(\_\text{i}^{t-j}\) represents the recorded value of the \(i\)th station at time t-j. Secondly, transform the R based on the correlation threshold \(\rho_{\text{th}}\), as shown in Equation (12). \[\text{R}^{\text{I}}=\text{I}(\text{R},\rho_{\text{th}}) \tag{12}\]  
where \(\text{I}(\cdot)\) represents the indicator function. When \(\text{R}_{(\text{i},\text{j})}\geq\rho_{\text{th}}\), \(\text{R}_{(\text{i},\text{j})}^{\text{I}}=1\). When \(\text{R}_{(\text{i},\text{j})}<\rho_{\text{th}}\), \(\text{R}_{(\text{i},\text{j})}^{\text{I}}=0\). Thirdly, apply the element-wise product to X and R\({}^{\text{I}}\). Then, the final output X\({}^{\prime}\) of STF layer is given, as presented in Equation (13).