## 3 Data and Methods
### Methods  
#### 3.3.1 Long Short-Term Memory (LSTM)  
After the data collection and preprocessing, the modeling algorithm could be applied to model the data and predict PM2.5 concentrations. In this paper, a correlation filtered spatial-temporal long  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Attribute** & **Value** \\ \hline Data content & PM2.5 concentrations \\ Temporal resolution & 1 h \\ Location & California State, the U.S. \\ Duration & 2016/01-2017/07 \\ Number of stations & 30 \\ Average 1 st quantile & 5.3933 \\ Average Mean & 11.3668 \\ Average Median & 9.3133 \\ Average 3 rd quantile & 14.7833 \\ Average Standard Deviation & 10.0275 \\ Unit & Micrograms/Cubic Meter \\ \hline \hline \end{tabular}
\end{table}
Table 1: A brief summary of the collected PM2.5 data.  
Figure 2: The distribution and code number of the 30 stations.  
short-term memory (CFST-LSTM) model was proposed to accomplish the task. Future air quality of a target station was predicted based on the historical data of itself and surrounding stations.  
The proposed CFST-LSTM network was developed based on the long short-term memory (LSTM) model. LSTM is a special architecture of recurrent neural network (RNN) proposed by [PERSON] and [PERSON] [30]. It overcomes the vanishing and exploding gradient problem of RNN and is capable of learning from long dependencies through a gating mechanism. Due to the strong ability of LSTM in modeling temporal sequential data, it has been reported to have state-of-the-art performance in many domains, such as speech recognition, language modeling, and sequence forecasting [37; 38; 39; 40].  
The network structure of LSTM consists of an input layer, an output layer, and a plurality of hidden layers. The specialty of LSTM lies in the compositions of its hidden layers, which are composed of one or more self-recurrent memory blocks. These blocks allow a value (forward pass) or gradient (backward pass) that flows into the block to be preserved and subsequently retrieved at the required time step [41]. The basic structure of the memory block is shown in Figure 3. It consists of three gates, including the forget gate \(\mathbf{f}_{t}\), the input gate \(\mathbf{i}_{t}\), the output gate \(\mathbf{o}_{t}\), and a recurrent connection cell \(\mathbf{C}_{t}\). \(\mathbf{x}_{t}\) is the input to the current block, \(\mathbf{h}_{t-1}\) is the hidden state of the last block, and \(\mathbf{h}_{t}\) is the state of the current block. At time step \(\mathbf{t}\), the LSTM memory block can be defined with the following set of equations:  
\[\mathbf{f}_{t}= \ \sigma(\mathbf{W}_{t}\cdot[\mathbf{h}_{t-1},\mathbf{x}_{t}]+ \ \mathbf{b}_{t}), \tag{2}\] \[\mathbf{i}_{t}\ = \ \sigma(\mathbf{W}_{i}\cdot[\mathbf{h}_{t-1},\mathbf{x}_{t}]+ \ \mathbf{b}_{i}),\] (3) \[\mathbf{\overline{C}}_{t}\ = \ \mathrm{tanh}(\mathbf{W}_{c}\cdot[\mathbf{h}_{t-1},\mathbf{x}_{t}]+ \ \mathbf{b}_{c}),\] (4) \[\mathbf{C}_{t}=\mathbf{f}_{t}\cdot\mathbf{C}_{t-1}+\mathbf{i}_{t} \mathbf{\overline{C}}_{t},\] (5) \[\mathbf{o}_{t}= \ \sigma(\mathbf{W}_{o}\cdot[\mathbf{h}_{t-1},\mathbf{x}_{t}]+ \ \mathbf{b}_{o}),\] (6) \[\mathbf{h}_{t}= \ \mathbf{o}_{t}\cdot\mathrm{tanh}\,\mathrm{h}(\mathbf{C}_{t}), \tag{7}\]  
where W represents the connection weights between neurons, b represents deflection, and \(\sigma\) denotes the sigmoid activation the gates used.