## 3 Methods
### Deep Learning Model  
#### 3.2.1 LSTM  
LSTM networks are a specialized type of Recurrent Neural Network (RNN) introduced by [PERSON] et al. in 1997 [PERSON] et al. (1997). They are designed to address the issues of gradient vanishing and gradient explosion that traditional RNNs encounter when processing long sequences. By incorporating memory cells and gating mechanisms, LSTMs can effectively capture and retain long-term dependencies within sequences, making them widely used in time series prediction, natural language processing, and other areas.  
The basic unit of an LSTM is called the LSTM cell, which consists of a cell state and three gating mechanisms: the input gate, the forget gate, and the output gate. The interplay of these components allows the LSTM to selectively remember or forget information, addressing the shortcomings of traditional RNNs. Figure 5 illustrates the structure of the LSTM cell.  
The forget gate determines which information in the cell state should be forgotten, with the following calculation formula:  
\[f_{t}=\sigma\Big{(}W_{f}\cdot[h_{t-1}]+b_{f}\Big{)} \tag{15}\]The forget gate processes the input \(x_{t}\) and the previous time step's hidden state \(h_{t-1}\) through a sigmoid function (\(\sigma\)), generating a value between 0 and 1 to control which information should be retained. \(f_{t}\) is the output of the forget gate. \(W_{f}\) is the weight matrix of the forget gate. \(x_{t}\) is the input at the current time step. The input gate controls the impact of the current input information on the cell state. The calculation formula is as follows:  
\[i_{t}=\sigma(W_{i}[h_{t-1},x_{t}]+b_{i}) \tag{16}\]  
\[\widetilde{C}_{t}=\tanh\left(W_{c}[h_{t-1},x_{t}]+b_{C}\right) \tag{17}\]  
The input gate uses both a sigmoid function and a tanh function to determine the importance of the new information and the extent to which it should be updated. \(i_{t}\) is the output of the input gate. The cell state is updated based on the outputs of the input gate and the forget gate. The calculation formula is as follows:  
\[C_{t}=f_{t}\cdot C_{t-1}+i_{t}\cdot\widetilde{C}_{t} \tag{18}\]  
The cell state is updated by combining the output of the forget gate \(f_{t}\) and the output of the input gate \(i_{t}\). The new cell state \(C_{t}\) contains the forgotten old information and the new input information. Here, \(C_{t}\) is the cell state at the current time step, \(C_{t-1}\) is the cell state at the previous time step, and \(\widetilde{C}_{t}\) is the candidate cell state at the current time step. The output gate controls the influence of the cell state on the output at the current time step. The calculation formula is as follows:  
\[o_{t}=\sigma(W_{o}\cdot[h_{t-1},x_{t}]+b_{o}) \tag{19}\]  
The output gate processes \(x_{t}\) and the cell state \(C_{t}\) through a sigmoid function to generate the output at the current time step. The hidden state is updated based on the cell state and the output of the output gate, and the calculation formula is as follows:  
\[h_{t}=o_{t}\cdot\tanh\left(C_{t}\right) \tag{20}\]  
The hidden state \(h_{t}\), which combines the cell state and the output of the output gate, serves as the final output of the LSTM unit and is passed to the next time step. \(o_{t}\) is the output of the output gate.