## 3 Data Description and Methods
### The Baseline Methodologies
#### 3.2.3 Cascade LSTM  
Cascade LSTM is an extended version of the basic LSTM structure, consisting of multiple stacked layers of LSTM cells. It combines the expressive power of deep neural networks with the inherent sequence processing capabilities of LSTM. While traditional single-layer LSTMs utilize cell states and gates to maintain important past information in long sequences, they are limited by a simple path where inputs and outputs are connected only once. In contrast, Cascade LSTM connects multiple layers of LSTM in series, allowing higher layers to further process and interpret the time-series patterns extracted by lower layers. This enables the model to learn more complex and abstract sequential structures. Each layer in a Cascade LSTM performs the same computational processes as a basic LSTM cell. At time step \(t\), let the input to layer \(l\) be \(\mathbf{x}_{t}^{(l)}\), and the previous hidden state and cell state be \(\mathbf{h}_{t-1}^{(l)}\) and \(\mathbf{C}_{t-1}^{(l)}\), respectively. First, the forget gate (forget gate) is calculated as follows:  
\[f_{t}^{(l)}=\sigma\Big{(}W_{f}^{(l)}[\mathbf{h}_{t-1}^{(l)},\mathbf{x}_{t}^{( l)}]+b_{f}^{(l)}\Big{)}. \tag{9}\]  
The forget gate determines which parts of the previously accumulated cell state should be erased to prevent excessive accumulation of information. Next, the input gate decides how much new input to accept, and the potential cell state candidate \(\hat{\mathbf{C}}_{t}^{(l)}\) is generated through the hyperbolic tangent function:  
Figure 2: The diagram of LSTM. \[rCl_{t}^{(l)} = \sigma\Big{(}W_{t}^{(l)}[\mathbf{h}_{t-1}^{(l)},\mathbf{x}_{t}^{(l) }]+b_{t}^{(l)}\Big{)}, \tag{10}\] \[\bar{\mathbf{C}}_{t}^{(l)} = \tanh\Big{(}W_{C}^{(l)}[\mathbf{h}_{t-1}^{(l)},\mathbf{x}_{t}^{(l) }]+b_{C}^{(l)}\Big{)}. \tag{11}\]  
Combining these, the new cell state \(\mathbf{C}_{t}^{(l)}\) is determined by summing the retained part of the previous cell state (\(f_{t}^{(l)}\odot\mathbf{C}_{t-1}^{(l)}\)) and the newly incoming part (\(i_{t}^{(l)}\odot\bar{\mathbf{C}}_{t}^{(l)}\)):  
\[\mathbf{C}_{t}^{(l)}=f_{t}^{(l)}\odot\mathbf{C}_{t-1}^{(l)}\ +\ i_{t}^{(l)}\odot\bar{\mathbf{C}}_{t}^{(l)}. \tag{12}\]  
Finally, the output gate (output gate) selects which information from the updated cell state should be exposed to the hidden state:  
\[rCl_{t}^{(l)} = \sigma\Big{(}W_{o}^{(l)}[\mathbf{h}_{t-1}^{(l)},\mathbf{x}_{t}^{ (l)}]+b_{o}^{(l)}\Big{)}, \tag{13}\] \[\mathbf{h}_{t}^{(l)} = o_{t}^{(l)}\odot\tanh\big{(}\mathbf{C}_{t}^{(l)}\big{)}. \tag{14}\]  
The resulting hidden state \(\mathbf{h}_{t}^{(l)}\) is immediately passed to the next time step \((t+1)\) within the same layer and, if the current layer is the \(l\)-th layer, it serves as the input \(\mathbf{x}_{t}^{(l+1)}\) for the upper layer. Therefore, in Cascade LSTM, as the layers deepen, they produce richer time-series representations and contextual information, which demonstrates excellent performance on data with high complexity. Figure 3 visualizes the overall structure of the Cascade LSTM. The input \(x_{t}\) at time step \(t\) passes through multiple layers of LSTM cells in sequence to produce the final output \(y_{t}\). The greatest advantage of the Cascade LSTM is that it provides a much more flexible and powerful representational capacity compared to a simple LSTM. This offers a significant benefit in tasks where understanding long-term context is crucial. By stacking layers in depth so that each layer further processes features extracted from the previous layer, the Cascade LSTM can learn richer and more complex patterns than a simple LSTM. However, as the number of layers increases, the number of model parameters can grow exponentially. Moreover, the final performance can vary greatly depending on hyperparameters such as the model architecture, learning rate, and batch size. Therefore, careful tuning is required to match the given problem and data characteristics.