## 3 Methodology  
### Convolutional Long Short-Term Memory (ConvLSTM)  
Originally introduced in 2015 by [PERSON] et al. [18], ConvLSTM is a neural network architecture that ingeniously combines convolutional neural networks (CNN) and long short-term memory networks (LSTM). This special blend of architectures allows ConvLSTM to leverage the benefaction of convolution operations in data feature extraction and characterize the recurrent information propagation inherent in LSTM. In comparison with conventional LSTM networks (the structure of the ConvLSTM cell is shown in Figure 3), ConvLSTM initially employs convolution operations to extract data features before training to better predict numerical trends. ConvLSTM, fundamentally a recurrent neural network, uses the output of the previous recurrent unit as the input to the subsequent one and hosts three LSTM gating units. With the inclusion of convolution operations, ConvLSTM exhibits the ability to capture spatial variations during object motion. Consequently, ConvLSTM has found extensive applications across various spheres like video prediction [19], image compression [20], and other general algorithmic tasks [18]. The internal structure of ConvLSTM comprises a fully connected layer where the input, cell output, and state vectors are all one-dimensional (1D). The entire input data, cell outputs, hidden layers, and all three gating units of ConvLSTM, however, maintain a representation in 3D tensors. The state of a unit at a specific moment is determined by the convolution operator in the ConvLSTM network, with the convolution kernel size outlining the speed of the detected action. Contrasting with traditional fully connected LSTM, ConvLSTM effectively handles the tokenization of pure numerical data. ConvLSTM is better suited for handling the tokenization of raw numerical data. Through the convolutional operation between data and convolution kernels, kernel parameters are shared across input \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Series**} & **ADF Test** & \multicolumn{4}{c}{**Critical Value**} & \multirow{2}{*}{**\(p\)-Value**} & **Stationarity** \\ \cline{3-3} \cline{5-6}  & **Statistic** & & **1\%** & **5\%** & **10\%** & **(5\%)** \\ \hline
**CLOSE** & \(-4.609\) & & & & 0.0001 & \\
**OPEN** & \(-4.825\) & & & & 0.0000 & \\
**HIGH** & \(-4.785\) & & & & 0.0001 & \\
**LOW** & \(-4.607\) & & & & 0.0000 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ADF test on the fitted spread series. Figure 2: Time series plot of the closing prices difference. data, enhancing the extraction of more generalized features. Equation (3) provides the mathematical representation of a single ConvLSTM network unit:  
\[\left\{ \begin{array}{l}i_{t}=\sigma(W_{xi}\cdot X_{t}+W_{hi}\cdot H_{t-1}+W_{ci} \odot C_{t-1}+b_{i})\\ f_{t}=\sigma(W_{xf}\cdot X_{t}+W_{hf}\cdot H_{t-1}+W_{cf}\odot C_{t-1}+b_{f})\\ C_{t}=f_{t}\odot C_{t-1}+i_{t}\odot\tan\ln(W_{xc}\cdot X_{t}+W_{hc}\cdot H_{t-1 }+b_{c})\\ o_{t}=\sigma(W_{xo}\cdot X_{t}+W_{ho}\cdot H_{t-1}+W_{co}\odot C_{t}+b_{o})\\ H_{t}=o_{t}\odot\tan\ln(C_{t})\end{array} \right. \tag{3}\]  
where \(X_{t}\) represents the input data and hidden state at sequential time \(t\), while \(f_{t}\), \(i_{t}\) and \(o_{t}\) denote the forget gate, input gate, and output gate, respectively. The function is a sigmoid function defined as \(\sigma(x)=1/(1+e^{-x})\). \(\odot\) signifies an element-wise dot product. W and b represent weight matrices and bias vectors, respectively. The forget gate layer determines the extent to which previous information, gathered from \(H_{t-1}\) combined with \(X_{t}\). If \(f_{t}\) equals 0, the model completely discards prior information; if \(f_{t}\) equals 1, the model retains all previous information. The input gate layer establishes which information to store in the cell state \(c_{t}\), in conjunction with the \(c_{t}\) vector. Last, the output layer identifies which information to transmit, incorporating the cell state \(c_{t}\) and filtered input \(o_{t}\) in the process.