# IeALL: Dam Deformation Prediction Model Based on Combination Model Method
## 5 Dam Deformation Prediction Model Based on Spatio-Temporal Correlation and IEALL
### Encoder  
#### 5.3.1 Spatial Feature Extraction  
GCN is utilized to extract spatial features since the input sequence in this part is organized as a graph. There are two common techniques to construct GCN. The first is the spatial domain method, which is comparable to the conventional convolution method in that it defines the convolution operation directly on each node's connection relationship. The other is a spectral domain graph convolution network, which performs spectral domain convolution operations using the graph's Fourier transform. In this section, the spectral domain graph method of graph convolution is used to perform convolution operation. The data are regarded as the signal on the graph, and the information on the graph is directly processed to capture the pattern and feature on the input sequence space, so as to accurately describe the spatial correlation between multiple measurement points and improve the prediction accuracy. By algebraically transforming the graph, spectrograms can be analyzed. In spectrogram theory, a Laplacian matrix L is used to describe the correlation of each node in  
Figure 5: Spatial attention mechanism. the graph structure, namely \(L=D-A\), and its normalized form after regularization is: \(L=I_{N}-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}\in\mathbb{R}^{N\times N}\), including \(I_{N}\) matrix for the unit, \(A\) as the adjacency matrix, \(D\in\mathbb{R}^{N*N}\) for degree matrix and node degree of diagonal matrix, \(D_{ii}=\sum JA_{ij}\). A further decomposition of the Laplacian matrix can be obtained: \(L=U\Lambda U^{T}\), where U is the Fourier basis and \(\Lambda=diag(\lambda_{0},\lambda_{1},\ldots,\lambda_{N-1})\in\mathbb{R}^{N \times N}\) is a diagonal matrix consisting of the Laplace matrix's eigenvalues. Using the deformation input \(X_{t}\)t at time \(t\) as an example, the input graph signal is \(X_{t}=\left(x_{t}^{1},x_{t}^{2},\ldots,x_{t}^{N}\right)\), and the picture information is Fourier transformed \(\tilde{X}_{t}=U^{T}X_{t}\). Because the nature of the available by Laplacian \(U\) is orthogonal matrix, \(X_{t}=U\tilde{X}_{t}\) can be obtained using the inverse Fourier transform of the figure signal processing operation. The convolution kernel \(g_{\theta}\) is used to convolution the input signal \(G(x)\) with the diagonalized linear operator defined in the Fourier domain, as shown in Equation (23):  
\[g_{\theta}*G(x)=g_{\theta}(L)X_{t}=g_{\theta}\left(U\Lambda U^{T}\right)X_{t}= Ug_{\theta}(\Lambda)U^{T}X_{t} \tag{23}\]  
The convolution operation of graph signals is equal to the product of these signals, so the cost of eigen factorization of the corresponding Laplacian matrix is higher when the scale is larger. For this problem, an approximate solution can be achieved by Chebyshev polynomial kernel, as shown in Equation (24):  
\[g_{\theta}*G(x)=g_{\theta}(L)X_{t}=Ug_{\theta}(\Lambda)U^{T}X_{t}\approx U \sum_{k=0}^{K-1}\theta_{k}T_{k}\left(\widetilde{L}\right)U^{T}X_{t} \tag{24}\]  
Among them, \(\theta_{k}\) is the coefficient of Chebyshev polynomials, \(\widetilde{L}=\frac{2L}{\lambda_{max}}\), \(\lambda_{max}\) is the largest eigenvalue of the Pierre-simon Laplace matrix. A Chebyshev Polynomials can be defined recursively as: \(T_{k}(x)=2 xT_{k-1}(x)-T_{k-2}(x)\),of which \(T_{0}(x)=1,T_{1}(x)=x\). The approximate solution is obtained by using this expansion, which is equivalent to extracting the periphery of every node in the graph by using convolution kernel \(0\sim K-1\). The neighborhood information of the order is calculated to obtain the spatial characteristics. In addition, for multi-channel graph signal that has multi-dimensional characteristics of the data, each node has \(D-1\) dimension, for the moment \(t\) use \(D-1\) Convolution check for convolution operations can obtain \(g_{\theta}*G(x)\). Of which \(\theta=(\theta_{1},\theta_{2},\ldots,\theta_{D-1})\) is the convolution kernel parameter, using the RELU linear correction unit as the activation function, \(\text{ReLU}(g_{\theta}*G(x))\).