# Graph Convolutional Spectral Clustering for Electricity Market Data Clustering
## 2 Integration and Mining of Electricity Market Data
### Electricity Information Data Mining Based on Clustering Algorithm
#### 2.2.2 Improved Spectral Clustering Algorithm Combining GCN and Spectral Filter  
Spectral clustering is a widely used unsupervised learning method that proves effective in many practical applications [33]. However, it faces certain challenges, such as the inability to learn effective representations between non-Euclidean data in traditional spectral clustering. Moreover, algorithms based on spectral clustering require eigenvalue decomposition of the Laplacian matrix, which is time-consuming, especially when dealing with large-scale data. Additionally, applying the k-means algorithm for spectral clustering in large-scale data can also be time-consuming. To address these issues, this paper combines GCN and Chebyshev polynomial approximation of spectral filters to improve the spectral clustering algorithm, as shown in Figure 3. Initially, GCN are used to represent and analyze non-Euclidean data, followed by the construction of the Laplacian matrix. Simultaneously, this chapter employs Chebyshev polynomial approximation of spectral filtering techniques to avoid eigenvalue decomposition of the Laplacian matrix. Graph Convolutional Network Module  
Graph convolutional networks typically utilize graph structure and node feature information to iteratively aggregate the feature information of local graph neighbor nodes. Due to their advantages in combining graph structure and node content information, GCNs have been widely applied for feature extraction from non-Euclidean data. The goal of GCN is to construct a weighted graph with the same number of nodes as the original graph. However, there may be different weights in the graph, encapsulating the topology of the original graph and node attributes. Let \(A\in R^{N\times N}\) be the adjacency matrix and its node attribute matrix \(X\in R^{N\times k}\), where \(k\) corresponds to the dimension of the node attribute vector. Given an undirected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\) with \(N=|\mathcal{V}|\) nodes and \(\mathcal{E}\) edges, let \(A\in R^{N\times N}\) be the adjacency matrix, and its node attribute matrix \(X\in R^{N\times k}\), where \(k\) corresponds to the dimension of the node attribute vector. GCN can be understood as a function \(f(\cdot)\), and  
Figure 3: The GCN and spectral clustering framework diagram with graph filtering. its output is a weighted graph \(\widetilde{\mathcal{G}}=f(\mathcal{V},\mathcal{E})\), represented by its adjacency matrix \(\widetilde{A}\in R^{N\times N}\), i.e., \(\widetilde{A}=f(A,X)\). For the learned representation weight matrix \(W\) in the \(l\)-th layer, \(Z^{l}\) can be obtained through the following convolution Equation (11):  
\[Z^{(l)}=f\Big{(}Z^{(l-1)},A\Big{)}=\varphi(\widetilde{D}^{-\frac{1}{2}} \widetilde{A}\widetilde{D}^{-\frac{1}{2}}Z^{(l-1)}W^{(l-1)}), \tag{11}\]  
where \(Z^{(l)}\) represents the \(l\)-th layer in GCN, denoted as \(\widetilde{A}=A+I\) and \(\widetilde{D_{ii}}=\sum_{j}\widetilde{A_{ij}}\). \(I\) is the identity matrix for each node's self-loop, and \(\varphi(\cdot)\) is the non-linear activation function. The representation \(Z^{(l-1)}\) traverses the normalized adjacency matrix \(\widetilde{D}^{-\frac{1}{2}}\widetilde{A}\widetilde{D}^{-\frac{1}{2}}\), resulting in a new representation \(Z^{l}\). Finally, the ReLU function is applied Equation (12) in the last layer of GCN:  
\[Z=f\Big{(}Z^{(l)},A\Big{)}=\varphi(\widetilde{D}^{-\frac{1}{2}}\widetilde{A} \widetilde{D}^{-\frac{1}{2}}Z^{(l)}W^{(1)}), \tag{12}\]  
where \(Z\) is a transformed node feature matrix, serving as both the output of GCN and the input to the spectral filtering module. The Chebyshev polynomial approximation of the spectral filtering module achieves convolution in the spectral domain, similar to Fourier domain filtering of signals. The eigenvalues of the Laplacian matrix of the graph represent the frequencies of signals defined on the graph. Assuming \(o_{i}\) is the \(i\)-th variable of \(U\),we obtain Equation (13). \[o_{i}=|U^{T}\delta_{i}|^{2},\delta_{i}(j)= \begin{cases}1&i=j\\ 0&i\
eq j\end{cases}