# 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
 \tag{13}\]  
It satisfies all \(o_{i}>0\), in addition \(O(i,i)=1/o_{i}\) and \(R=(r_{1},r_{2},...,r_{s})\in R^{N\times s}\) which consists of \(s\) random signals. Let \(r\) be a vector with independent random Gaussian variables that follow a standard normal distribution. These vectors have a mean of \(0\) and a variance of \(1/s\). Then, \(f_{i}\) is a feature vector, and the estimated value feature vector \(\tau_{i}\) of each node that is very similar to \(f_{i}\) can be defined according to the following Equation (14):  
\[\tau_{i}=\big{(}OH_{\lambda_{k}}R\big{)}\delta_{i}. \tag{14}\]  
According to the above Equation (14), it is known that \(H_{\lambda_{k}}=U_{k}U_{k}^{k}\), and it satisfies \(U^{\prime}=OU\). Based on the JL lemma and the \(U\) has orthogonal columns, we can obtain Equation (15). \[|U\big{(}f_{i}-f_{j}\big{)}|^{2}=|\big{(}f_{i}-f_{j}\big{)}|^{2}, \tag{15}\]  
Let \(r\in R^{n}\) be a vector with independent random terms and follows a standard normal distribution. Filtering \(r\) through \(H_{\lambda_{k}}\), we obtain \(r_{\lambda_{k}}\) as shown in Equation (16). The \(i\)-th element of \(r_{\lambda_{k}}\) is Equation (17), and the mean of \(\big{(}r_{\lambda_{k}}\big{)}_{i}^{2}\) satisfies Equation (18). Therefore, a practical method to estimate \(OH_{\lambda_{k}}R\) is to first compute \(H_{\lambda_{k}}R\), and then normalize its rows to unit length. \[r_{\lambda_{k}}=U\text{diag}(\lambda_{1},\lambda_{2},\ldots,\lambda_{k},0, \ldots,0)U^{T}r=UU^{T}r \tag{16}\]  
\[\big{(}r_{\lambda_{k}}\big{)}_{i}=r_{\lambda_{k}}^{T}\delta_{i}=r^{T}UU^{T} \delta_{i}, \tag{17}\]  
\[E\big{(}r_{\lambda_{k}}\big{)}_{i}^{2}=|U^{T}\delta_{i}|^{2}. \tag{18}\]  
The k-means sampling module implements spectral clustering on a small-scale data set, reducing the algorithm's time complexity. Let \(\Omega=\{\tau_{1},\tau_{2},\ldots,\tau_{n}\}\) represent the extracted \(n\) data points from the total \(N\) data points for clustering. Then, the k-means clustering algorithm is applied to this set, clustering it into \(k\) clusters. Considering that k-means can also be applied to the original vector set \(\{f_{1},f_{2},\ldots,f_{N}\}\), a sampling matrix \(G\) can be formulated to satisfy \(y_{i}^{\tau}=Gy_{i}\). \(G\) is an \(nN\) dimensional matrix. Ideally, \(G\) satisfies Equation (19). To recover \(y_{i}\) from the clustering results, a quadratic programming Equation (20) is utilized. \[G_{ij}= \begin{cases}1&i=\tau_{j}\\ 0&i\
eq\tau_{j}\end{cases}  \tag{19}\]  
\[\min_{y_{i}\in R^{N}}\frac{1}{2}y_{i}^{\top}\Big{(}G^{\top}G+\gamma g(L)\Big{)} y_{i}-y_{i}^{\top}G^{\top}y_{i}^{\prime}. \tag{20}\]  
Then, Equation (20) can be solved using Equation (21), where \(G\) is the sampling matrix, \(\gamma\) is the regularization parameter, and \(g(L)\) is a positive non-decreasing polynomial:  
\[\Big{(}G^{\top}G+\gamma g(L)\Big{)}y_{i}=G^{\top}y_{i}^{\prime}. \tag{21}\]  
A method independent of the entire optimization formula is proposed for computing \(g(L)\). Pseudo-eigenvectors replace first \(k\) eigenvectors, providing a high-order approximation and retaining distances between original eigenvectors. Pseudo-eigenvectors aid in reconstructing \(g(L)\) by computing Euclidean distances between them. This reconstruction using pseudo-eigenvectors helps reduce the time complexity. The adjacency matrix used for reconstruction is defined as in Equation (22): The reconstructed Laplacian matrix \(L\) can be expressed as Equation (23):  
\[\widetilde{A}_{ij}=\exp\left(-\frac{q^{2}(Z_{i}-Z_{j})}{\sigma_{i}\sigma_{j}} \right), \tag{22}\]  
\[\widetilde{L}=I-\widetilde{D}^{-\frac{1}{2}}\widetilde{A}\widetilde{D}^{-\frac {1}{2}}, \tag{23}\]  
where \(Z_{i}\) represents the \(i\)-th row of \(Z\), and \(q^{2}\big{(}Z_{i}-Z_{j}\big{)}\) represents the square of the Euclidean distance between \(Z_{i}\) and \(Z_{j}\). \(\sigma_{i}\) represents the size of a range, denoted as \(\sigma_{i}=q\big{(}Z_{i},Z_{j}\big{)}\).