# Graph Convolutional Spectral Clustering for Electricity Market Data Clustering
## 2 Integration and Mining of Electricity Market Data
### Electricity Information Data Fusion
#### 2.1.2 Exceptional Data Reconstruction Based on RBF Neural Network  
The power data of the user is first preprocessed by the abnormal data identification algorithm based on isolated forest to distinguish the normal data from the abnormal data caused by harmonic interference, fault and other factors. Subsequently, these preprocessed data are used as input values and substituted into the RBF neural network for network training. The RBF neural network consists of an input layer, a hidden layer, and an output  
Figure 1: Electricity consumption information fusion flow chart. layer, representing a forward three-layer structured network. The network topology architecture is illustrated in Figure 2.  
The core elements of the network architecture include center vectors, the number of nodes in the hidden layer, radial basis function width, and weight matrices. The network parameters, namely the radial basis function width, weight matrix, number of hidden layer units, and centers, are explicitly determined through network training. The key to network training is specifying an appropriate number and orientation of center points. The expression for the radial basis function is denoted as Equation (1):  
\[F\left(\|\theta-W_{j}\|^{2}\right)=\exp\left(-\frac{\psi}{d_{\text{max}}}\| \theta-W_{j}\|\right), \tag{1}\]  
where \(W_{j}\) represents the center of the radial basis function and the input vector \(\theta\) has the same dimensions. The symbol \(\psi\) represents the number of center points, and \(\theta-W_{j}\) represents the distance between the input vector and the center point, with \(d_{\text{max}}\) representing the maximum distance between center points. The width parameter \(\lambda\) of the radial basis is represented as Equation (2) and the expansion constant \(\beta\) of the radial basis is expressed Equation (3):  
\[\lambda=\frac{d_{\text{max}}}{\sqrt{2\psi}}, \tag{2}\]  
\[\beta=\delta d_{min}, \tag{3}\]  
where \(d_{\text{min}}\) represents the distance between each center point, and \(\delta\) represents the overlap index. A cost function \(\omega\) is defined Equation (4) to measure the network's output \(y_{z}\). \[\omega=\frac{1}{2}\sum_{i=1}^{n}(\,\varphi_{i}\,-\,y_{z_{i}})^{2}, \tag{4}\]  
where \(\varphi_{i}\) and \(y_{z_{i}}\) represent the expected output and actual output of the RBF neural network output nodes, respectively. The adjustment process of the network weight matrix, radial basis function width parameter, and adjustment of the center points of the hidden layer units during the \(t\) time period is estimated using the gradient descent method. Assuming new data samples are input into the network and network parameters need to be calibrated, parameters can be modified through gradient descent. After a finite number of adjustments, the output error of the network for consistent data mining can be kept within an acceptable range. If the error \(\omega\) is less than the allowable error, the sample does not need adjustment. Finally, the formula for calculating the output of the RBF neural network training is expressed as  
Figure 2: RBF neural network topology architecture. Equation (5), completing the entire algorithm process and obtaining the results of consistent data mining. \[f(x)=\hat{\eta}\theta, \tag{5}\]  
where \(x\) represents the input data of RBF network and \(f(x)\) represents the output result of RBF network, \(\hat{\eta}\) represents the hidden layer output matrix of RBF network, and \(\hat{\theta}\) represents the output weights of RBF network. The fundamental idea of network learning is based on the assumption that there is an inherent law of consistency in associated data. A corresponding network is created to mimic this pattern, thereby reconstructing abnormal data. The sample sequence of data is denoted as \(\{\sigma(l),u(l)\}_{l=1}^{\infty}\), where \(\sigma(l)\) is the input sample, and \(u(l)\) is the expected output relative to \(\sigma(l)\).