# 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
#### 2.2.1 Feature Extraction of End-User Load Data  
In the face of the high dimensionality of terminal user load data, this section begins by describing the fluctuation of load curves using the concept of entropy from statistics. Based on the numerical values of entropy, the load curve is segmented, and then segment-wise aggregation is employed to achieve numerical re-expression of each segment. This process is aimed at reducing the dimensionality of load data and extracting features.  
Analysis of Load Time-Series Fluctuations  
The fluctuation degree of the load time-series curve is analyzed using information entropy, where a higher information entropy value corresponds to a greater fluctuation degree. Suppose \(X\) represents the load data of a time-series segment. By statistically analyzing the load values within this time period, let us assume there are n different load values denoted as \(x_{1},x_{2},\ldots,x_{n}\), and the probability of each load value occurring is represented by \(p_{1},p_{2},\ldots,p_{n}\). The entropy value \(H_{n}\) of this segment of time-series is then calculated as Equation (6):  
\[H_{n}(p_{1},p_{2},\cdots,p_{n})=-\sum_{i=1}^{n}p_{i}\ln p_{i}\,,0<p_{i}<1,\sum_ {i=1}^{n}p_{i}=1. \tag{6}\]  
From the above formula, it can be observed that, when \(p_{1}=p_{2}=p_{3}=\cdots=p_{n}\), \(p_{i}=1/n\), \(H_{n}(p_{1},p_{2},\cdots,p_{n})=H_{max}=\ln n\) which is the maximum value for the entropy of this time-series segment. Simultaneously, the average entropy is defined as Equation (7):  
\[\overline{H_{n}}=\sum_{i=2}^{n}\frac{1}{i}. \tag{7}\]  
In order to comprehensively consider the entropy value, maximum entropy, and average entropy of the load data, a new concept \(\tau_{m}\) is introduced to describe the fluctuation degree of the load curve for \(m\) loads in this time-series segment. When its value is 1, it indicates a significant fluctuation, and the data mean cannot be used as a substitute. The specific definition of \(\tau_{m}\) is given as Equation (8):  
\[\tau_{m}=\begin{cases}1,&\frac{H_{n}(X_{m})}{\ln n}>\omega\overline{H_{n}}(X_ {m})\\ 0,&\frac{H_{n}(X_{m})}{\ln n}\leq\omega\overline{H_{n}}(X_{m})\end{cases}. \tag{8}\]  
In the formula, \(m=1,2,\ldots,M\); \(M\) represents the total number of loads; the scale factor is \(\omega\), its value range is [1; 2], and in this article, \(\omega=1\).  
By synthesizing the \(\tau_{m}\) of the time section of multiple loads, the proportion of the number of loads of \(\tau_{m}=1\) in M loads is obtained, and if the value exceeds the threshold, it means that the curve of this section of M loads is very different, and it needs to be described in segments. The specific definitions of \(\rho\) as Equation (9):  
\[\rho=\frac{1}{M}\sum_{m=1}^{M}\tau_{m}. \tag{9}\]  
Determining the number of segments based on the fluctuation level of the load curve, we then employ segmented aggregation approximation to interpret high-dimensional time-series through low-dimensional data features. This approach achieves the reduction and re-expression of load time-series while preserving the individual characteristics of the load curve shape.  
Take the load data \(Q=\{q_{1},q_{2},\ldots q_{L}\}\) of a complete time-series; its time-series length is \(L\). By segmenting the elements in it, a new data sequence is \(\overline{Q}=\{\overline{q}_{1},\overline{q}_{2},\cdots,\overline{q_{l}}\}\) with a length of \(l\), where \(l\) ( \(L\) and \(l\) are divisible by \(L\). The specific calculation process of \(\overline{q_{i}}\) is as Equation (10):\[q_{i}=\frac{l}{L}\sum_{j=\frac{l}{l}(i-1)+1}^{L_{i}}q_{j},i\in\{1,l\},j\in\{1,L\}. \tag{10}\]