## 2 Methodology
### Statistical Analysis
#### 2.4.2 Calculation of Correlation Coefficient Based on Energy  
The second step of the statistical analysis is to calculate the Spearman correlation coefficient of energy, that is, to calculate the correlation coefficient between the energy under gravity of all rock blocks at any time and the energy under gravity of all rock blocks at \(T_{1}\).  
The Spearman correlation coefficient, also known as the Spearman rank correlation coefficient, is a kind of rank correlation coefficient. The Spearman correlation coefficient is a measure of the synergy relationship. It is different from the Kendall correlation coefficient, which describes the synergy relationship between two independent and identically distributed binary distributions. The Spearman correlation coefficient describes one of the three independent and identically distributed binary distributions. In this paper, the Spearman correlation coefficient is used as a statistical indicator due to its wide applicability and monotonic transformation invariance, and it is not restricted by dimensions, and is not affected by outliers. At the same time, it could also describe a certain non-linear relationship. Therefore, the Spearman correlation coefficient is chosen as a statistical indicator. The calculation equation used is as follows.  
\[CC=1-\frac{6\sum d_{i}^{2}}{n(n^{2}-1)} \tag{3}\]  
The calculation process is as follows. By first sorting the data of the two variables \((X,Y)\), and then writing down the position \((X_{1},Y_{1})\) after sorting, the value of \((X_{1},Y_{1})\) is called the rank, the difference of the rank is the \(d_{i}\) in the above formula, n is the number of the variable data, and finally the formula could be used to solve the result. The following examples illustrate the meaning of each letter as shown in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Original Location** & **Original X** & **After Sorting** & **Rank X\({}_{\mathbf{1}}\)** & **Original Y** & **After Sorting** & **Rank Y\({}_{\mathbf{1}}\)** & **Square of Rank Difference \(d_{i}\)** \\ \hline
1 & 11 & 490 & 5 & 2 & 75 & 6 & 1 \\
2 & 490 & 43 & 1 & 75 & 44 & 1 & 0 \\
3 & 14 & 30 & 4 & 3 & 42 & 5 & 1 \\
4 & 43 & 14 & 2 & 44 & 7 & 2 & 0 \\
5 & 30 & 11 & 3 & 7 & 3 & 4 & 1 \\
6 & 3 & 3 & 6 & 42 & 2 & 3 & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 1: A simple example of explaining the meaning of \(d_{i}\).  
Figure 3: A simple illustration of the displacement of a rock block under gravity.  
A relationship curve is drawn between each CC value and the corresponding number of iteration steps. When the number of iterations reaches a certain value, the relationship curve between CC and the number of iterations begins to converge, and the iteration process could be terminated.  
The proposed method based on the energy correlation coefficient in this paper considers the mass or weight and displacement of each block. The proposed method considers both the temporal correlation and spatial correlation. Therefore, compared with the displacement-based method, the proposed method could be effectively applied to investigate the deformations and failures of jointed rock slopes in which the potential sliding surface could not be determined easily.