## 3 Correlation Analysis of Seismic _IMs_
### Correlation Analysis between Seismic IMs  
The correlation coefficient is a statistical indicator used to evaluate the degree of correlation between two variables. Currently, the most commonly used correlation coefficients include the Pearson correlation coefficient, Spearman correlation coefficient, and Kendall correlation coefficient. Among them, the Pearson correlation coefficient is commonly used to evaluate the correlation between two continuous variables that follow a normal distribution and have a linear relationship; the Spearman correlation coefficient is a rank correlation coefficient, which does not require variables to follow a normal distribution and can describe the nonlinear correlation between two variables; the Kendall correlation coefficient is suitable for measuring the correlation between two ordinal categorical variables and evaluates the correlation according to the consistency of the ordered pairs between the two variables. Since the data of various _IMs_ may not conform to a normal distribution, and the calculation method of the Kendall correlation coefficient is relatively complicated, the authors of this study ultimately decided to adopt the Spearman correlation coefficient to evaluate the correlation between each _IM_. The Spearman correlation coefficient between each initial candidate seismic _IM_ is calculated according to Equation (1):  
\[\rho_{s}=1-\frac{6\sum\limits_{k=1}^{n}{(IM_{i,k}-IM_{j,k})^{2}}}{n(n^{2}-1)} \tag{1}\]  
where \(n\) is the number of selected ground motions, herein \(n=100\); \(IM_{i,k}\) is the rank of the \(i\)-th \((i=1,2,\ldots,29)\) seismic _IM_ in sample \((IM_{i,1},IM_{i,2},\ldots,IM_{i,100})\); \(IM_{i,k}\) is the rank of the \(j\)-th \((j=1,2,\ldots,29)\) seismic _IM_ in sample \((IM_{j,1},IM_{j,2},\ldots,IM_{j,100})\).  
According to the absolute value of the Spearman correlation coefficient, the degree of correlation between seismic _IMs_ can be divided into three levels: when \(|\rho_{s}|=0.0\,\sim 0.3\), this indicates that the correlation degree between two seismic _IMs_ is low; when \(|\rho_{s}|=0.6\,\sim 1.0\), this indicates a significant correlation between two seismic _IMs_; when \(|\rho_{s}|=0.6\,\sim 1.0\), this indicates that the two seismic _IMs_ are highly correlated. Therefore, this study first calculates the values of 29 _IMs_ for the 100 pulse-like ground motions discussed in Section 2.1, and then calculates the Spearman correlation coefficients between the 29 seismic _IMs_ according to Equation (1), as shown in Figure 6.  
Figure 6: Bubble chart of the correlation coefficients of the 29 initial candidate _IMs_.  
As can be seen from Figure 6, the correlation coefficient values among the 29 initial candidate _IMs_ have a wide distribution. Notably, the correlation coefficient values between PGV and Vp, as well as between Arms and \(\alpha_{\text{RMS}}\), are both 1.0, indicating that these _IMs_ are completely correlated. The correlation coefficient value between \(\alpha_{\text{RMS}}\) and I\({}_{\text{C}}\) is the smallest at only 1.2 \(\times\) 10\({}^{-4}\), close to zero, indicating that \(\alpha_{\text{RMS}}\) and I\({}_{\text{C}}\) are almost completely uncorrelated. This further illustrates that the 29 initial candidate _IMs_ in Section 3.1 are relatively comprehensive and are suitable as initial candidates for seismic _IMs_.