## 5 Correlation Analysis
### Correlation Coefficient  
The Pearson correlation coefficient \(\rho\) is calculated for four IM-DI pairs in the logarithm space, including _IM_MS vs. _DI_1 MS-AS_, _IM_AS vs. _DI_1 MS-AS_, _IM_1 MS-AS vs. _DI_1 MS-AS_, and _IM_2 MS-AS vs. _DI_2 MS-AS_, respectively. The basic system with \(R=2\) at \(T=0.2\) s, 1.0 s, and 5.0 s are taken herein for illustration to represent the structure with short (acceleration-sensitive), medium (velocity-sensitive), and long (displacement-sensitive) vibration periods, respectively (Riddell [38]).  
Figure 8 shows the scatters in terms of ln_PGA_MS vs. ln_DI_1 MS-AS_, ln_PGA_AS vs. ln_DI_1 MS-AS_, ln_PGA_1 MS-AS vs. ln_DI_1 MS-AS_, and ln_PGA_2 MS-AS vs. ln_DI_2 MS-AS_. It is clear that the correlation between the MS (or AS) intensity alone (i.e., _PGA_MS or _PGA_AS) and the structural cumulative damage (i.e., _DI_1 MS-AS) is insignificant with the \(\rho\) values less than 0.3. Moreover, there is no clear improvement in the correlation between ln_PGA_1 MS-AS and ln_DI_1 MS-AS in which the entire earthquake sequence is regarded as one nominal ground motion. Yet, _PGA_2 MS-AS shows a much stronger interdependence with _DI_2 MS-AS, with correlation coefficients exceeding 0.5. Besides _PGA_, similar observation can also be obtained from the other IM-related data, which are summarized in Figure 9. As observed, the correlations between ln_IM_2 MS-AS and ln_DI_2 MS-AS for different considered IMs are always much larger than those of the other pairs (e.g., between ln_IM_MS and ln_IM_1 MS-AS). These results confirm that _IM_2 MS-AS performs much better than the other three types of _IM_s, i.e., _IM_M_, _IM_AS, and _IM_1 MS-AS, for the prediction of the damage potential of earthquake sequences.  
Figure 8: _Cont._Figure 10 shows the correlations associated with the considered classic IMs between \(IM_{2}\)MS-AS and \(DI_{2}\)MS-AS. Among the acceleration-related IMs, \(I_{A}\) is the most optimal one to generate \(IM_{2}\)MS-AS since \(I_{A,2}\)MS-AS has higher correlation with \(DI_{2}\)MS-AS than \(E_{a,2}\)MS-AS, \(PGA^{\text{MS-AS}}\), \(I_{a,2}\)MS-AS, \(a_{\text{rms},2}\)MS-AS, \(I_{c,2}\)MS-AS, \(a_{\text{rs},2}\)MS-AS, and \(P_{a,2}\)MS-AS. As for the velocity-related and displacement-related IM groups, the corresponding optimal IMs are \(v_{rs}\) and \(PGD\), respectively.  
According to the results of Figure 10, \(I_{A}\), \(v_{rs}\) and \(PGD\) are found as the optimal IMs to formulate \(IM_{2}\)MS-AS in the acceleration-, velocity-, and displacement-based IM group, respectively. Then the \(\rho\) values between the three pairs, including \(I_{A,2}\)MS-AS vs. \(DI_{2}\)MS-AS, \(v_{rs,2}\)MS-AS vs. \(DI_{2}\)MS-AS, and \(PGD_{2}\)MS-AS vs. \(DI_{2}\)MS-AS, are plotted together at the entire period range, as shown in Figure 11. It is noteworthy that only the results related to the basic SDOF model (i.e., Model-1) are used. It is clear that there is no single IM exhibiting the best correlation throughout the entire period range, and thus, a transition period (i.e., \(T_{c}\)) exists. In fact, before and after \(T_{c}\), \(I_{A}\) and \(v_{rs}\) are the most optimal IMs (with the highest correlation to \(DI_{2}\)MS-AS) to formulate \(IM_{2}\)MS-AS, respectively.  
Figure 8: Interdependence of four PGA-DI pairs.  
Figure 9: Interdependence of four PGA-DI pairs interdependence between the considered four pairs in terms of \(\ln\)\(IM^{\text{MS}}\) vs. \(\ln\)\(DI_{1}\)MS-AS, \(\ln\)\(IM^{\text{AS}}\) vs. \(\ln\)\(DI_{1}\)MS-AS, \(\ln\)\(IM_{1}\)MS-AS vs. \(\ln\)\(DI_{1}\)MS-AS and \(\ln\)\(IM_{2}\)MS-AS by using different IMs for SDOF systems at varying periods.