## 7 Empirical Prediction Equations  
In this section, empirical equations are developed to predict the correlation coefficients, \(\rho\), between the most optimal \(IM_{2}\)\({}^{\text{MS-AS}}\) and the resulting \(DI_{2}\)\({}^{\text{MS-AS}}\). As seen from Figure 11, \(I_{A,2}\)\({}^{\text{MS-AS}}\) and \(v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}},2}\)\({}^{\text{MS-AS}}\) are separately the most optimal IM to formulate \(IM_{2}\)\({}^{\text{MS-AS}}\) before and after the transition period \(T_{c}\). This conclusion is verified with considering different hysteretic models and varying \(R\) levels in this section. Therefore, a step-wise relationship is feasible to represent the relationship between \(\rho\) and \(T\). Figure 15 collects the calculated \(\rho\) values between the most optimal \(IM_{2}\)\({}^{\text{MS-AS}}\) and \(DI_{2}\)\({}^{\text{MS-AS}}\) corresponding to different hysteretic models with varying \(R\) values. From a viewpoint of median prediction, the median \(\rho\) values, i.e., \(\rho_{50\%}\), at different \(T\) values are obtained by statistics; a stepwise equation consisting of a linear and a nonlinear section is regressed by Equation (2). It is clear that Equation (2) can appropriately fit the calculated \(\rho_{50\%}\) values with a coefficient of determination as \(R^{2}=0.9\).  
\[\rho_{50\%}=\left\{\begin{array}{cc}-0.143\times T+0.87&T\leq T_{c}=0.45\; \text{s}\\ 0.92\times exp\!\left[-\left(\frac{T-4}{10}\right)^{2}\right]&T_{c}=0.45\; \text{s}\leq T\leq 6.0\;\text{s}\end{array}\right. \tag{2}\]  
where the transition period \(T_{c}\) is taken as 0.45 s. When \(T\leq T_{c}\), \(I_{A}\) is recommended as the best candidate to formulate \(I_{A,2}\)\({}^{\text{MS-AS}}=I_{A}\)\({}^{\text{AS}}/I_{A}\)\({}^{\text{MS}}\) with the corresponding correlation with \(DI_{2}\)\({}^{\text{MS-AS}}\) approximated as \(\rho_{50\%}=-0.143\times T+0.87\). For the case of \(T<T_{c}<6.0\) s, \(v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}},2}\)\({}^{\text{MS-AS}}=v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}}}\)\({}^{\text{AS}}/v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}}}\) is taken as the best IM for MS-AS sequences, whose median correlation with \(DI_{2}\)\({}^{\text{MS-AS}}\) can be evaluated as \(\rho_{50\%}=0.92\times\exp\{-[(T-4)/10]^{2}\}\). Based on Equation (2), the predicted correlation \(\rho_{50\%}\) is generally higher than 0.8, showing a significant interdependence between \(I_{A,2}\)\({}^{\text{MS-AS}}\) (or \(v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}},2}\)\({}^{\text{MS-AS}}\)) and \(DI_{2}\)\({}^{\text{MS-AS}}\). This result provides a basis that it is feasible to develop a linear relationship between \(I_{A,2}\)\({}^{\text{MS-AS}}\) (or \(v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}},2}\)\({}^{\text{MS-AS}}\)) and \(DI_{2}\)\({}^{\text{MS-AS}}\) in logarithm space. This log-linear relationship is valuable for the quick assessment of structural damage caused by MS-AS sequences.  
Figure 14: The correlation related to \(I_{A}\), \(v_{\text{\tiny{$\tau$}}\text{\tiny{$S$}}}\) and \(PGD\) between \(\text{ln}IM_{2}\)\({}^{\text{MS-AS}}\) and \(\text{ln}DI_{2}\)\({}^{\text{MS-AS}}\) at the entire period range with consideration of varying \(R\) values.  
The variability of \(\rho\) in terms of the standard deviation \(\sigma_{\rho}\) is further calculated based on the obtained \(\rho\) values. Figure 16 shows the \(\sigma_{\rho}\) values along with the period. It is clear that a stepwise type equation is suitable to fit the \(\sigma_{\rho}\) values. Then the stepwise function shown in Equation (3) is developed, which shows an acceptable goodness-of-fitting with a value of \(R^{2}\) equal to 0.84.  
\[\sigma_{\rho}=\left\{\begin{array}{cc}0.024\times T+0.021&T\leq 1.0\;\mathrm{s} \\ 0.003\times T^{2}-0.0254\times T+0.021&1.0\;\mathrm{s}\leq T\leq 6.0\;\mathrm{s} \end{array}\right. \tag{3}\]