## 4 Recommendation on Improving the Efficiency of Intensity Measures
### Analysis of Functional Forms  
Thus far, we have used the standard functional model (single parameter regression model) to define relationships between IMs and EDP (Equation (1)). However, this model does not sufficiently capture the complex interaction of the two. We observe that by including the seismological parameters in the functional form, the efficiency in predicting EDPs increase on average by 10% for the selected IMs. [PERSON] et al. [51] considered a multiple IM approach by including _PGA_, _CAV_, _Sa(T)_ and _Sp(T)_ within a frequency range close to the fundamental period of the structure. [PERSON] et al. [6] included seismological  
Figure 10: Logarithmic ratio of observed EDP and elastic limit for SP, IP and LP structures (**a–c**). The black solid line is the elastic limit for SP, IP and LP structures respectively. The colors show ground motion grouped into different magnitude bins. The solid lines are the moving averages of logarithmic ratio of observed EDP and elastic limit. As epicentral distance increases, the records rather produce elastic responses and hence show larger correlations. parameters (\(M_{w}\) and \(R_{epi}\)) directly in the functional forms (Equation (24)), noticing an increased efficiency and sufficiency with respect to \(M_{w}\) and \(R_{epi}\). \[log\ (EDP)=a+b^{*}log\ (IM)+c^{*}log\ (M_{w})+d^{*}log\ (R_{epi})+\delta \tag{24}\]  
By extending Equation (24), we show that IMs can be made even more efficient and sufficient by including soil stiffness parameter (_Vs30_) and structural period (_Tj_) in the functional forms (Equations (25) and (26)). However, different combinations of these parameters may generate a series of models with the risk of either over- or under-fitting. We therefore apply the Akaike information criteria (_AIC_) to determine the statistical quality of all possible models arising from these parameters. Table 6 lists _AIC_ scores for different input parameters while using _PGA_ and _PGV_ as IMs. As expected, _AIC_ scores are lowest when all parameters are considered in the regression model (bold values in Table 6). Similar results are found considering other IMs. \[log\ (EDP)=a+b^{*}log\ (IM)+c^{*}log\ (M_{w})+d^{*}log\ (R_{epi})+e^{*}log\ (Vs30)+\delta \tag{25}\]  
\[log\ (EDP)=a+b^{*}log\ (IM)+c^{*}log\ (M_{w})+d^{*}log\ (R_{epi})+e^{*}log\ (Vs30)+f^{*}log\ (T_{\rho})+\delta \tag{26}\]  
Figure 11: Dependence of correlations of intensity measures with site classes B, C, D and E (based on NEHRP recommendation), along with number of records per site class (in brackets). Rows represent variations in correlations of peak measures, spectral measures and energy measures for three different types of structures (SP, IP, LP). For short-period structures, the correlations of PGA appear to be independent of site class (**a**). For IP/LP structures, correlations of _PGV_ are invariant with site class (**b**,_c_). _PGD_ correlations decrease with site class (**c**). For spectral measures, _ASI_ and _VSI_ exhibit correlations similar to _PGA_ and _PGV_ (**d–f**). For energy measures, \(A_{RMS}\) and \(I_{A}\) show good correlations and least dependence with site class (**g–i**). Figure 12: Variation of inter-story drift (ISD) ratio with _Vs30_ for SP, IP and LP structures (**a**–**c**). Black line indicates a moving window average of ISD. The mean ISD (black line) does not change significantly for different ranges of _Vs30_.