## 4 Recommendation on Improving the Efficiency of Intensity Measures
### Analysis of Functional Forms
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input Parameters** & **No. of Parameters** & **AIC, IM = PGA** & **AIC, IM = PGV** \\ \hline _IM_ & 1 & 10258.27 & 9604.36 \\ \hline _IM_, _M\({}_{w}\)_ & & 10128.76 & 9519.32 \\ _IM_, _R\({}_{spi}\)_ & & 10258.20 & 9482.93 \\ _IM_, _Vs30_ & & 10186.23 & 9604.32 \\ _IM_, _T\({}_{f}\)_ & & 9001.96 & 8033.23 \\ \hline _IM_, _M\({}_{w}\)_, _R\({}_{spi}\)_ & & 10032.44 & 9480.98 \\ _IM_, _M\({}_{w}\)_, _V\({}_{30}\)_ & & 10053.73 & 9519.09 \\ _IM_, _M\({}_{w}\)_, _T\({}_{f}\)_ & & 8818.62 & 7902.50 \\ _IM_, _R\({}_{spi}\)_, _V\({}_{30}\)_ & & 10188.03 & 9484.23 \\ _IM_, _R\({}_{spi}\)_, _T\({}_{f}\)_ & & 9003.86 & 7845.15 \\ _IM_, _Vs30_, _T\({}_{f}\)_ & & 8901.22 & 8035.17 \\ \hline _IM_, _M\({}_{w}\)_, _R\({}_{spi}\)_, _Vs30_ & & 9930.11 & 9482.91 \\ _IM_, _M\({}_{w}\)_, _R\({}_{spi}\)_, _T\({}_{f}\)_ & & 8677.92 & 7840.91 \\ _IM_, _R\({}_{spi}\)_, _Vs30_, _T\({}_{f}\)_ & & 8902.92 & 7846.04 \\ \hline _IM_, _M\({}_{w}\)_, _R\({}_{spi}\)_, _Vs30_, _T\({}_{f}\)_ & 5 & **8526.51** & **7842.79** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis of Akaike information criterion. Table 7 summarizes efficiencies and proficiencies for various IMs and the different functional forms defined in Equations (24 and 26). We find that the suggested modifications to Equation (1) increase the efficiency and proficiency. We also find that the fundamental period of the structure plays an important role in predicting EDP, hence including the structure's period (Equation (26)), and soil stiffness parameter (_Vs30_) improves the predictions remarkably (bold values in Table 7). We observe significant increases of 45% and 19% for _Sa(T)_ and _ASI_, respectively, with respect to the standard implementation. Table 8 lists the steadfastness index for different functional forms and IMs. For our dataset, we observe lower steadfastness for [PERSON] et al. 's [6] functional form (Equation (24)) compared to the standard form (Equation (1)). We observe a significant increase in steadfastness (by 100%) for _Sa(T)_ when structural period is considered. IMs such as _PGD_ and _CAV_ show good steadfastness, which could be due to their invariance over distance (Figure 2). The steadfastness of _PGV_, _VSI_ and _Sa(T)_ are observed to be larger than those of _PGA_ and _ASI_, which we attribute to the large number of long period frames used in our dataset whose responses correlate with _PGV_ and _VSI_ more than _PGA_ and _ASI_. Next, we examine the sufficiency (\(p\) value) of the different functional forms following Luco and Cornell [5]. Residuals between predicted and observed EDP are plotted with respect to seismological parameters (\(M_{w}\), \(R_{q\pi}\) and _Vs30_), and a \(p\) value is computed. Recall that \(p<0.05\) implies a statistically significant relation, hence indicating in the present cases that the IM is insufficient. Figure 13 depicts \(p\) values with respect to \(M_{w}\), \(R_{q\pi}\) and _Vs30_ (\(p_{M}\), \(p_{R}\), \(p_{Vs30}\)) for different IM-EDP pairs using different functional forms. The dotted line marks the threshold of \(p=0.05\) for classifying sufficient (\(p>0.05\)) and insufficient IMs (\(p<0.05\)). Very low \(p\) values (\(p<10^{-4}\)), indicating a strong relationship between residuals and seismological parameters, are shown at \(p=10^{-4}\) level. With event magnitude (\(M_{w}\)), both Equations (24) and (26) show sufficiency for all IMs, whereas Equation (1) is insufficient.