# An Alternative Procedure for the Description of Seismic Intensity Parameter-Based Damage Potential
## 3 Results and Discussion  
### Training the Multiple Regression Models  
As previously stated, a correlation study was initially conducted to determine the intensity parameters that correlate the most with the associated Damage Indices. Figure 2 illustrates the rank correlation coefficients of all IMs for a given DI in descending order. A lower limit (threshold) of \(\rho_{\text{lim}}=90\%\) was adopted to select the IMs used in the multiple regression analyses. The selected IMs and their respective rank correlation coefficients are presented in Table 5. The most highly correlated IMs come from the Spectral and Energy categories, with only two out of the ten selected belonging to the Peak category (PGV and Sustained Maximum Velocity). An important observation regarding the Peak-based IMs is that they are velocity-related. Velocity parameters often characterize a pulse-like strong motion, in which the pulse is usually found within the velocity time history. Specifically, PGV has been identified as a key parameter in pulse-like excitations with forward directivity effects [47]. Using the selected 10 IMs, a series of Multiple Regression Analyses were performed. Ordinary Least Squares, Forward Stepwise Selection, and Backward Stepwise Selection iterative procedures [25] were employed both on the conventional regression models (Model Set 1) and the proposed models (Model Set 2) by raising each IM to an exponent dependent \begin{table}
\begin{tabular}{c c c} \hline \hline
\begin{tabular}{c} Ordinary Least Squares \\ Forward Stepwise Selection \\ Backward Stepwise Selection \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Examined regression expressions and methods. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Intensity Measure** & **Normalized** & **Maximum** & **Modified** \\  & **MISDR** & **Softening** & **Park–Ang** \\ \hline Housner Intensity & 96.79\% & 95.32\% & 94.71\% \\ S\({}_{\text{a,avg}}\) & 95.98\% & 95.12\% & 94.12\% \\ Specific Energy Density & 93.75\% & 95.31\% & 95.79\% \\ Velocity Spectrum Intensity & 96.47\% & 92.81\% & 92.84\% \\ PGV & 94.85\% & 92.00\% & 92.20\% \\ S\({}_{\text{v,T1}}\)(\%) & 95.54\% & 91.88\% & 90.49\% \\ S\({}_{\text{a,T1}}\)(\%) & 93.76\% & 93.38\% & 90.09\% \\ S\({}_{\text{d,T1}}\)(\%) & 93.57\% & 93.32\% & 90.07\% \\ Sustained Maximum Velocity & 91.43\% & 91.01\% & 93.03\% \\ Velocity RMS & 91.72\% & 91.73\% & 91.69\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Selected Intensity Measures and rank correlation coefficients with the examined Damage Indices. As an example, the application of Equations (5), (6a) and (6b) for the prediction of the Maximum Softening DI for Model 1.2 and Model 2.2 is provided below in Equations (7) and (8), respectively:  
\[\delta_{\rm M}=-0.00396\cdot\mathrm{SI}_{\rm H}+2.27422\cdot\mathrm{S}_{\rm a,avg }+0.00147\cdot\mathrm{PGV}+0.00317\cdot\mathrm{SMV} \tag{7}\]  
\[\delta_{\rm M}=-0.03541\cdot\mathrm{SI}_{\rm H}\frac{0.952-0.9}{1-0.9}+2.01052 \cdot\mathrm{S}_{\rm a,avg}\frac{0.9512-0.9}{1-40.9} \tag{8}\]  
\[+0.00079\cdot\mathrm{SED}\frac{0.953-0.9}{1-0.9}-0.06588\cdot\mathrm{SI}_{ \rm v}\frac{0.952-0.9}{1-0.9}\]