## 5 A New Prediction Model for Vertical DRFs
### Nonlinear Regression of the Mean DRFs of All the Selected Ground Motions  
Based on several studies [32; 33; 35; 36] and the observations in this study, the polynomial function of natural logarithms of the natural vibration period and damping ratio parameters can be used to describe the relations of the DRFs.  
Considering the theoretical boundary constraints of the DRF with respect to the natural vibration period, the DRF tends to approach unity when the natural vibration period is close to zero or infinity. The proposed functions for predicting the acceleration, displacement, and velocity DRFs are expressed by Equations (2)-(4), respectively.  
\[\mathrm{DRFa}=\mathrm{a}_{0}+\mathrm{a}_{1}\mathrm{cos}(\mathrm{T}\times \mathrm{w})+\mathrm{b}_{1}\mathrm{sin}(\mathrm{T}\times\mathrm{w})+\mathrm{a}_ {2}\mathrm{cos}(2\times\mathrm{T}\times\mathrm{w})+\mathrm{b}_{2}\mathrm{sin}( 2\times\mathrm{T}\times\mathrm{w}) \tag{2}\]  
\[\mathrm{DRFd}=\mathrm{a}_{0}+\mathrm{a}_{1}\mathrm{cos}(\mathrm{T}\times \mathrm{w})+\mathrm{b}_{1}\mathrm{sin}(\mathrm{T}\times\mathrm{w})+\mathrm{a}_ {2}\mathrm{cos}(2\times\mathrm{T}\times\mathrm{w})+\mathrm{b}_{2}\mathrm{sin}( 2\times\mathrm{T}\times\mathrm{w}) \tag{3}\]  
\[\mathrm{DRFv}=\begin{cases}\mathrm{a}_{0}+\mathrm{a}_{1}\mathrm{cos}(\mathrm{T }\times\mathrm{w})+\mathrm{b}_{1}\mathrm{sin}(\mathrm{T}\times\mathrm{w})+ \mathrm{a}_{2}\mathrm{cos}(2\times\mathrm{T}\times\mathrm{w})+\mathrm{b}_{2} \mathrm{sin}(2\times\mathrm{T}\times\mathrm{w}),&\mathrm{For}\ \mathrm{T}<0.10\ \mathrm{s}\\ \mathrm{a}\times\mathrm{T}^{\mathrm{b}}+\mathrm{c},&\mathrm{For}\ \mathrm{T}\geq 0.10\ \mathrm{s} \end{cases} \tag{4}\]  
Tables 2-4 show the values of the fitting coefficients for the prediction of the mean DRFs. The \(\mathrm{R}^{2}\) between the calculated and predicted mean DRFs for the database is greater than 0.99. The RMSE is predominantly less than 0.02 and, reaching a maximum value of 0.065, is very low, still implying strong correlation and low error. In addition, the SSE values are also given in the corresponding tables for comparison of each model. According to the calculations, the SSE ranges around 0.001-0.089, 0-0.09, and 0-0.329 for the DRFa, DRFd, and DRFv, respectively. It seems that the SSE is slightly higher, resulting from the simplified equation, for \(\mathrm{T}\geq 0.10\ \mathrm{s}\). Nevertheless, the \(\mathrm{R}^{2}\) is higher than 0.993, and the RMSE is quite low and equal to 0.013, which implies the accuracy of the equation. Based on these statistical results, it can be said that the proposed DRF models can capture the main features of the DRFs as a function of the vibration period and the damping ratio.  
A comparison of the prediction of the proposed DRF models with the real mean values of the DRFs is given in Figure 13, and the figure clearly presents the prediction accuracy of the proposed equations. Despite the small differences between the real and predicted values for very short periods (\(\mathrm{T}<0.03\ \mathrm{s}\)), it is apparent that the proposed DRF models have a strong robustness and accuracy.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & **1\%** & **3\%** & **10\%** & **15\%** & **20\%** & **30\%** & **40\%** \\ \hline \multirow{8}{*}{
\begin{tabular}{c} **Courier** \\ **