## 5 Results
### Variation in Amplification Factors for Specific Period Ranges Using RBF  
Design codes take into account site effects via multiplying the design spectra by amplification factors (i.e., \(F(T)\) in the CNBC). These amplification factors vary with the period of the structure and soil properties. For the sake of simplicity and with respect to earlier versions of codes (i.e., CNB2005), we consider averaged site factors for three specific ranges: \(\mathrm{F_{a}}\) for short periods; \(\mathrm{F_{v}}\) for intermediate periods; and \(\mathrm{F_{I}}\) for long periods. In this section, we compute these amplification factors, \(\mathrm{F_{a}}\), \(\mathrm{F_{v}}\), and \(\mathrm{F_{I}}\), from the RBF model based on the triplet \((PGA,f_{0},V_{S30})\), which proved to be efficient and accurate. Figures 8-11 display the dependence of these three factors, \(PGA\), \(f_{0}\), and \(V_{S30}\). The [0.15, 40 Hz] interval is considered for \(f_{0}\) in all cases, covering the full range of the database. In Figure 8, the results are computed for a \(V_{S30}\) value varying within the frequency range studied, while in the other figures, typical low (150 m/s), mid (270 m/s), high (450 m/s), and very high (640 m/s) values of \(V_{S30}\) are considered. The intensity of the seismic excitation is displayed for three (03) discrete levels: \(PGA\) = 0.05 g, 0.25 g, and 0.75 g.  
Figure 8: Variations in the short period amplification factors, \(\mathrm{F_{a}}\), with \(f_{0}\) for different \(PGA\) values: 0.05 g, 0.25 g, and 0.75 g. (**a**) Clay soil profiles; (**b**) sand soil profiles. Table 5 presents a summary of the statistical parameters obtained by the RBF triplet predictions. As shown, the initial standard deviation of the residuals is greatly reduced by a factor ranging roughly from 3 to 30. Excellent coefficients of determination, R\({}^{2}\), are obtained, and are greater than 0.91, ranging from 0.96 to 0.99 for F\({}_{\rm a}\) and F\({}_{\rm v}\) for both soil types. Figure 10: Variation in amplification factor at mid period F\({}_{\rm v}\) with \(f_{0}\) and _PGA_ for four values of \(V_{\rm s30}\) (150, 270, 450, and 640 m/s) (**a**) for clay soil profiles and (**b**) for sand soil profiles. Figure 9: Variation in amplification factor at short period Fa with \(f_{0}\) and _PGA_ for four values of \(V_{\rm s30}\) (150, 270, 450, and 640 m/s) (**a**) for clay soil profiles and (**b**) for sand soil profiles. The values of \(\rm F_{a}\), \(\rm F_{v}\), and \(\rm F_{I}\) may be predicted as a function of \(f_{0}\), \(V_{S30}\), and \(PGA\) using the following explicit equation based on RBF models:  
\[\log(\rm Fa)=LW^{T}a1+b2 \tag{26}\]  
where \(a1\) is a vector of \(\rm t\) lines output by the hidden layer, with \(\rm t\) being the number of key neurons in the hidden layer, determined from the training phase. It is estimated from their Euclidian distance (see Section 3 of Supplementary File) using the following equation:  
\[a1=\exp\left(-\left[b1\sqrt{\left[\left(we1\right)^{2}+\left(we2\right)^{2}+ \left(we3\right)^{2}\right]}\right]^{2}\right) \tag{27}\] \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Statistical Summary for Amplification at** & \(\rm\bf F_{a}\) & \(\rm\bf F_{v}\) & \(\rm\bf F_{I}\) \\
**Specific Period Ranges** & & & \\ \hline Overall standard deviation \(\varepsilon_{\rm RBF,m}\) (Clay, \(\theta=0\)) & 0.0441 & 0.0436 & 0.0363 \\ \hline Initial standard deviation (\(\theta=0\)) & 0.3759 & 0.1632 & 0.0917 \\ \hline \(\rm R^{2}\) (all database) (\(\theta=0\)) & 0.9931 & 0.9637 & 0.9184 \\ \hline Number of key neurons & 16 & 10 & 10 \\ \hline Standard deviation for all databases & & & \\ \(\varepsilon_{\rm RBF,m}(\theta=1)\) (soil type: sand) & 0.0504 & 0.0437 & 0.0341 \\ \hline Initial standard deviation (\(\theta=1\)) & 0.4551 & 0.2228 & 0.0967 \\ \hline \(\rm R^{2}\) (all database) (\(\theta=1\)) & 0.9939 & 0.9805 & 0.9358 \\ \hline Number of key neurons (\(\theta=1\)) & 12 & 10 & 9 \\ \hline \hline \end{tabular}
\end{table}