## 5 Results
### Determination of Site Proxies Using GRNN  
A total of 200 GRNN models using different combinations of input parameters were derived, and their results were analyzed and compared in terms of the standard deviation of residuals (predicted-actual values) to the initial standard deviation values for each period, i.e., \(\sigma_{0}(\theta,T_{i})\), and the overall variability \(\sigma_{0\text{{m}}}(\theta)\) defined earlier. Equations (22) and (23) are used to compute the error between predicted and actual values, which in turn are compared with the initial variabilities found using Equations (18)-(21). The standard deviation of residuals for each period and each neural network model, for comparison with the initial variability term \(\sigma_{0}(\theta,T_{i})\), is given by the following:  
\[\varepsilon_{\text{ANN}}(\theta,T_{i})=\sqrt{\frac{1}{11*n_{P}}\sum_{lo=1}^{11 }\sum_{k=1}^{n_{P}}[\log(AF_{\text{ANN}}(P_{k},\theta,T_{i},PGA(lo)))-\log(AF_ {m}(P_{k},\theta,T_{i},PGA(lo)))]^{2}} \tag{22}\]  
where the ANN is GRNN or RBF. The maximum standard deviation over all periods is defined as  
\[\varepsilon_{\text{ANN,max}}(\theta)=Max_{T_{i}}[\varepsilon_{\text{ANN}}( \theta,T_{i})] \tag{23}\]  
The overall error is defined as the average over all periods of the error term, that is,  
\[\varepsilon_{\text{ANN,m}}(\theta)=\frac{1}{n_{T}}\sum_{i=1}^{n_{T}}\varepsilon _{\text{ANN}}(\theta,T_{i}) \tag{24}\]  
To obtain a statistically meaningful insight into the relative performances of each site proxy considered in controlling the amplification factor, the standard deviation for each soil type and period error term \(\varepsilon_{\text{GRNN}}(\theta,T_{i})\) is plotted, together with the initial variability, \(\sigma(\theta,T_{i})\), in Figure 7a,b for clay and sand soil profiles, respectively. Figure 7: Variations in root mean square error and standard deviation of residuals \(\varepsilon_{\text{GRNN}}(\theta,T_{i})\) for various GRNN models with various sets of input site parameters compared to initial variability \(\sigma_{0}(\theta,T_{i})\) (**a**) for clay soil profiles and (**b**) for sand soil profiles. Data are displayed as function of real period. In order to better identify the importance of the site proxy, the reduction in the overall standard deviation of residuals is defined by:  
\[\text{RS}_{\text{ANN,m}}(\theta)=1-\frac{\varepsilon_{\text{ANN,m}}(\theta)}{ \sigma_{\text{ANN,m}}(\theta)} \tag{25}\]  
Table 4 presents the overall standard deviation of residuals as well as the reduction in the standard deviation of residuals for different GRNN models using different input parameter combinations. Figure 7a,b exhibit several noticeable features:  
1. The \(PGA\) is common to all input parameter combinations. The model did not converge when not considering the \(PGA\) nor when considering only a single parameter. This means that the \(PGA\) is a predominant input parameter, and at the very least, a couple of \(PGA\)s with another parameter are needed to achieve convergence. 2. The \(PGA\) and \(f_{0}\) constitute the best couple for the prediction of the amplification factor, producing 64% to 65% reductions in the standard deviation for clay and sand  
soil profiles, respectively. The \(PGA\) performs well for all periods and soil types. In comparison, all other couples of parameters offer a lower reduction in variability, capped at 33%. 3. The triplet (\(PGA\), \(\mathrm{Cv},f_{0}\)) is the most pertinent triplet for predicting the \(AF\), with a standard deviation reduction of more than 71% for clay and 73% for sand profiles. The other triplets (\(PGA\), \(\mathrm{Cv2},f_{0}\)), (\(PGA\), \(V_{smt},f_{0}\)), and (\(PGA\), \(V_{s30},f_{0}\)), which present interesting but slightly lower performances, are also worthy of consideration. However, because parameters such as \(\mathrm{Cv}\) and \(\mathrm{Cv2}\) are difficult to measure in practice and have less physical meaning, the triplet (\(PGA\), \(V_{s30},f_{0}\)) is the triplet retained for predicting the \(AF\).