# Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures
## 3 Results
### Fuzzy Linear Regression Results  
Fuzzy linear regression analysis was applied to predict the rotation of every rocking structure using 18 seismic parameters as inputs. Particularly, the fuzzy linear regression equation with triangular fuzzy numbers was derived as follows:  
\[\mathrm{Y}=A_{0}+A_{1}\mathrm{X}_{1}+\ldots+A_{18}\mathrm{X}_{18} \tag{21}\]  
in which \(\mathrm{X}_{\mathrm{i}}\) were the seismic parameters, and \(\mathrm{Y}\) was the rotation of the structure. The results of triangular fuzzy coefficients obtained through the method for every specific case were represented in Table 1 in the Appendix A. In order to evaluate the similarity ratio of the three fuzzy linear regression models containing the same input variables, we used the following equation [67]:  
\[\lambda^{2}(\mathrm{d},\mathrm{d}^{\prime})=\min\biggl{\{}\kappa^{2},\frac{1}{ \kappa^{2}}\biggr{\}} \tag{22}\]  
in which \(\mathrm{d}\) and \(\mathrm{d}^{\prime}\) were different sets of data and \(\kappa\) was a ratio expressed as  
\[\kappa^{2}=\left[\sum_{\mathrm{i}=0}^{\mathrm{n}}{\left(\mathrm{c_{i}}\right) ^{2}\mathrm{d}}\right]\Bigg{/}\left[\sum_{\mathrm{i}=0}^{\mathrm{n}}{\left( \mathrm{c_{i}}\right)^{2}\mathrm{d}}\right] \tag{23}\]  
The results of the similarity ratio \(\lambda_{\mathrm{i},\mathrm{j}}\) of the three models mentioned above are summarized in Table 2. The subscript notation represents the parameter calculated with respect to structure \(\mathrm{i}\) and \(\mathrm{j}\). By evaluating the results, it was concluded that the models of the first and third structures had the highest similarity ratio value, and thus, the regression models were closer in contrast to the other sets. For determining the accuracy of the predicted outputs that emerged from the fuzzy linear regression method, the root mean square error (RMSE) parameter was evaluated with the following equation:  
\[\mathrm{RMSE}=\sqrt{\frac{\sum_{\mathrm{d}=1}^{\mathrm{n}}{\left(\mathrm{e_{t }}\right)^{2}}}{\mathrm{n}}}=\sqrt{\frac{\sum_{\mathrm{t}=1}^{\mathrm{n}}{ \left(\mathrm{A_{t}-F_{t}}\right)^{2}}}{\mathrm{n}}} \tag{24}\]  
in which \(\mathrm{A_{t}}\) and \(\mathrm{F_{t}}\) represented the observed and predicted values, respectively, and \(\mathrm{n}\) corresponded to the number of data sets. The results of the RMSE parameter for the finite rocking response of every structure are depicted in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline \(\lambda\) & **Value** \\ \hline \(\lambda_{1,2}\) & 0.853 \\ \(\lambda_{1,3}\) & 0.99 \\ \(\lambda_{2,3}\) & 0.851 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Estimated similarity ratios for FLR analysis. In order to assess the impact of individual input variables on the similarity ratio of our model, we conducted a comprehensive analysis by constructing a total of 18 fuzzy linear regression models. Each model was formulated to include a distinct input variable, while the resulting output corresponded to the rotation of the structure. This approach allowed us to thoroughly evaluate each input variable's impact on our models' similarity ratio. The results of the triangular fuzzy coefficients are demonstrated in Table A2 in the Appendix A section, and the values of the similarity ratio \(\lambda\) and the RMSE parameter for every model are represented in Tables 4 and 5, respectively. The subscript notation of RMSE indicates the structures under examination for which the parameter is calculated. It has to be highlighted that due to the normalization of IMs, a self-similar response between seemingly different rocking systems is achieved. That fact justifies the observed