# Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures
## 4 Conclusions  
The assessment of rocking response presents a formidable challenge due to its significant nonlinearity. In this paper, two distinct methodologies to evaluate the finite rocking rotations and rocking overturn of three typical rocking systems were studied. Specifically, we employed fuzzy linear regression with triangular fuzzy numbers and a hybrid model that combined logistic regression and fuzzy logic. These approaches were chosen to achieve our research objectives effectively. The fuzzy linear regression method was applied to determine the finite rocking rotations and the similarity ratio of the three structures. The model consisted of 18 seismic parameters, which were extracted from the recorded seismic accelerograms and considered as inputs. Extensive analysis was carried out to evaluate each input variable's influence on the model's similarity ratio. This procedure involved creating a set of 18 fuzzy linear regression models. This comprehensive approach highlighted the significance of each parameter in shaping the model's outcomes. According to the results, high similarity ratios \begin{table}
\begin{tabular}{c c c c} \hline
**IM** & **Log Loss\({}_{1}\)** & **Log Loss\({}_{2}\)** & **Log Loss\({}_{3}\)** \\ \hline PCA/(g tan(\(\alpha\))) & 0.013 & 0.035 & 0.04 \\ p PGV/(g tan(\(\alpha\))) & 0.014 & 0.022 & 0.027 \\ p\({}^{2}\) PGD/(g tan(\(\alpha\))) & 0.013 & 0.037 & 0.045 \\ p T\({}_{m}\) & 0.014 & 0.036 & 0.044 \\ p PGV/PGA & 0.012 & 0.034 & 0.044 \\ p PGD/PGV & 0.013 & 0.037 & 0.050 \\ A\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & 0.051 & 0.032 & 0.035 \\ p V\({}_{\text{RMS}}\) /(g tan(\(\alpha\))) & 0.011 & 0.042 & 0.036 \\ p\({}^{2}\) D\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & 0.013 & 0.040 & 0.045 \\ p\({}^{2}\) L\({}_{\text{cs}}\)/(g tan(\(\alpha\))) & 0.011 & 0.020 & 0.031 \\ p\({}^{2}\) L\({}_{\text{cc}}\)/(g tan(\(\alpha\))) & 0.011 & 0.023 & 0.034 \\ p\({}_{1}\)/(g tan(\(\alpha\))) & 0.012 & 0.031 & 0.052 \\ p\({}^{8.0}\) IC/(g tan(\(\alpha\))) \({}^{3/2}\) & 0.012 & 0.031 & 0.042 \\ p\({}^{1.25}\) I\({}_{\text{f}}\)/(g tan(\(\alpha\))) & 0.010 & 0.021 & 0.032 \\ p\({}^{5}\) SED/(g tan(\(\alpha\))) \({}^{2}\) & 0.013 & 0.037 & 0.045 \\ p\({}^{4}\)ig & 0.013 & 0.037 & 0.046 \\ p CAV/(g tan(\(\alpha\))) & 0.019 & 0.030 & 0.040 \\ p\({}^{2}\) CAD/(g tan(\(\alpha\))) & 0.015 & 0.037 & 0.020 \\ \hline \end{tabular}
\end{table} 
Table 8: The results of the log loss function for every model of logistic regression. \begin{table}
\begin{tabular}{c c c c} \hline
**IM** & \(\mathbf{\lambda_{1,2}}\) & \(\mathbf{\lambda_{1,3}}\) & \(\mathbf{\lambda_{2,3}}\) \\ \hline PCA/(g tan(\(\alpha\))) & 0.83 & 0.54 & 0.65 \\ p FGV/(g tan(\(\alpha\))) & 0.78 & 0.61 & 0.77 \\ p\({}^{2}\) PGD/(g tan(\(\alpha\))) & 0.004 & 0.34 & 0.01 \\ p T\({}_{m}\) & 0.59 & 0.35 & 0.58 \\ p PGV/PGA & 0.56 & 0.34 & 0.60 \\ p PGD/PGV & 0.75 & 0.97 & 0.73 \\ A\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & 0.76 & 0.38 & 0.50 \\ p V\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & 0.28 & 0.46 & 0.62 \\ p\({}^{2}\) D\({}_{\text{RMS}}\)/(g tan(\(\alpha\))) & 0.02 & 0.04 & 0.56 \\ p\({}^{2}\) L\({}_{\text{c}}\)/(g tan(\(\alpha\))) & 0.61 & 0.14 & 0.62 \\ p\({}^{2}\) L\({}_{\text{cc}}\)/(g tan(\(\alpha\))) & 0.52 & 0.32 & 0.61 \\ p\({}_{1}\)/(g tan(\(\alpha\))) & 0.89 & 0.18 & 0.22 \\ p\({}^{0.50}\) IC/(g tan(\(\alpha\))) \({}^{3/2}\) & 0.79 & 0.39 & 0.49 \\ p\({}^{1.25}\) I\({}_{\text{f}}\)/(g tan(\(\alpha\))) & 0.81 & 0.68 & 0.85 \\ p\({}^{3}\) SED/(g tan(\(\alpha\))) \({}^{2}\) & 0.00 & 0.00 & 0.03 \\ p t\({}_{\text{RMS}}\) & 0.86 & 0.15 & 0.18 \\ p CAV/(g tan(\(\alpha\))) & 0.95 & 0.73 & 0.81 \\ p\({}^{2}\) CAD/(g tan(\(\alpha\))) & 0.42 & 0.12 & 0.27 \\ \hline \end{tabular}
\end{table} 
Table 9: The results of estimated similarity ratios for every model of hybrid analysis. were observed among the various structures in every case.