# Fuzzy Multivariate Regression Models for Seismic Assessment of Rocking Structures
## 4 Conclusions
More specifically, in the case where the fuzzy model included all input variables, the similarity ratio of the structures had a value of 0.99 between structures 1 and 3, which indicated that these regression models were closer, in contrast to the other sets. Similarly, a hybrid model that combined logistic regression and fuzzy logic was applied to evaluate the rocking binary overturn of three typical rocking systems and, thus, their similarity ratio. The application of this model necessitated the computation of probability \(\pi\) of structure failures with logistic regression. Then, the computed probability was subjected to the FLR method to determine the similarity ratio of every system. In this analysis, the similarity ratios in every case were lower, contrary to the first method, mainly when the fuzzy model included all input variables. The highest similarity ratio was observed between the 2 and 3 structures (\(\lambda_{2,3}=0.533\)). The model that exhibited the highest similarity ratio among the structures was the model that included only the variable p PGD/PGV as an input, with a similarity ratio value of \(\lambda_{1,3}=0.97\). This result demonstrated that these regression models were closer for structures 1 and 3. In order to assess the outputs that emerged from the fuzzy linear regression and the logistic regression, the root mean square error (RMSE) parameter and the log loss function were determined, respectively. As indicated, both methods demonstrated satisfactory outcomes, showing minimal deviations from the observed values. However, the results revealed that in both methods, in cases where the models were composed of all input variables, the coefficients expressing the error of the methods provided better results compared to the cases where the models consisted of only one input. For example, in the fuzzy linear regression method, the smallest RMSE parameter (RMSE = 3.54) was observed where the model was composed of all input parameters of the third structure. Similarly, in the logistic regression method, the smallest log loss function (long loss = 0.008) was evaluated in the model consisting of all seismic input parameters of the first structure. This result indicated a small deviation between the predicted and observed values, highlighting the method's accuracy and precision. In conclusion, the fuzzy linear regression method and the hybrid model that combined logistic regression and fuzzy logic are effective modeling tools that can be used accurately for regression and classification problems, respectively. Also, the parameters of triangular fuzzy numbers can be utilized to determine the applied model's similarity ratio. Thus, these models result in accurate seismic assessment of rocking structures and serve as valuable modeling tools in the realm of engineering. The high similarity ratios, stemming from the dimensionless IMs, indicate that the proposed methodology may result in a universal assessment tool that can be applied to evaluate the response of different rocking structures. However, a thorough investigation, including systems with various geometries and boundary conditions, should be conducted in order to extend the potential application of the adopted methodology. Conceptualization, F.I.G., A.E. and B.K.P. ; methodology, F.I.G., K.E.B., I.E.K., A.E. and B.K.P. ; software, F.I.G., K.E.B., I.E.K., A.E. and B.K.P. ; validation, F.I.G., K.E.B. and I.E.K. ; formal analysis, F.I.G. and B.K.P. ; investigation, F.I.G., K.E.B., I.E.K., A.E. and B.K.P. ; resources, K.E.B., I.E.K. and A.E. ; data curation, F.I.G., K.E.B., I.E.K., A.E. and B.K.P. ; writing--original draft preparation, F.I.G., K.E.B. and I.E.K. ; writing--review and editing, A.E. and B.K.P. ; visualization, F.I.G. and K.E.B. ; supervision, A.E. and B.K.P. ; project administration A.E. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The data presented in this study are available on request from the corresponding author.