## 4 Proposed Functions and Validation  
The predictive models developed for the inelastic displacement ratio and residual displacement demand of structural systems are useful tools in terms of the displacement-based design philosophy. The current study mainly focuses on the inelastic displacement ratio and the residual displacement demand of self-centering systems with soil-structure interaction under near-fault pulse-type ground motions; thus, two predictive models for these structural response parameters were developed. A comprehensive nonlinear regression analysis was conducted to derive simplified functions for the inelastic displacement ratio and residual displacement. As the first step of regression analysis, the relations of the inelastic displacement ratio and the residual displacement ratio of interacting systems with the normalized period, ductility, energy dissipation ratio, post-yield stiffness ratio, and aspect ratio were regressed from statistical analysis, and the most effective parameters were obtained. Using the Levenberg-Marquardt method [61], nonlinear regression analyses were conducted to derive simplified functions to estimate the inelastic displacement ratio and the residual displacement of self-centering interacting SDOF systems individually. The proposed function for inelastic displacement ratio (\(\mathrm{C_{\mu}}\)) can be expressed as  
\[\mathrm{C_{\mu}}=\mathrm{a_{0}}+\mathrm{e}^{(\mathrm{a_{1}}+\mathrm{a_{2}} \cdot(\frac{\widetilde{\mathrm{  
Figure 16 shows the comparison of the inelastic displacement ratios (left) and residual displacement ratios (right) for the self-centering structure between the nonlinear analysis results and the predictive models. Comparisons are presented for all values of considered variables. It is evident that the developed predictive models can accurately estimate the inelastic displacement ratio and residual displacement ratios for self-centering structures subjected to near-fault pulse-like ground motions. In other words, the results indicate that the statistical analysis and the calculation results fit well for considered earthquake database.  
Two commonly used metrics of the \"goodness of fit\" of the results of the nonlinear regression analyses are computed for the proposed equations. Table 4 reports the correlation coefficient of the proposed functions and standard error values calculated for the cases under consideration. The standard error definition and correlation coefficient are presented by Equations (11) and (12). It is clear from the measurements that the suggested functions for estimating residual displacements and inelastic displacement ratios yield accurate results.  
\[\text{SE}_{\text{of prediction}}=\sqrt{\left\{\frac{\Sigma\Big{(}\text{x}_{ \text{exact}}-\text{x}_{\text{predicted}}\Big{)}^{2}}{\text{n}}\right\}} \tag{11}\]  
Figure 16: Comparison of the predicted values with the statistical results. Figure 15: Fitness of the regressed functions. In order to verify the accuracy of the methodology, the mean inelastic displacement ratio and residual displacement demands obtained from direct time history analysis using the self-centering flag-shaped hysteresis model for interacting systems is compared with the proposed equations in this study. To this purpose, new numerical analyses are conducted for a new ensemble of five ground motions selected from the PEER database representing pulse-type ground motions. Previously, these records were not utilized in the statistical analyses. Near-field pulse-like ground motions used for verification are given in Table 5. Figure 17 illustrates the comparisons of the numerical results of the new database with the results computed using the proposed formulas for the residual displacement ratios and inelastic displacement ratios for self-centering structures. From the comparisons, it can be perceived that the proposed formulas can accurately predict the mean C\({}_{\mu}\) and D\({}_{\rm s}\) spectrum.