## 3 Results
### Regression Using SHAP
Figure 2: Normalized SHAP values \(\bar{\phi}_{j}\) as in Equation (5). First, it can be readily observed that while the log-transformed period has a significant variance for a fixed value of number of stories, the corresponding SHAP values exhibit a much smaller variance. Thus, an analytical curve fitted to the SHAP values instead of the raw data provides a much more accurate representation of the contribution of this feature to the output. The corresponding SHAP values for the opening percentage exhibit a somewhat larger variance. This is attributed to feature interaction, i.e., variation of the contribution of one feature in dependence on the value of a different feature [73]. Such interaction effects can be more accurately captured using SHAP interaction values [70]. However, this adds significant complexity to the derived equations. As shown below, in our case the derived equations achieved excellent accuracy; thus, as our aim was to derive simple analytical equations, this additional complexity was not justified. Given the shape of each individual curve, we experimented with a variety of analytical curves for each one, including quadratic polynomials, square roots, logarithms, and exponentials. The results are shown in Table 3. Note that the contribution of the \"number of spans\" feature was negligible and was ignored; similarly, the curves with the best fits for each feature are presented in Figure 4.  
The equations in Table 3 are valid for non-bare frames, i.e., cases where the opening percentage is <100%. For bare frames, the contribution of the wall stiffness is set to 0 due to  
Figure 4: Analytical curves with the best fit for each feature. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Feature**} & \multirow{2}{*}{**Fitted Curve**} & \multicolumn{2}{c}{**MAE**} & \multicolumn{2}{c}{**RMSE**} \\  & & **Training** & **Test** & **Training** & **Test** \\ \hline \multirow{2}{*}{Number of stories} & \(-0.0056x^{2}+0.233x-1.71\) & 0.0593 & 0.0623 & 0.0763 & 0.0801 \\  & \(0.652\sqrt{x}-2.1\) & 0.0823 & 0.0837 & 0.1011 & 0.1045 \\  & \(0.86\ln x-1.9\) & 0.0546 & 0.0575 & 0.0740 & 0.0797 \\ \hline \multirow{2}{*}{Opening percentage} & \(-0.653x^{2}+1.75x-0.67\) & 0.0758 & 0.0778 & 0.0888 & 0.0914 \\  & \(1.05\sqrt{x}-0.7\) & 0.0725 & 0.0763 & 0.0916 & 0.0955 \\ \hline \multirow{2}{*}{Length of spans} & \(0.066x-0.33\) & 0.0246 & 0.0308 & 0.0330 & 0.0441 \\  & \(0.3\sqrt{x}-0.64\) & 0.0358 & 0.0350 & 0.0420 & 0.0414 \\ \hline \multirow{2}{*}{Wall stiffness} & \(0.0008x^{2}-0.037x+0.254\) & 0.0450 & 0.0439 & 0.0544 & 0.0533 \\  & \(0.515e^{-0.11x}-0.213\) & 0.0431 & 0.0438 & 0.0535 & 0.0538 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitted curves of the SHAP values of each feature. The provided equations for opening percentage and wall stiffness are valid for non-bare frames. the missingness property (Equation (3)). Similarly, when the \"opening percentage\" is equal to 1, as in the bare frame case, the SHAP values of this feature range from 0.27 to 0.325. Thus, the contribution of this feature for bare frames can be obtained using a single value. As is well known, the median minimizes MAE, while the mean minimizes RMSE. In our case, both the mean and median were very close, approximately equal to 0.33. From the above equations, we selected those with the overall best performance. Finally, it should be noted that the above equations only approximate the SHAP values \(\phi_{i}\) in Equation (4). In order for the local approximation (Equation (2)) to be accurate, it is necessary to include the constant \(\phi_{0}\), i.e., the average model prediction. In our case, this was approximately equal to \(-0.21\).