## 4 Summary and Conclusions  
In the framework of the present study, a novel regression approach is presented based on ML explainability methodologies, in particular, the so-called SHapley Additive exPlanations (SHAP values). SHAP values are commonly employed in the literature as an explainability tool. However, in the present study, they are used as a basis to provide interpretable analytical equations that approximate the behavior of the underlying machine learning model. We have implemented our proposed methodology on the problem of estimating the fundamental period of buildings, which is a very important parameter \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Log-Transformed**} & \multicolumn{3}{c}{**Original Space**} \\  & **Training** & **Test** & **Full Dataset** & **Training** & **Test** & **Full Dataset** \\ \hline MAE & 0.1185 & 0.1217 & 0.1193 & 0.1174 & 0.1193 & 0.1179 \\ RMSE & 0.1485 & 0.1515 & 0.1493 & 0.1725 & 0.1707 & 0.1721 \\ \(R^{2}\) & 0.9706 & 0.9681 & 0.9700 & 0.9522 & 0.9510 & 0.9520 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression metrics of the simplified analytical equations. in the context of seismic design of new buildings and evaluating the seismic capacity of existing buildings. Similar problems of great importance in the engineering community have been addressed in various design codes by providing simplified analytical equations that approximate the target variable. On the other hand, ML models have demonstrated the ability to tackle such problems and produce state-of-the-art results with excellent accuracy. This is demonstrated in the present paper, where our fully trained ML model achieved a coefficient of determination \(R^{2}\approx 0.99\). In addition, our model obtained MAE \(\approx 0.05\) in both the log-transformed and original spaces, while the corresponding RMSE ranged from \(\approx\)0.065-0.085. However, ML models often lack interpretability, which hinders their full adoption by the engineering community. Our proposed novel methodology can help to bridge this gap. On the one hand, we employ fully trained ML models to achieve state-of-the-art results; on the other, we utilize an explainability technique based on SHAP values as a basis for providing interpretable analytical equations, which offers distinct advantages compared to the previously established literature. First, due to the additive nature of SHAP values, our proposed methodology allows for the incremental addition of terms in simplified analytical equations. This in turn allows for a controlled tradeoff between model complexity and the desired level of accuracy according to the application at hand. Second, as Figure 2 demonstrates, SHAP values allow the percentage of influence of each individual parameter to be quantified. Specifically, we found that approximately 50% on average of the total model predictions were based on the \"Number of Stories\" feature, while an additional 30% were attributed to the \"Opening Percentage\" feature. Approximately 10% of predictions were attributed to each of the \"Length of Spans\" and \"Masonry Wall Stiffness\" features, while the effect of the \"Number of Spans\" feature was found to be negligible. Thus, the controlled tradeoff between accuracy and model complexity in our proposed methodology is not carried out blindly, but rather in an informed manner guided by the feature importance ranking provided by the SHAP values. Due to the fact that SHAP values approximate the behavior of the underlying ML model, the proposed methodology can provide simplified equations with comparable accuracy to that of the fully trained ML model, as demonstrated in Tables 2 and 4. Indeed, the fully trained model achieved \(R^{2}\approx 0.99\), while the simplified equations achieved \(R^{2}\approx 0.97\) and \(R^{2}\approx 0.95\) in the log-transformed and original spaces, respectively.