## 1 Introduction
They compared a feed forward neural network model with one hidden layer and a Neuro Fuzzy (ANFIS) model [62] and they tried to derive equations for computing the FUP values. One of their most remarkable conclusions was that the most effective parameter for calculating the fundamental period was the masonry infill ratios of the frames (opening ratios). [PERSON] et al. [63], also working on the Asteris training dataset [56], focused on the so-called boosting ensemble technique and employed three ML models, namely, LightGBM, Gradient Boosting Decision Trees, and CatBoost. Moreover, by applying the \"_Multivariate Adaptive Regression Spline_\" technique, they arrived at an approximate equation for computing the FUP values. From the comparisons they made with similar relations from the literature, they concluded that the combined use of the _LightGBM_ model and the Multivariate Adaptive Regression Spline technique leads to the most efficient relation both in terms of the level of accuracy and avoidance of overfitting effects. [PERSON] et al. [64] compared the outcome of four ML algorithms (artificial neural network, \(k\)-nearest Neighbors, support vector regression, and random forest). Emphasizing interpretability, they employed three different explainability techniques, namely, SHAP, Local Interpretable Model-Agnostic Explanations (LIME), and Partial Dependence Plots (PDPs), to analyze the predictions of their models. It should be clarified that such post hoc explainability methods analyze and interpret the decision-making process of a trained model after it has made predictions, providing insights into how the model arrived at its outputs. In addition, the authors developed an (explainability) XAI-based user interface [65] through which it is possible to input the appropriate structural data/parameters and automatically extract the estimated fundamental period. Using this application, it is possible to achieve the selection of structural parameters in real time in order to optimize the design of the infilled RC frames and interpret how each structural parameter influences the final result. This research has strong elements of novelty, as it introduces an explainable modeling approach using the SHapley Additive exPlanations (SHAP values) technique. The SHAP machine learning explainability methodology is used to provide interpretable analytical equations that approximate the behavior of the underlying ML model. They quantify each feature's importance based on the predictive power that it contributes, thereby accounting for complex feature interactions. They employ game theory to assign credit to each feature or feature value, resulting in good prediction capacity on the part of the developed model. The way SHAP works is to decompose the output of a model by summing the impacts of each feature. As shown in Section 2.4, SHAP values are feature attribution methods, i.e., they analyze the predictions of ML models into components pertaining to each individual feature. However, in the currently established literature this methodology is employed as an explainability tool, i.e., to quantify the effect of each input parameter on the model's prediction [60; 64]. In our proposed formulation, SHAP values are not only used in this regard; in addition, departing from the currently established literature, they are also employed as a basis for the creation of interpretable and simplified analytical equations that approximate the behavior of the underlying ML model. Thus, the benefit of the proposed formulation is threefold: first, the analytical equations are derived independently and sequentially; second, again by utilizing the SHAP values, features are inserted into the simplified model in an informed manner starting from the most important; lastly, as determined by the specific application at hand, this allows for the determination of a controlled tradeoff between accuracy and model complexity.