## 2 Data-Driven Shear Wave Velocity (Vs) Prediction Models
### Extreme Gradient Boosting (XGBoost) Algorithm  
XGBoost algorithm is a powerful machine learning algorithm known for its efficiency, scalability, and ability to handle large datasets with high-dimensional feature spaces and non-linear relationships [31]. In this study, XGBoost was selected to develop a Vs predictive model using seismic cone penetration tests (SCPTs) and SCPTs with a pore water pressure measurement (SCPTu) dataset. Each observation in the dataset represents a measurement from SCPT or SCPTu (e.g., cone resistance (qc), sleeve friction (fs), and depth, among others). These observations (features) are used as an input to the XGBoost model for the prediction of output (in this case the Vs). The XGBoost algorithm has the capability to discover patterns and relationships in the data without requiring pre-defined mathematical models. For instance, the relationships between Vs and geotechnical parameters (e.g., qc and fs) may involve non-linearities and are difficult to express mathematically. XGBoost overcomes this limitation by learning directly from the data, making it particularly suited for SCPT-based Vs predictions where the physical or mathematical relationships may be unknown or too complex to model explicitly. The model uses an ensemble of decision trees, built iteratively to minimize the prediction error by optimizing a loss function. Specifically, XGBoost combines weak learners (individual trees) in a sequential manner, where each tree corrects the errors of its predecessors. In XGBoost, a regularization parameter was incorporated to reduce model complexity and prevent overfitting. The objective function of XGBoost is defined as follows [31]:  
\[\mathcal{L}(\phi)=\sum_{i=1}^{N}l(y_{i},y_{i})+\sum_{i=1}^{T}\Omega(f_{t}) \tag{9}\]  
where \(\mathcal{L}(\phi)\) is objective function, \(l(. )\) is loss function that measures difference between actual value \(y_{i}\) and predicted value \(y_{i}\), \(N\) is number of observations, \(T\) is number of estimators, and \(\Omega(f_{t})\) is regularization parameter defined as  
\[\Omega(f_{t})=\gamma T+\frac{1}{2}\lambda\omega_{t} \tag{10}\]  
where \(T\) is the number of nodes, and \(\omega_{t}\) weight of the leaf nodes, and \(\gamma\) and \(\lambda\) are regularization coefficients that control the model's complexity. The predictive performance and generalization capability of XGBoost depends on the values of its hyperparameter. Therefore, the optimal hyperparameter values must be determined through hyperparameters tuning techniques. A next-generation optimization framework, Optuna, is widely used [32]. To prevent overfitting during the tuning process, k-fold cross validation techniques are commonly employed. This process helps the identification of the optimal parameter configuration that maximizes the model's coefficient of determination (\(R^{2}\)) while avoiding overfitting. To analyze the influence of input features on the optimized XGBoost model, partial dependence plots (PDPs) and SHAPs were utilized. SHAP, introduced by [PERSON] [33], offers a unified framework for interpreting feature importance in the model. SHAP assignsa unique Shapley value (\(\phi_{\text{j}}\)) to each feature based on its contribution to the model output. For a prediction \(f(x)\), SHAP is expressed as [33]:  
\[f(x)=\phi_{0}+\sum_{j=1}^{N}\phi_{\text{j}} \tag{11}\]  
where \(\phi_{0}\) is base value (average model output for the dataset), \(\phi_{\text{j}}\) is SHAP value for feature \(j\) and it is defined as  
\[\phi_{\text{j}}=\sum_{S\subseteq N\smallsetminus\{j\}}\frac{|S|!(|N|-|S|-1)!}{|N |!}[f(S\cup\{j\})-f(S)] \tag{12}\]  
where \(N\) is total number of input features, \(S\) is a subset of features excluding \(j\), \(f(S)\) is the model prediction for the subset \(S\), \(|S|\) is the number of features in subset \(S\), and \(S\subseteq N\smallsetminus\{j\}\) means that \(S\) is a subset of the set of features \(N\) excluding feature \(j\). The feature with the highest absolute SHAP value is considered the most influential in the model's decision-making process.