## 2 Data Preprocessing
### The XGB Algorithm  
#### 2.2.1 Principle of XGB  
The gradient boosted decision tree (GBDT) algorithm employs gradient descent to iteratively create new trees, aiming to minimize the objective function [41]. The XGB algorithm is derived from GBDT and is widely applied to classification and regression tasks. It is a potent machine learning algorithm, characterized by notable advantages such as high predictive performance, robust modeling capabilities for complex relationships, and excellent performance on large-scale datasets [42; 43; 44]. XGB possesses an inherent advantage over models like neural networks in terms of mitigating overfitting. This advantage stems from the algorithm's built-in regularization terms and pruning mechanisms, allowing for effective control of model complexity. This characteristic makes XGB particularly adept at handling relatively small datasets or situations with high data noise, contributing to its widespread adoption in various practical \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Input Variable** & **Minimum Value** & **Maximum Value** & **Average Value** \\ \hline Moisture content (\%) & 10.48 & 51.2 & 29.98 \\ \hline Dry density (g-cm\({}^{-3}\)) & 1.02 & 2.08 & 1.49 \\ \hline Temperature (\({}^{\circ}\)C) & \(-\)0.5 & \(-\)25 & \(-\)7.15 \\ \hline Strain rate (s\({}^{-1}\)) & \(1.05\times 10^{-7}\) & 0.07 & - \\ \hline Sand content (\%) & 0 & 90 & - \\ \hline Powder content (\%) & 5 & 94.66 & - \\ \hline Sticky grain content (\%) & 3.43 & 89.5 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data distribution for the XGB model. applications. By incorporating regression trees, the model continuously learns feature partitions for the data at each internal node. It generates a new classification and regression tree (CART) based on the residuals of the previous model, keeps track of the total number of trees in the training model, and assigns weights to the predictions from each tree to obtain the final prediction value. In this study, Python 3.9 was utilized to construct the XGB model for data training. As an integrated algorithm, XGB combines multiple weak learners to produce the final learning outcome. During the processing of the objective function, a regularization term is introduced to decrease the model's variance and prevent overfitting. Additionally, XGB employs a second-order Taylor expansion to enhance the convergence speed and improve the prediction accuracy. The predicted value for the \(i\)-th sample at the \(t\)-th iteration is given by the following equation:  
\[\hat{g}_{i}^{(t)}=\hat{g}_{i}^{(t-1)}+f_{t}(x_{i}) \tag{2}\]  
where \(\hat{g}_{i}^{(t)}\) is the predicted value of sample \(i\) after the \(t\)-th iteration, \(\hat{g}_{i}^{(t-1)}\) is the predicted value of the \(t-1\)st tree, \(f_{t}(x_{i})\) is the model of the \(t\)-th tree, and \(x_{i}\) is the input value. The XGB objective function form is shown in Equation (3):  
\[\mathcal{O}\text{b}\hat{g}^{(t)}=\sum_{i=1}^{n}l(y_{i},\hat{y}_{i}^{(t)})+ \Omega(f_{t})+c \tag{3}\]  
where \(l\left(y_{i},\hat{y}_{i}^{(t)}\right)\) is the loss function of the model, \(y_{i}\) is the true value of the \(i\)-th sample, \(n\) is the total number of samples, \(\Omega(f_{t})\) is the regularization term, and \(c\) is a constant term. The specific form of the regularization term is  
\[\Omega(f_{t})=\gamma T+\frac{1}{2}\lambda||\omega||^{2} \tag{4}\]  
where \(\gamma\) and \(\lambda\) are penalty coefficients, \(T\) is the number of leaves, and \(\omega\) is the weight of the leaf node. The second-order Taylor approximation is expanded for the objective function, omitting higher-order infinitesimal and constant terms:  
\[\mathcal{O}\text{b}\hat{g}^{(t)}=\sum_{i=1}^{n}\left[l(y_{i},\hat{y}_{i}^{(t)} )+g_{i}f_{t}(x_{i})+\frac{1}{2}h_{i}\hat{f}_{i}^{2}(x_{i})\right]+\Omega(f_{t} )+c \tag{5}\]  
where \(g_{i}\) and \(h_{i}\) are Taylor expansion constant terms.