## 3 Prediction Model of Pile Axial Stress
### Selection of Algorithms  
In this study, the authors conduct a comparative analysis of the predictive models of pile axial stress derived in each case in terms of their performance, utilizing various machine-learning algorithms. Specifically, eight algorithms are considered, namely, Linear Regression (LR), Lasso (L1), Ridge (L2), Multi-layered Perceptron (MLP), Support Vector Machine (SVR), Random Forest (RF), Extreme Gradient Boosting (XGB), and Light Gradient Boosting (LightGBM).  
LR is a straightforward, rapid, and comprehensible algorithm used for the linear prediction of regression problems. However, it has the drawback of being less accurate. L1 and L2 are algorithms designed to address the shortcomings of LR by imposing additional constraints to prevent overfitting of the predictive models derived through LR. L1 and L2 are restricted via L1-norm and L2-norm penalties, respectively. A detailed explanation of the LR, L1, and L2 algorithms can be found in the following references: [23], [24], and [25].  
MLP is a multilayer neural network structure with multiple perceptrons in various layers, and is a fundamental algorithm of artificial neural networks, containing one or more hidden layers between the input and output layers. For further details on the MLP algorithm, the reader is referred to [26].  
SVR is one of the most popular algorithms used in machine learning and separates data by choosing the hyperplane that is furthest from the data. It is less sensitive to erroneous data, less prone to overfitting, and more user-friendly than neural network algorithms such as MLP. However, it has the drawbacks of taking longer to learn and being less interpretable. A detailed explanation of the SVR algorithm can be found in [27].  
Random Forest is one of the representative ensemble algorithms that combines multiple decision-tree algorithms to derive a predictive model. It uses a bagging method for training and has the advantages of good generalization performance, ease of parameter ad  
Figure 9: Correlation between the features: (**a**) case 1; (**b**) case 2; (**c**) case 3; (**d**) case 4.  
justment, and poor overfitting. However, it also has the disadvantages of difficult analysis for individual trees and poor model performance when the dimensionality is large and the amount of data is not significant. A description of RF can be found in [28].  
XGB is a boosting-based algorithm that has won several competitions and is one of the ensemble algorithms that, like RF, uses decision trees. For a description of XGB, the reader should refer to [29].  
LightGBM, like XGB and RF, is an ensemble algorithm that utilizes a decision tree. Unlike XGB, which expands the tree in a level-wise manner, LightGBM expands the tree leaf-wise. This reduces the number of losses when expanding the same leaf, and the algorithm has the advantage of being faster than XGB. For an explanation of LightGBM, the reader can refer to [30].  
In machine learning, the data are typically divided into training, validation, and test data in a ratio of 5:3:2 to prevent overfitting, with only the training data being used for algorithm selection. The learning results of each algorithm were evaluated using the root-mean-square error (RMSE), and the algorithm results are summarized in Table 4.  
The results reveal that the algorithms RF, XGB, and LightGBM exhibited the best performance, while MLP and SVR, which are based on artificial neural networks (ANN), demonstrated a lower performance. Therefore, hyper-parameter tuning was performed only on RF, XGB, and LightGBM to enhance the predictive model performance.