## 3 Methodology
### Machine-Learning Methods  
Machine-learning methods have gained wide application in various prediction tasks in recent years. To predict the enclosure structure's horizontal displacement, six mainstream machine-learning algorithms were selected to test the ability of prediction shown in Table 3.  
The extra trees algorithm builds upon the conventional decision trees approach. In the traditional decision tree algorithm, data objects are segregated into different branches based on their attribute values. The critical aspect lies in selecting the most suitable data features and their splitting points to facilitate decision making [34]. [PERSON] et al. [35] ingeniously introduced additional steps to the traditional decision tree algorithm to enhance its randomness. This augmentation not only improves the accuracy of suboptimal solutions but also enhances the flexibility of solution computation. This refinement yields particularly notable benefits in classification problems. The random forest (RF) algorithm combines the bagging method with CART decision trees and creates a novel classification algorithm [36]. RF, in essence, constitutes numerous randomly constructed decision trees. In the creation of each tree, a sample set is generated through randomized extraction, and the node set is also formed by random extraction, leading to distinct trees. When a new sample enters the forest, each decision tree makes a classification judgment, culminating in an outcome based on the majority rule. Specifically, the random forest formation exhibits the following characteristics:  
1. Bootstrap sampling method:  
A new set of samples is created by randomly selecting N samples out of all N samples in a back-and-forth manner, facilitating the generation of a decision tree;
2. Random selection of features: \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & \
\begin{tabular}{c} **Machine-Learning** \\ **Algorithms** \\ \end{tabular} &
\begin{tabular}{c} **Clarification** \\ \end{tabular} \\ \hline
1 & Extra tree Regressor [28] & A regressor with multiple highly randomized decision trees and limited to integration methods. \\
2 & Random Forest (RF) & The algorithm builds multiple decision trees by random sampling and averages the results of all the trees to obtain an overall prediction. \\
3 & Gradient Boosting & An algorithm for combining multiple simple models into a composite model. \\
4 & Decision Tree (DT) & A method for approximating the value of a discrete function. Generalization is used to generate readable rules and decision trees, and decision making is used to analyze new data. \\
5 & CatBoost Regressor [32] & Asymmetric decision tree-based algorithm that can effectively and rationally handle categorization features. \\
6 & Multilayer Perceptron (MLP) Regressor [33] & A feed-forward neural network comprising a plurality of neuron layers forming a fully connected structure between the layers. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Brief description of machine-learning algorithms. During each node's split while growing the decision tree, m features (where m << M) are randomly chosen from the total M features. The best-performing feature from the selected m is used as the basis for splitting;
3. Formation of diverse decision trees:  
A multitude of unique decision trees are generated following the aforementioned steps to construct a random forest. The gradient boosting regression [37; 38] (GBR), akin to a collaborative learning technique, combines multiple weaker learning algorithms in an attempt to learn from errors. It functions by iteratively creating numerous weak learners, each aiming to fit the negative gradient of the previous cumulative model's loss function. The primary notes are:  
1. Weak learners integration:  
Individual learning algorithms might not be highly accurate, but their integration yields higher accuracy when combined;
2. Sequential application:  
These learning algorithms are applied sequentially, with each learning from the errors of its predecessor.