## 2 Material and Methods
#### 2.2.3 Random Forest Algorithm  
The decision tree is a tree-like prediction model in machine learning that is widely used in various fields because its output results are easy to understand [21]. It has a flowchart-like structure in which each branch represents a choice. Moreover, each leaf node corresponds to a classification and produces a rule that consists of the conditions along all the paths from the root node to the leaf node. The conclusions presented on the leaf node represent the conclusions derived from the corresponding rules. In machine learning, the decision tree is a prediction model that represents a mapping relationship between the attributes and values of an object. Random forests are classifiers that use multiple decision trees to train and predict samples [22]. Each classifier in a random forest is a decision tree. At each node, a decision tree is generated by splitting randomly selected attributes. Subsequently, when performing classification, the outcomes are determined through a voting process involving multiple decision trees. Based on the above mechanism, random forests are less prone to overfitting and are more stable when dealing with error points and outliers. The random forest algorithm builds each tree according to the following steps:  
1. Let \(N\) represent the number of training cases (samples), and \(M\) represent the number of features. The number of input features \(m\) is used to determine the decision result of a node on the decision tree, where \(m\) should be much smaller than \(M\). 2. From the \(N\) training samples in the way of sampling with replacement, sampling \(N\) times to form a training set, and using the unsampled samples as predictions to evaluate the error. 3. For each node, \(m\) features are randomly selected, and the decision of each node on the decision tree is determined based on these features. According to the \(m\) features, calculate the optimal splitting method. 4. Each tree grows fully without pruning. Feature Importance Evaluation Based on Random Forest Algorithm  
Since the random forest algorithm uses random sampling with replacement, about one-third of the data is not used and does not participate in the establishment of the decision tree. This part of the data can be used to evaluate the performance of the decision tree and calculate the prediction error rate of the model, which is called out-of-bag (OOB). The importance of a characteristic factor can be judged by calculating the change in OOB. Let \(x=\left[x_{i}\right]_{i=1}^{n}\) be the training dataset, where \(x_{p}=\left(a_{1}^{(p)},a_{2}^{(p)},\ldots,a_{m}^{(p)}\right)\). In order to study the importance of the \(i\)th feature, we are essentially studying the influence of this feature on the overall classification effect. Specifically, the method used to calculate the importance of a feature in the random forest is as follows:  
1. First train a random forest classifier \(y=f(x)\) and calculate \(OOB_{1}\). 2. Apply random perturbation \(\varepsilon\) to the \(i\)th feature, that is, \(x^{\prime}=\left[x_{i}^{\prime}\right]_{i=1}^{n}\), where \(x_{p}^{{}^{\prime}}=\left(a_{1}^{(p)},a_{2}^{(p)},\ldots,a_{i}^{(p)}+ \varepsilon_{p},\ldots,a_{m}^{(p)}\right)\). 3. Train a random forest classifier \(y=g(x^{\prime})\) and calculate \(OOB_{2}\). Then the importance of the \(i\)th feature is:  
\[\text{imp}(i)=OOB_{2}-OOB_{1} \tag{5}\]  
After calculating the importance of each feature, the ranking of all feature factors can be assigned. The complete algorithm flow is shown in Algorithm 1. Major Limitations of the Model  
The main limitations of the model are mainly reflected in the following three aspects: For high-dimensional data, the results of factor importance analysis may be affected by dimensional disaster, leading to inaccurate analysis results. For nonlinear data, the expression ability of the BPNN model may be insufficient, leading to inaccurate factor importance analysis results.