## 3 Modeling and Results
### Development and Assessment of PSVE Model  
In order to achieve optimal performance of the PSVE model, this study is conducted in two steps. Firstly, preliminary training on commonly used machine learning algorithms are performed to ensure the best fitting effect. Subsequently, we use grid search methods to find the best hyperparameters, and use 10-fold cross-validation to evaluate the performance of the model. Three machine learning models are selected: Decision Tree (DT), Random Forest (RF), K-Nearest Neighbors (KNN), and Multilayer Perceptron (MLP), which is a neural network model. 1. DT is a hierarchical structure where each internal node represents an attribute-based judgment; in binary prediction tasks, the ultimate objective is to assign data points  
Figure 2: Evaluation of correlation between the independent variable and response variable. to one of two categories. The training of a decision tree specifically involves the segmentation decision of the tree. Figure 3 shows a simple decision tree model [30] consisting of a single binary target variable Y (0 or 1) and two continuous variables X1 and X2, ranging from 0 to 1. The main components of the decision tree model are nodes and branches, and the most important steps to build the model are splitting, stopping and pruning. The most important part is feature selection, which is related to the degree of \"purity\" of the resulting child nodes and is used to choose between different potential input variables. These features include entropy, Gini index, classification error, information gain, gain ratio, and two criteria. Additionally, in this study, the Gini impurity method is employed for making decisions. Gini impurity is a measure of the impurity of datasets and is used in decision trees to evaluate the quality of segmentation. The lower the Gini impurity, the purer the category in the dataset. When constructing the decision tree, the goal of selecting segmentation points is to minimize the Gini impurity of the child nodes; this refers to making each child node contain a single class of data as much as possible. \[Gini(T)=1-\sum_{i=1}^{c}p_{i}^{2}, \tag{2}\]  
Here, \(p_{i}\) represents the relative frequency of class \(i\) in dataset \(T\), while \(c\) denotes the total number of classes. 2. RF is an ensemble learning algorithm based on decision trees, aiming to enhance prediction accuracy by constructing multiple decision trees and aggregating their predictions [31]. As is shown in Figure 4, it achieves this by randomly selecting subsets of features from the dataset for training individual decision trees, which may yield different prediction results. For a new data point, each tree provides a prediction result. The final classification result in random forest is determined through a voting mechanism, where the category with the highest number of votes becomes the predicted result. Figure 3: Sample decision tree based on binary target variable Y. 3. KNN is an instance-based learning method that classifies instances by measuring distances between feature values. It does not involve explicit training but rather stores training datasets only. During prediction, given a new data point, KNN identifies the closest K points in the dataset; the classification decision is based on the distance measurement, and the formula is: \[d(x,y)=\sqrt{\sum_{i=1}^{n}{(x_{i}-y_{i})}^{2}},\] (3)
4. As shown in Figure 5, MLP refers to a feedforward artificial neural network comprising an input layer, one or more hidden layers, and an output layer; every layer consists of multiple neurons interconnected via weighted connections between adjacent layers [32]. Applying certain thresholds or activation functions at various nodes within MLP's architecture determines its categorization assignment. MLP uses backpropagation and gradient descent to update the weights to implement the training process, which can be expressed as: \[\omega_{new}=\omega_{new}-l\frac{\partial L}{\partial\omega},\] (4)  
Figure 4: The entire process of classification based on random forest.