## 3 The Proposed Feature Selection Framework
### Feature Clustering with SC  
The aim of the feature clustering step is to group highly correlated features while minimizing redundancy across groups. Traditional clustering methods like k-means, which assume spherical data distributions, often struggle to capture complex, non-linear relationships among crash data features. To achieve this, we adopted the spectral clustering (SC) algorithm [4; 5], a graph-based algorithm capable of capturing non-linear correlations through pairwise similarity analysis. Specifically, SC divides the original feature set into multiple clusters, where features within the same cluster exhibit high correlation, whereas features in different clusters are minimally overlapping. The technical details of SC are elaborated below. Assuming the feature set of the crash data as \(F(f_{1},f_{2},\ldots,f_{n})\) (\(n\) is the number of features), we construct an undirected graph \(G(V,E)\), in which \(V\) is a set of points \((v_{1},v_{2},\ldots,v_{n})\) corresponding to each feature and \(E\) is a set of edges \((e_{1},e_{2},\ldots,e_{n})\). Considering one point pair \((v_{i},v_{j})\), we define \(w_{ij}\) as the weight of the edge connecting \(v_{i}\) and \(v_{j}\). If there is a connection between \(v_{i}\) and \(v_{j}\), we set \(w_{ij}>0\), otherwise, \(w_{ij}=0\). To quantify the similarity between features, we first establish the adjacency matrix \(W\) by applying the fully connected technique in which the weights between all point pairs are non-negative. The Gaussian radial basis function was chosen to define the weight of the pair, offering a continuous and differentiable metric for quantifying similarity, as shown in Equation (1). \[w_{ij}=exp\left(-\frac{||v_{i}-v_{j}||_{2}^{2}}{2\delta^{2}}\right) \tag{1}\]  
where \(\delta\) is a free parameter denoting the width of the Gaussian kernel and \(\left|\left|\cdot\right|\right|_{2}\) denotes the \(L_{2}\)-norm. \(\delta\) controls the scale of similarity measurement, where smaller \(\delta\) values emphasize local relationships by assigning higher weights to closer features, while larger \(\delta\) values capture broader relationships with reduced local sensitivity. In this study, \(\delta\) was selected through cross-validation to balance local and global feature relationships. This function ensures that the adjacency matrix \(W\) effectively represents the relationships between features in the feature space. Based on the adjacency matrix \(W\), we compute a degree matrix \(D\) by defining degree \(d_{i}\) corresponding to \(v_{i}\) as \(d_{i}=\sum_{j=1}^{n}w_{ij}\). This degree matrix reflects the overall connectivity of each feature within the graph, summarizing the influence of each node. Next, using \(D\) and \(W\), we derive the unnormalized Laplacian matrix, \(L=D-W\), and the normalized Laplacian matrix, \(L_{norm}=D^{-1/2}LD^{-1/2}\). The normalized Laplacian matrix ensures that features with varying connectivity are appropriately scaled, making the clustering results less sensitive to the magnitude of node degrees. Subsequently, to obtain the clustering results, SC adopts the idea of graph cutting and makes the weight of the edges between different subgraphs as low as possible; meanwhile, the weight of inner edges of the same subgraph is as high as possible. Different subgraphs represent different groups, and points in the same subgraph represent features in the same group with a higher similarity. For each feature, \(f_{i}(i=1,2,\ldots,n)\), we have the formula shown in Equation (2). \[f^{\mathrm{T}}L_{norm}f=\frac{1}{2}\sum_{i,j=1}^{n}w_{ij}\Big{(}f_{i}/\sqrt{d_ {i}}-f_{j}/\sqrt{d_{j}}\Big{)}^{2} \tag{2}\]  
The eigenvectors corresponding to the smallest \(s\) eigenvalues capture the most significant structural information of the graph, allowing the features to be effectively projected into a reduced \(s\)-dimension. These eigenvectors are then used to construct the final eigen-matrix, \(M\in\mathcal{R}^{n\times s}\).