## 2 Methodology
### Impact Analysis Model
#### 2.3.3 Categorical Boosting  
Due to the assumption of independence and identical distribution between the variables in the ordered logit model, the direct application of this model in our study faces a challenge, as the time buffer values come from multiple driving tasks of 15 drivers, leading to substantial correlations between the data samples that are hardly compatible with the assumption of independence and identical distribution. To solve this problem, we introduce the nonparametric and learning-based CatBoost (Categorical Boosting) method for further analysis. We also use SHAP (SHapley Additive exPlanations) [38; 39; 40] to interpret the results of the machine learning approach and gain insights into the predictions of the model.  
The data used in this study primarily comprise categorical variables with a small proportion of numerical variables and thus represent extensive high-dimensional data. The CatBoost algorithm incorporates an innovative approach that automatically converts categorical features into numerical features. In addition, the algorithm uses combinatorial categorical features that enable the discovery of relationships between features, which significantly enriches the structure and hierarchy of the feature space. Due to these properties, the CatBoost algorithm is used to analyze the relationship [41; 42].  
The output formula of CatBoost is shown in Equation (13):  
\[\hat{x}_{k}^{i}=\frac{\sum_{j=1}^{p-1}\Big{[}x_{\sigma_{j,k}}=x_{\sigma_{p,k}} \Big{]}Y_{\sigma_{j}}+a\times p}{\sum_{j=1}^{p-1}\Big{[}x_{\sigma_{j,k}}=x_{ \sigma_{p,k}}\Big{]}+a} \tag{13}\]  
where \(\hat{x}_{k}^{i}\) represents the predicted value for feature \(k\) and category \(i\);  
\(Y_{\sigma_{j}}\) represents the target value (label) for sample \(j\);  
\(p\) represents the added prior distribution term;  
\(a\) is typically a positive weight coefficient.  
SHapley Additive exPlanations (SHAP) is a methodology for interpreting the results of machine learning models. Based on game theory, a prediction model outputs a predicted value for a specific person. The SHAP method assumes that each feature makes a \"contribution\" to the target prediction and assigns it a Shapley value to quantify its importance.  
The sum of the Shapley values of all characteristics for a particular person corresponds to the target prediction, with adjustments made for the average predicted value. We adopt Tree SHAP for feature importance analysis.  
The Shapley value represents the average marginal contribution of a particular feature \(i\) to the prediction. For a model with n features, the Shapley value of each feature is calculated using Equation (14) [43]:  
\[\Phi_{i}(v)=\sum_{S\in N}\frac{[(|S|-1)!(n-|S|)!]}{n!}\times[v(s)-v(s\backslash \left\{i\right\})] \tag{14}\]  
where \(\Phi_{i}(v)\) represents the Shapley value of feature \(i\);  
\(N\) is the set of all features;  
\(S\) is any subset of \(N\) that does not contain feature \(i\);  
\(v(S)\) denotes the contribution of the feature set (\(S\)) to the model's output.  
A key property of SHAP values is additivity, meaning that the sum of SHAP values across all features equals the model's output. For a given prediction model \(f(x)\), its output can be expressed as Equation (15):  
\[f(x)=\Phi_{0}+\sum_{i=1}^{n}\Phi_{i} \tag{15}\]  
where \(\Phi_{0}\) is the model's output at the baseline value, while \(\Phi_{i}\) is the SHAP value of feature \(i\).