## 3 Results
### Interpretable Machine Learning
#### 3.2.2 SHAP (Shapley Additive exPlanations)  
SHAP analysis can be said to be a technique based on the Shapley value of game theory. Based on the idea that the Shapley value expresses how much each participant contributed, SHAP makes it possible to calculate the influence of each variable during data analysis using a machine learning model. Using these numerical values, SHAP allows us to interpret the extent to which each variable contributed to the derivation of the model's result [48]. \[\varnothing_{i}=\sum_{S\subseteq F\backslash\{i\}}^{n}\frac{|S|!(F-|S|-1)!}{|F |!}\Big{[}g_{S\cup\{i\}}\Big{(}x_{S\cup\{i\}}\Big{)}-g_{S}(x_{S})\Big{]} \tag{9}\]  
\(\varnothing_{i}\) is the Shapley value of the \(i^{th}\) input variable, and \(F\) is the set of all input variables. \(S\) is all subsets of input variables without the \(i^{th}\) input variable. In the above equation, values, \(x_{S}\) and \(x_{S\cup\{i\}}\), represent the values of the input variables in set \(S\) without the \(i^{th}\) input variable and the dataset that includes the \(i^{th}\) input variable. \(g_{s}(x_{S})\) represents the prediction based on input \(x_{S}\). Therefore, the \(i^{th}\) input variable contribution value could be  
Figure 5: Individual PDP plot of volume and length: (a) PDP plot for length; (b) PDP plot for length. calculate as subtracting the sum of contribution excluding the \(i^{th}\) variable from the total contribution [48]. Figure 6 is a feature importance plot of the SHAP value. In the case of Figure 3, the permutation technique is used to measure the effect of a variable on the model. If each variable is dependent on the other, the result may be distorted, and critically, the negative influence of variable importance is not calculated. Since the possibility of mutual influence between variables is taken into account, the negative influence can be calculated. Therefore, it is possible to obtain different results from the existing feature importance plot [53]. Compared with Figure 3, it can be seen that Heavy Vehicle Volume is selected as an important variable instead of Land Use 3. As we mentioned earlier, the SHAP value could be able to express the variable's positive and negative influences on the result. This graph visualizes how all variables affect the distribution of the SHAP value [4; 54]. In Figure 7, the x-axis shows the SHAP value of each variable, and the y-axis lists the variables in order of importance in Figure 6. Each dot represents each sample used for the analysis, and it can be seen that the color of the dots is higher for red and lower for blue through the 'Feature value' color bar on the right y-axis. According to the graph, Heavy Vehicle Volume and Volumes, it can be seen that the occurrence of accidents increases as the values increase. In the case of Length, the longer the length, the more accidents occur, but after a certain section, the colors are mixed, and it is difficult to clearly distinguish their correlation. On the other hand, Lane width and Shoulder width can be seen as the occurrence of accidents decreases as the width increases. Through the distribution of SHAP values for all variables in the Figure 8, the overall interpretation of the effect of each variable on accident occurrence was possible. Figure 8 shows the specific correlations [55]. However, in the case of the Length variable, difficulties in interpretation might occur. Figure 8 shows the SHAP value graph for each variable that was drawn so that the characteristics not seen in Figure 7 could be identified. For each graph, the x-axis means the value of the variable, and the y-axis means the SHAP value of the variable. In the case of Heavy Vehicle Volume and Volume graphs, it can be seen that as the value increases, the SHAP value also increases. In particular, the SHAP value of the Volumes variable is the highest when Volumes are almost 30,000 veh/day. Furthermore, in the case of the Length variable, it is possible to figure out that the SHAP value increases  
Figure 6: Feature importance plot using Shapley value.