## 5 Discussion
### Driving Style Discussion
#### 5.1.2 The Transition Probabilities of Driving Behavior  
The previous subsection analyzed the frequency and duration proportion of driving style, intuitively demonstrating the driving preference of different driving styles. This section will further discuss the transfer characteristics of driving behavior and reveal the relationship between driving style and transition probability of the driving behavior. The  
where \(N\) is the number of driving behavior \(a_{i}\) transferring to \(a_{j}\), \(m\) is the total number of driving behavior transferring from \(a_{i}\) to other driving behaviors; in this paper \(m=12\), satisfying \(\sum_{j=1}^{m}a_{ij}=1\). The dimension of the transition probability matrix is 12 \(\times\) 12 due to 12 driving behaviors, and the diagonal is zero, and each driver has 144 transition probability features. Because the probability of some driving behaviors is 0, in some cases, \(a_{ij}=0\), or the transition probability is very small. In order to select the feature set with the strongest correlation with driving style from 144 transition features, the joint mutual information maximization (JMIM) algorithm was used to remove redundant features [40]. This method combines feature correlation and redundancy concepts together to select the optimized subset by the forward greedy search algorithm. It is expressed by  
\[f_{JMIM}=\arg\max_{f_{i}\in F-S}\left(\min_{f_{i}\in S}(I(f_{i},f_{s};C)) \right), \tag{38}\]  
\[I(f_{i},f_{s};C)=\left[-\sum_{c\in C}p(c)\log(p(c))\right]-\left[\sum_{c\in C }\sum_{f_{i}\in F-S}\sum_{f_{s}\in S}\log\left(\frac{p(f_{i}f_{s},c|f_{s})}{p( f_{i}|f_{s})p(c|f_{s})}\right)\right], \tag{39}\]  
where \(F\) represents the set of all candidate features, \(S\) represents the selected feature set, \(f_{i}\) is the candidate features, satisfying \(f_{i}\in F-S,f_{s}\) is the selected features, satisfying \(f_{s}\in S\), \(C\) is a discrete variable, \(C\in\) [1,2,3], where 1,2,3 represent cautious, normal, and aggressive driver groups. Equation (23) employs joint mutual information and the'maximum of the minimum' approach, which means for a feature \(f_{i}\), if JMI is larger than that of all other features \(f_{i}\), where \(f_{i}\in F-S\) (\(i\
eq j\)), then it is the most relevant feature to the class label \(C\) in the context of the subset \(S\). For more details, please refer to [40]. The driver data with the same subjective score labeling result and objective risk indices labeling result were selected, and data from each driver were divided into five segments randomly, and finally, 5 \(\times\) 28 = 140 groups of data were obtained. The JIMI method was used to select an optimized sub-feature set from 144 transition probabilities. Figure 16 shows the top ten transition probabilities with the greatest joint mutual information. Finally, the first seven features with the greatest mutual information were selected, which were (1) from middle following with normal deceleration to middle following with normal acceleration, (2) from near following with aggressive deceleration to near following with normal deceleration, (3) from near following with normal deceleration to near following with normal acceleration, (4) from near following with normal deceleration to near following with aggressive deceleration, (5) from far following with normal deceleration to far following with normal acceleration, (6) from middle following with normal deceleration with far following with normal acceleration, and (7) from far following with normal acceleration to far following with normal deceleration. These seven features will be used for driving style analysis and driving style recognition in the following subsection. Figure 16: The top ten transition probabilities with the greatest joint mutual information. The significance analysis of the seven features is shown in Table 4. The significance levels of the seven features were less than 0.05, indicating that there were significant differences among the seven features of the three types of drivers.