## 2 The Elementary Principle and Design of the Method
### Quantification Method
#### 2.2.3 Quantification Analysis  
Based on the previous analysis, the \(\mathcal{ESC}\) of a state is determined by three factors: the probability of exposure \(\mathcal{E}(s,a)\), severity \(\mathcal{S}(s,a)\) and controllability \(\mathcal{C}(s,a)\). Therefore, the three factors should be quantified to describe \(\mathcal{ESC}[(s_{i},a_{i})]\) and \(\mathcal{ESC}(s_{0})\) in Equation (3).  
* The value of \(\mathcal{E}(s,a)\) is determined by naturalistic driving data (NDD) to reflect the probability of exposure. NDD describes the natural driving environment and can provide exposure information. By analyzing the samples in NDD, we can obtain the frequency of state as \(\mathcal{Samples}(s)\) and the frequency of completing an action in the current state as \(\mathcal{Samples}[(s,a)]\). Therefore, \(\mathcal{E}(s,a)\) can be defined as: \[\mathcal{E}(s,a)=P(a|s)=\frac{\mathcal{Samples}[(s,a)]}{\mathcal{Samples}(s)}.\] (5)
* The value of \(\mathcal{S}(s,a)\) should reflect the level of severity of completing an action a in a current state s. As the direct consequence of completing an action in the current state leads to the next state, \(\mathcal{S}(s,a)\) can be quantified by analyzing the next state caused by the action. As the definition is similar to reward function \(\mathcal{R}\), the \(\mathcal{S}(s,a)\) can be seen as the reward \(\mathcal{R}(s,a)\) in exploration. Time to collision (TTC) is regarded as a useful metric to obtain the driver perception of collision risk [47] and was selected to evaluate the forward collision warning (FCW) warning threshold time. TTC is defined as Equation (6) [48] where \(D_{r}\) is the related distance and \(V_{r}\) is the related velocity. We quantified the severity according to TTC, as shown in Equation (7). \[TCC=\frac{D_{r}}{V_{r}},\] (6) \[\mathcal{S}(s,a)=\mathcal{TTC}[(s,a)].\] (7)
* The definition of \(\mathcal{C}(s,a)\) differs from that of ISO26262, because controllability aimed at hazardous events focuses on external measures after traffic accidents, whereas testing aimed at scenarios does not. However, the primary goal considers the potential consequences and long-term challenge effects of the current state. Based on this, we defined \(\mathcal{m}athcalc(s,a)\) as the expected _esc_ from actions in the next state to reflect the long-term value of actions in the current state, as shown in Equation (8). \[\mathcal{C}(s,a)=E[\mathcal{ESC}[(s,a),a]]=\frac{1}{m}\sum_{i=1}^{m}\mathcal{ ESC}[(s,a),a_{i}].\] (8)  
After the three factors are quantified, the \(\mathcal{ESC}[(s_{i},a_{i})]\) and \(\mathcal{ESC}(s_{0})\) can be described by them:  
\[\begin{split}\mathcal{ESC}[(s_{t},a_{t})]&=\mathcal{ E}(s_{t},a_{t})\cdot\mathcal{S}(s_{t},a_{t})+\mathcal{C}(s_{t},a_{t})\\ &=P(a_{t}|s_{t})\cdot\mathcal{TTC}(s_{t},a_{t})+\frac{1}{m}\sum _{i=1}^{m}\mathcal{ESC}[(s_{t+1},a_{i})],\end{split} \tag{9}\]  
Figure 2: Scenario generating process can be described as a decision tree based on MDPs.  
\[\mathcal{ESC}(s_{0})=\mathcal{ESC}[(s_{0},a_{1})]+\mathcal{ESC}[(s_{0},a_{2})]+ \cdots+\mathcal{ESC}[(s_{0},a_{m})]=\sum_{i=1}^{m}\mathcal{ESC}[(s_{0},a_{i})]. \tag{10}\]