## 3 Experiments
### Cut-In Case  
Figure 8 shows the primary illustration of a typical cut-in case. As in the previous experiment, we first modeled scenarios in specific applications based on the interaction between HV and BV. In the cut-in case, we focused on horizontal interactions and vertical interactions. There are six state variables in total which are velocity, relative velocity, and relative distance in both directions. However, since the longitudinal distance is short and the two datasets have different levels of detail in the longitudinal direction. HighD provides detailed motion statuses, such as distance, speed, and acceleration in the longitudinal direction, while NGSIM only provides a single distance in the longitudinal direction. So, the modeling and implementation of cut-in case experiments need to be adjusted and improved. We set the horizontal axis as the x-axis and the vertical as the y-axis. A state was modeled as follows:  
\[s=state=(v^{x},v^{x}_{r},d^{x}_{r},d^{y}_{r}) \tag{22}\]  
The modeled state consists of relative velocity in x-axis \(v^{x}\), relative velocity in x-axis \(v^{x}_{r}\), relative distance in x-axis \(d^{x}_{r}\), and relative distance in y-axis \(d^{y}_{r}\). In many studies, longitudinal motion is simplified. For example, [PERSON] et al. [57] simplified the longitudinal movement of the background vehicle by assuming that the longitudinal acceleration is 0. Our experiment used a relatively minor simplification. Because the longitudinal distance is short and only the longitudinal creative distance is in the state, the longitudinal velocity was used as the longitudinal action rather than the longitudinal acceleration. We need to further process the difference in longitudinal distance between two frames extracted from the NGSIM dataset to calculate the longitudinal speed. The action of testing scenarios was both the horizontal acceleration \(acc^{x}\) and vertical velocity \(v^{y}\) of the BV:  
\[a=action=(acc^{x},v^{y}) \tag{23}\]  
Figure 7: The simulation results of the car-following case and cut-in case using (**a,c**) HighD and (**b,d**) NGSIM datasets. Therefore, a testing scenario is modeled as:  
\[\mathbb{S}=\Big{(}s_{0},(acc^{x}_{0},v^{y}_{0}),\cdots,(acc^{x}_{k-1},v^{y}_{k-1}) \Big{)}, \tag{24}\]  
where \(s_{0}\) is the initial state and \((acc^{x}_{0},v^{y}_{0})\), \((acc^{x}_{1},v^{y}_{1})\), \(\cdots,(acc^{x}_{k-1},v^{y}_{k-1})\) is the action sequence of the BV. After analyzing HighD and NGSIM, we obtained the distribution and density of these parameters, as shown in Figure 9.Then, their value range was determined and shown in Table 4, where the discrete precision of velocity, distance, and acceleration in the x-axis was also 1 m/s, 1 m, and 0.1 m/\(s^{2}\). In contrast, the relative distance and velocity in the y-axis was 0.1 m and 0.1 m/s. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter (HighD)** & \(\mathbf{v^{x}}\) & \(\mathbf{v^{x}_{r}}\) & \(\mathbf{d^{x}_{r}}\) & \(\mathbf{d^{y}_{r}}\) & \(acc^{x}\) & \(v^{y}\) \\ \hline
**unit** & **(m/s)** & **(m/s)** & **(m)** & **(m)** & **(m/s2)** & **(m/s)** \\ \hline maximum & 42 & \(16\) & 345 & 5 & \(1.9\) & \(1.8\) \\ minimum & 23 & \(-19\) & 0 & 0 & \(-1.1\) & \(0.1\) \\ \hline
**Parameter (NGSIM)** & \(\mathbf{v^{x}}\) & \(\mathbf{v^{x}_{r}}\) & \(\mathbf{d^{x}_{r}}\) & \(\mathbf{d^{y}_{r}}\) & \(acc^{x}\) & \(v^{y}\) \\ \hline
**unit** & **(m/s)** & **(m/s)** & **(m)** & **(m)** & **(m/s2)** & **(m/s)** \\ \hline maximum & 29 & \(17\) & 224 & 13 & \(3.4\) & \(2.0\) \\ minimum & 0 & \(-15\) & 2 & 0 & \(-3.4\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Value range of parameters in cut-in case. Figure 8: An illustration of cut-in case. Figure 9: An illustration to show the distribution and density of parameters in the cut-in case based on the HighD and NGSIM. As in the previous experiment, based on our preprocessed data and modeling method, the \(\mathcal{E}(s,a)\) of cut-in case was designed as:  
\[