## 3 Proposed Method
### Operation Characteristic Estimation  
The proposed method considers a driver's behavior as the stimulus-response system; consequently, the acceleration or deceleration is derived from the distance and relative speed with respect to the preceding vehicle of that vehicle. Among the various methods to model the lane-keeping movement, the proposed method employs the GM model, as numerous studies have been conducted to calibrate the GM model as representing human-like car-following behavior. Thanks to previous works, the GM model is considered as one of the best models, and the effectiveness was demonstrated by field experiments [3; 27]. At any time step, \(t\), let the longitudinal position of the corresponding vehicle be represented by \(x_{i}^{t}\). Here, \(i\) is an index of adjacent vehicles, and \(i+1\) represents the preceding vehicle of the vehicle \(i\).  
\[\hat{a}_{i}^{t}=\left[\frac{\alpha_{l,m}(v_{i}^{t})^{m}}{(x_{i+1}^{t-\Delta T }-x_{i}^{t-\Delta T})^{t}}\right](v_{i+1}^{t-\Delta T}-v_{i}^{t-\Delta T}), \tag{5}\]  
where \(x_{i+1}^{t}\) indicates the position of the vehicle \(i+1\), and \(v_{i+1}^{t}\) denotes its speed at time step \(t\). Similarly, \(x_{i}^{t}\) represents the position of the vehicle \(i\) in the operation characteristic estimation, and \(v_{i}^{t}\) corresponds to its speed at time step \(t\). \(\alpha\), \(l\), and \(m\) are the model parameters to determine the operation characteristics, and \(\Delta T\) is the reaction time. These four variables are considered as the operation characteristic variables in this paper.  
The proposed method performs the optimization of the three model parameters using the Levenberg-Marquardt algorithm [24; 25]. Although there are many iterative optimization algorithms, such as the gradient descent or the Newton method, the Levenberg-Marquardt algorithm is generally used to solve nonlinear problems. Employing this algorithm, the proposed method estimates the optimized values at the current time based on information on the acceleration at the previous step. To perform optimization of the three model parameters, the reaction time is used at the previous step. If optimization fails, or the derived acceleration value is too large or small, the value of the previous step is used as a fallback value.  
After optimization of the three model parameters (\(\alpha\), \(l\), and \(m\)), the estimation of the reaction time is performed. According to the previous study, the reaction time is distributed in the range between 0.92 and 1.94 s [8]. Including the room for distribution, the proposed method identifies the optimal value of the reaction time in the range from 0.5 to 2.5 s with increments of 0.1 s. The value can be derived as follows,  
\[\Delta T=\underset{\Delta T}{\arg\min}\ |a_{i}^{t-1}-\hat{a}_{i}^{t-1}(\Delta T)|, \tag{6}\]  
where \(a_{i}^{t-1}\) represents the ground truth of acceleration and \(\hat{a}_{i}^{t-1}\) denotes the derived value obtained using the proposed method.  
Estimation of the operation characteristic variables is performed following the above process, and the optimal values are derived at each time step. However, the operation characteristics may not drastically change in a short time period. Therefore, the proposed method defines a sliding window of a constant size. Consequently, the values within the window are modified according to a moving average.