# Driving Behavior Modeling Based on Consistent Variable Selection in a PWARX Model
## 5 Model Evaluation  
### Prediction Performance  
The proposed model is first evaluated for modeling accuracy using prediction performance against the Gipps driver model [55]. This prediction performance was carried out on a data set different from the training data set. The Gipps driver model is chosen for comparison here because it is one of the most-used microscopic driver model for traffic flow simulation and the model output, speed is the same as that of the proposed model. Intuitively speaking, the [PERSON] model assumes that the ego car's driver selects his speed to ensure that the car can be safely brought to a stop if the leading vehicle suddenly comes to a stop. The Gipps car-following driver model is given as follows:  
\[v_{n}(t+\tau)=\min(v_{following},v_{free}), \tag{13}\]  
\[v_{following}=v_{n}(t)+2.5a_{n}\tau\Big{(}1-\frac{v_{n}(t)}{V_{n}}\Big{)}\sqrt{ 0.025+\frac{v_{n}(t)}{V_{n}}}, \tag{14}\]  
\[v_{free}=b_{n}\tau\ +\ \sqrt{b_{n}2\tau^{2}}-\sqrt{b_{n}\Big{(}2\big{(}x_{n-1}(t) -s_{n-1}-x_{n}(t)\big{)}-v_{n}(t)\tau-c\Big{)}} \tag{15}\]  
where \(c=v_{n-1}^{2}(t)/b\), and the variables in (14) and (15) are defined as in [55].  
The result of prediction performance comparison is plotted in Figure 9 for Drivers B and C. The root mean square error (_rmse_), which measures the deviation of the predicted values from the actual values, is shown in Table 5 for all the drivers and is given as  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Driver** & \(\mathit{opt_{s}}\) & **Mode (\(i\))** & \(\boldsymbol{u_{1}}\) & \(\boldsymbol{u_{2}}\) & \(\boldsymbol{u_{3}}\) & \(\boldsymbol{u_{4}}\) & \(\boldsymbol{u_{5}}\) & \(\boldsymbol{u_{6}}\) \\ \hline \multirow{3}{*}{A} & \multirow{3}{*}{3} & 1 & 1.4078 & \(-\)0.1736 & \(-\)0.0468 & 0 & \(-\)0.  
\[rmse=\sqrt{\frac{\sum_{i=1}^{N}(\hat{y}_{k}-y_{k})^{2}}{N}} \tag{16}\]  
where \(\hat{y}_{k}\) and \(y_{k}\) are the predicted and actual values, respectively, for the \(k\)th observation. Generally speaking, the lower the \(rmse\), the better the model and, as can be seen from Table 5, the \(rmse\) of the proposed model is significantly lower compared to the Gipps model for all the drivers and across all modes. Additionally, in the proposed model, the \(rmse\) of the dangerous region (mode 1) is relatively lower compared to the other modes. Therefore, it can be said that the proposed model shows a better modeling accuracy than the Gipps driver model in modeling real-road driving.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Driver** & **Mode (i)** & **Gipps Model** & **Proposed Model** \\ \hline \multirow{4}{*}{A} & 1 & 6.8086 & 0.0426 \\  & 2 & - & - \\  & 3 & 6.1663 & 0.3003 \\ \hline \multirow{4}{*}{B} & 1 & 0.4176 & 0.1144 \\  & 2 & 0.1421 & 0.2902 \\  & 3 & 0.2351 & 0.0796 \\ \hline \multirow{4}{*}{C} & 1 & 1.7185 & 0.0711 \\  & 2 & - & - \\  & 3 & 0.6308 & 0.0713 \\  & 4 & 1.8837 & 0.1043 \\ \hline \multirow{4}{*}{D} & 1 & 1.0527 & 0.1129 \\  & 2 & 1.2829 & 0.1011 \\  & 3 & 0.9368 & 0.1187 \\ \hline \multirow{4}{*}{E} & 1 & 5.34527 & 0.0528 \\  & 2 & - & - \\  & 3 & 5.0658 & 0.3154 \\ \hline \multirow{4}{*}{F} & 1 & 4.0194 & 0.1489 \\  & 2 & 3.5276 & 0.1669 \\ \cline{1-1}  & 3 & 4.9792 & 0.1377 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Root mean square error (\(rmse\)) prediction performance.  
Figure 9: Prediction of real world test data by the proposed model and Gipps model: (**a**) using the test data from driver B’s driving data; (**b**) using the test data from driver C’s driving data.