## 5 Parameter Estimation for Trajectory Prediction
### Parameter Estimation Equation  
In this research, the terminal point was defined as mentioned in Section 3. The position of the terminal point was decided as a point on the central part of the lane of the escaping road of the intersection with the distance \(d_{pre}\) from Point B. From this definition, the accuracy of the trajectory prediction is highly dependent on the distance parameter \(d_{pre}\). To predict the turning trajectory with high accuracy, the distance \(d_{pre}\) has to be estimated before generating the turning trajectory. We hypothesized that there is a correlation between the distance parameter \(d_{pre}\) and the shape of the intersections. Here, as shown in Figure 9, the distance from the central part of the driving lane to the right-side edge of the entering road was defined as \(l_{in}\), and the distance from the driving center lane to the right-side edge of the escaping road was defined as \(l_{out}\), and the crossing angle of the entering road and the escaping road was defined as \(\theta_{cross}\). \begin{table}
\begin{tabular}{c c} \hline
**Trajectory Prediction Method** & **95\% Confidence Interval of Population Mean [m]** \\ \hline Triclothoidal curve & 0.532 \(\leq\mu\leq\) 0.843 \\ Biarc curve & 0.824 \(\leq\mu\leq\) 1.157 \\ Clothoidal curve & 0.813 \(\leq\mu\leq\) 1.119 \\ \hline \end{tabular}
\end{table} 
Table 1: Ninety-five percent confidence intervals of the population mean \(\mu\) of the maximum absolute prediction error. Figure 9: Definitions of intersection shape related parameters. The correlations between the parameters were estimated by linear regression \(\hat{d}_{pre}=ax+b\) by using the results from the data acquisition experiments at 31 intersections. The size of the intersections and the angles of right turns were measured using Google Earth Pro. Some assumptions for variables \(x\) were prepared as listed in Table 2. From the experiments at 31 intersections, the parameters \(d_{pre}\) in each experimental scenario were calculated from the acquired driving data by optimizing the predicted trajectory to minimize the error from the recorded actual driving path. From the assumptions \(x\) and calculated \(d_{pre}\), the coefficients \(a\), \(b\) and correlation coefficient in \(\hat{d}_{pre}=ax+b\) were decided by the linear regression as shown in Table 2.  
From the table, \(x=l_{in}+l_{out}\) and \(x=l_{in}\cdot l_{out}\) had a slightly strong linear correlation between \(d_{pre}\) and \(x\). \(x=l_{in}\cdot l_{out}\cdot\sin(|\theta_{cross}|)\) is an area of a parallelogram consisting of \(l_{in}\) and \(l_{out}\); however, because the correlation by \(\sin(|\theta_{cross}|)\) had a negative correlation, the correlation coefficient by \(x=l_{in}\cdot l_{out}\cdot\sin(|\theta_{cross}|)\) became lower than the coefficient by \(x=l_{in}\cdot l_{out}\). In contrast, the correlation coefficient of \(x=l_{in}\cdot l_{out}/\sin(|\theta_{cross}|)\) was the highest in the assumptions, and there was a slightly strong linear correlation between \(d_{pre}\) and \(x\). A scatter plot of the correlation between \(d_{pre}\) and \(x=l_{in}\cdot l_{out}/\sin(|\theta_{cross}|)\) is shown in Figure 10. In this study, choosing \(x=l_{in}\cdot l_{out}/\sin(|\theta_{cross}|)\), an equation to estimate a distance parameter \(\hat{d}_{pre}\) was defined as  
\[\hat{d}_{pre}=0.129l_{in}\cdot l_{out}/\sin(|\theta_{cross}|)+12.5. \tag{25}\] \begin{table}
\begin{tabular}{c c c c} \hline \(x\) & \(a\) & \(b\) & **Correlation Coefficient** \\ \hline \(l_{in}\) & 1.01 & 11.6 & 0.472 \\ \(l_{out}\) & 0.755 & 12.0 & 0.591 \\ \(\sin(|\theta_{cross}|)\) & \(-55.9\) & 71.8 & \(-0.489\) \\ \(l_{in}+l_{out}\) & 0.812 & 7.45 & 0.735 \\ \(l_{in}\cdot l_{out}\) & 0.139 & 12.3 & 0.733 \\ \(l_{in}\cdot l_{out}\cdot\sin(|\theta_{cross}|)\) & 0.146 & 12.2 & 0.714 \\ \((l_{in}+l_{out})/\sin(|\theta_{cross}|)\) & 0.687 & 8.68 & 0.731 \\ \(l_{in}\cdot l_{out}/\sin(|\theta_{cross}|)\) & 0.129 & 12.5 & 0.743 \\ \hline \end{tabular}
\end{table}