# Lateral Oscillation Characteristics of Vehicle Trajectories on the Straight Sections of Freeways
## 4 Data Analysis and Result  
### Speed Distribution  
The speed of each vehicle type was summarized separately, and the distribution of the number of vehicles in different speed intervals was recorded, as shown in Figure 6. Furthermore, each vehicle type's velocity frequencies were fitted using Gaussian curves to gain insight into the distribution mechanism of speed. The Gaussian fitting equation was denoted as follows:  
\[y=y_{0}+\frac{A}{w\sqrt{\pi/2}}\exp\left(\frac{-2(x-x_{c})^{2}}{w^{2}}\right) \tag{6}\]  
where \(A\),\(w\),\(x_{c}\) and \(y_{0}\) are model parameters, and \(x_{c}\) represents the expected value of the speed. A Kolmogorov-Smirnov test confirmed the validity of the normal distribution, and the significance was 0.081, 0.142, and 0.193 (>0.05) for each vehicle type. The results of the fitting coefficients are shown in Table 4. From Table 4 and Figure 6, we found the following:  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Vehicle Types**} & \multicolumn{6}{c}{**Gauss Fitting Coefficients**} & \multicolumn{6}{c}{**Speed (km/h)**} \\ \cline{2-11}  & \(A\) & \(w\) & \(x_{c}\) & \(y_{0}\) & \(\mathbf{R^{2}}\) & **Min** & **Max** & **Mean** & **85 th** & **SD** \\ \hline Passenger cars & 95,735.5 & 32.5 & 113.1 & 51.5 & 0.961 & 16.6 & 203.3 & 108.6 & 127.8 & 22.1 \\ Medium-large Peak1 & 5517.8 & 10.0 & 87.0 & 6.54 & & & & & & \\ cars  Peak2 & 3270.9 & 29.5 & 106.5 & 0.17 & & & & & & \\ Extra-large trucks & 4268.4 & 8.3 & 86.8 & 2.48 & 0.992 & 17.4 & 112.4 & 85.1 & 90.8 & 8.9 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Basic information on speed and the coefficients of curve fitting.  
Figure 5: The number of clusters, using the SSE index.  
(1) The speeds of all the vehicles satisfied the Gaussian distribution when the speed intervals of the three different types of vehicles were separated into steps of 4, 2, and 1 km/h. Among them, the goodness-of-fit for passenger cars was 0.961, and the expected speed was 113.1 km/h. The speed distribution of medium-large cars resulted from the accumulation of two Gaussian distribution curves, so there were two peaks, which were reached at \(x_{c}\) = 87.0 km/h and \(x_{c}\) = 106.5 km/h, with a goodness-of-fit of 0.987. The extra-large trucks had the best fit, with a goodness-of-fit of 0.992, peaking at \(x_{c}\) = 86.6 km/h and then declining as speed rose.  
(2) The one-way analysis of variance (ANOVA) showed that the speeds of the three vehicle types were significantly different at a significance level of 0.05 (F = 3273.64, \(p<0.01\)). The speed of passenger cars was the most prominent (mean=108.6 km/h), followed by medium-large cars (mean = 92.4 km/h) and the most minor for extra-large trucks (mean = 85.1 km/h). The different vehicles' speed minima did not differ significantly; however, the speed maxima of the two adjacent vehicle types differed by as much as 45 km/h. It demonstrated that the lower-speed threshold of the three vehicle types was generally the same (perhaps due to road construction or traffic congestion). In contrast, the upper-speed threshold had a major difference.  
(3) With an overall speed distribution of 80-150 km/h, passenger cars had the biggest speed dispersion (SD = 22.1 km/h). Extra-large vehicles had the weakest speed dispersion (SD = 8.9 km/h), and their speed distribution was more tightly focused, mostly between 80 and 95 km/h. The speed dispersion of medium-large cars was intermediate between the other two (SD = 17.7 km/h), concentrated within 80-125 km/h. Therefore, only the speed data in the concentration region was considered in the subsequent analysis.  
Figure 6: (**a**) Speed distribution of passenger cars; (**b**) Speed distribution of medium-large cars; (**c**) Speed distribution of extra-large trucks; (**d**) Relative frequency of speed for all types of vehicles.