## 5 Simulation Results and Discussion
### Verification of Model  
In order to verify our \(AR\) model, we tested the model against three realistic datasets. We chose three vehicular mobility traces from OpenStreetMap [30]. The first trace (\(dataset\) 1), which is illustrated in Figure 16, represents a single vehicle trip along the route between Northeast Union Hill Road and  
Figure 14: Zoom in on the longitudinal positioning error produced by the \(Autoregression\)-Kalman filter for vehicle \(\mathbf{v_{1}}\).  
Figure 13: Latitudinal positioning error produced by the standard Kalman filter and the \(Autoregression\)-Kalman filter for vehicle \(\mathbf{v_{2}}\).  
Figure 15: Zoom in on the latitudinal positioning error produced by the \(Autoregression\)-Kalman filter for vehicle \(\mathbf{v_{1}}\).  
Bravern 1, Bellevue, WA. The total trip time was 30 min and 45 s (totaling 1845 data points), and vehicle speed ranged from 0 km/h to 104.4 km/h. The second trace (_dataset_ 2), illustrated in Figure 17, represents a single vehicle trip between East Main Street and the Gelder Park, Derry, PA, USA. The total trip time was 5 min and 45 s (totaling 343 data points), and speed ranged from 0 km/h to 96.7 km/h. The third trace (_dataset_ 3), illustrated in Figure 18, depicts a vehicle's trip around a new residential area in Stockholm, Sweden. The total vehicle trip time was 25 min and 56 s (totaling 1313 data points). There is no mention of the vehicle speeds for the third dataset. For the three datasets, location measurements were taken every second.  
As with the previous experiment set, the model parameters and process noise were extracted from each of the datasets using the autoregression model of order 1. The Kalman filter was integrated with the model parameters and applied to the location measurements. Figures 19-24 show, respectively, the longitudinal and latitudinal position measurement errors produced by the Kalman filter when the Gauss-Markov model is used for the prediction step when applied to the OpenStreetMap traces. The margin of error for all three datasets stays within a very small and negligible range (0.005 m), which indicates the high level of accuracy produced by the localization model. The larger variation in longitudinal error in comparison to the latitudinal error in \(dataset\) 1 is due to the larger variation rate in longitudinal measurements compared to a small overall change in the latitudinal measurements.  
The localization model performs well regardless of the road scenario--highway, residential, or a combination of both. Error variation is miminal at the intervals during which the vehicles do not change their positions, as can be noticed in the interval between the seconds 1101 and 1550 in \(dataset\) 1 and between the seconds 310 and 650 for \(dataset\) 3, for example. We notice that error in the third section of the position measurements for \(dataset\) 3 exhibits more fluctuations. When the corresponding vehicle's route is closely inspected, it can be evident that this variation is due to the rather unsmooth movement along some sections of the residential area. We can assume that this error variation can be controlled if vehicle speed is incorporated into the prediction model, which is part of our future work.  
Figure 19: Longitudinal positioning error produced by \(Autoregression\)-Kalman filter for \(dataset\) 1.  
Figure 18: OpenStreetMap route for verification dataset 3.