## 5 Simulation Results and Discussion
### Simulation Results  
Three sets of experiments were conducted to investigate two modes of operation for the proposed model. First, we investigated the model performance when used as a standalone model for prediction, for example when GPS is not available. Second, we investigated the performance of our model when integrated into a localization algorithm such as a Kalman filter algorithm. The Kalman filter was chosen because of its popularity as a localization scheme, as well as its ease of implementation. Third, we applied the Kalman filter that was adjusted by the Gauss-Markov model parameters to a real-life datasetin order to assess the accuracy of the enhanced localization scheme and rule out overfitting. The first two sets of experiments study the tradeoff between reducing the complexity of the \(p\) -order Gauss-Markov model and the location prediction accuracy, as well as assess the increase in accuracy of localization when using the model in the prediction step of the Kalman filter. The third experiment was used for validation of the model. Each model is designated \(AR(p)\) for ease of reference. The model parameters \(A_{i}\) that were produced by the Yule Walker equations for the two mobility traces for vehicles \(v_{1}\) and \(v_{2}\) are listed in Table 1. For the first set of experiments (standalone prediction), the results in Figures 6 and 7 show the errors in the longitudinal (\(x\)) and latitudinal (\(y\)) position estimates made by the Gauss-Markov model for \(p=1,2,4\) for vehicle \(v_{1}\). The figures show that higher model orders correspond to reduced estimation error of the future location coordinates of the vehicle; \(p=2\) and \(p=4\) can predict the vehicle's location with an average error of 11 m for \(x\) and 16 m for \(y\), with inconsiderable improvement made by the model with \(p=4\) compared to the model with \(p=2\). However, when \(p=1\), the average error is 55 m for both \(x\) and \(y\). This is further corroborated by the results produced when applying the same Gauss-Markov model on the location measurements of vehicle \(v_{2}\), which has been designated for validation, as shown in Figures 8 and 9. We can conclude that \(p=2\) is a reasonable compromise between complexity and accuracy and therefore is sufficient for deployment of the error model as a standalone prediction tool. The tradeoff performance provided by the model when \(p=2\) will make the model's prediction robust even when used in fast-fading channel scenarios, which may exhibit lower correlation windows, since the correlation window in this case is small enough to allow for correlation to be manifested, while still producing accurate prediction. This makes the model applicable in different scenarios with potentially varying correlation windows. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Vehicle Traces**} & \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Longitude**} & \multicolumn{6}{c}{**Latitude**} \\ \cline{3-10}  & & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) & \(A_{1}\) & \(A_{2}\) & \(A_{3}\) & \(A_{4}\) \\ \hline \multirow{4}{*}{\(trace\)\(v_{1}\)} & \(AR(1)\) & 0.996 & & & & 0.9961 & & & \\  & \(AR(2)\) & 0.9978 & 0.0018 & & & 0.9977 & 0.0016 & & \\  & \(AR(4)\) & 0.9978 & 0 & 0 & 0.0018 & 0.9977 & 0.0001 & 0 & 0.0016 \\ \hline \multirow{4}{*}{\(trace\)\(v_{2}\)} & \(AR(1)\) & 0.9963 & & & & 0.996 & & & \\  & \(AR(2)\) & 0.9992 & 0.0029 & & & 0.9976 & 0.0016 & & \\  & \(AR(4)\) & 0.9992 & 0.0001 & 0.0002 & 0.0028 & 0.9976 & 0.0001 & 0.0001 & 0.0015 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Autoregression model parameters for the two vehicle mobility traces used for experiments. Figure 6: Longitudinal position error for vehicle \(v_{1}\) using autoregression model with \(p=1,2,4\). For the second experiment, we want to investigate the integration of the model into a localization algorithm.