## 5 Simulation Results and Discussion
### Simulation Results
The Gauss-Markov model was integrated into the Kalman filter localization scheme proposed in [29] in order to illustrate how error modeling and parameter estimation enhance the accuracy of localization. We will experiment with \(p=1,2,4\) to assess whether reducing the model complexity using \(p=1\) will provide comparable results to higher orders. The Kalman filter involves two steps: a prediction step and a measurement update step. The prediction step assumes that the state of a system at time \(t\) evolved from the prior state at \(t-1\) according to:  
Figure 8: Longitudinal position error for vehicle \(\
u_{2}\) using autoregression model with \(p=1,2,4\). Figure 7: Latitudinal position error for vehicle \(\
u_{1}\) using autoregression model with \(p=1,2,4\). Figure 9: Latitudinal position error for vehicle \(\
u_{2}\) using autoregression model with \(p=1,2,4\). \[x_{t}=F_{t}x_{t-1}+B_{t}u_{t}+\vartheta_{t} \tag{17}\]  
where \(x_{t}\) is the state vector containing system information at time \(t\), \(u_{t}\) is a vector containing control inputs, \(F_{t}\) is a state transition matrix that applies the effects of the system state at \(t-1\) to its state at time \(t\), \(B_{t}\) is a control input matrix that serves a similar purpose to \(F_{t}\) but for the control inputs, and \(\vartheta_{t}\) is the process noise. In our experiments, the transition matrix \(F_{t}\) was composed using the parameter values \(A_{p}\) that were produced by the Yule Walker equations, which are listed in Table 1. A different matrix was constructed for each model using the longitude and latitude parameter values. The size of the transformation matrix was the same size of the dataset. The control matrix \(B_{t}\) was set to zero since there are no control inputs. The measurement step performs corrections to the predicted location, according to the equation:  
\[z_{t}=H_{t}x_{t}+w_{t} \tag{18}\]  
where \(z_{t}\) is the vector of measurements, \(H_{t}\) is a transformation matrix which is set to default value of 1 in our experiments, and \(w_{t}\) represents the measurement noise. The process and measurement noise in Kalman filter-based localization algorithms are usually assumed to be zero-mean Gaussian white noise. Our proposed error model can compute the noise variance in location information. Therefore, we incorporate it into the prediction step of the Kalman filter and study how this will improve the localization algorithm. The \(AR\)-integrated Kalman filter is then applied to the two time series location information of vehicles \(v_{1}\) and \(v_{2}\), and the results shown in Figures 10-13 compare the localization error for the three error models; \(p=1\), \(p=2\), and \(p=4\). We note that the computational complexity of the Kalman filter and the _autoregression_-integrated Kalman filter is dominated by the matrix multiplication, and therefore grows as \(O(n^{2})\), where \(n\) is the number of location measurements. We also note that the computational complexity of the AR-integrated Kalman filter implementation and the standard Kalman filter implementation are identical. This is because the integration of the AR model into the Kalman filter involves the replacement of the transition and error matrices and the extension of prediction step to accommodate computations when \(p=2\) and \(p=4\). We can see that incorporating our error model into the Kalman filter outperforms the results produced by the Kalman filter with zero Gaussian error, which we call the standard Kalman filter for convenience. Similar accuracy gains are obtained when the error model is integrated into the Kalman filter and applied to the location measurements of vehicle \(v_{2}\) whose location information is used for validation. We can also observe that there is no observable gain in accuracy when using a higher Gauss-Markov order, since \(p=1\) provides almost identical results to \(p=4\).