# Credible Navigation Algorithm for GNSS Attack Detection Using Auxiliary Sensor System
## 4 Performance Evaluation
### Parameter Analysis  
This section analyzes the impact of the jump parameter settings (\(n_{\text{j}}\) and \(Th_{\text{j}}\)) and slow-change parameter settings (\(n_{\text{s}}\) and \(Th_{\text{s}}\)) on \(P_{\text{D}}\) and \(P_{\text{FA}}+P_{\text{MD}}\) mentioned in Section 3.1. First, we focus on the impact of jump parameter settings. Data set \(\widetilde{\Psi}_{2}\) is selected for analysis in order to shield the influence of slow-change parameter settings on \(P_{\text{D}}\), \(P_{\text{FA}}\) and \(P_{\text{MD}}\). According to Equations (16)-(18), the value range of \(n_{\text{j}}\) is set from 1 to 20, and the value range of \(Th_{\text{j}}\) is set as follows:  
\[ \begin{split}& Th_{\text{j}}=0.1t\text{ m}\\ &\text{s.t. }t=0\,\sim 60\end{split}  \tag{30}\]  
The simulation results are shown in Figure 9. It can be seen that when \(n_{\text{j}}\) remains unchanged, with the increase in \(Th_{\text{j}}\), \(P_{\text{D}}\) becomes stable to 1 after its wavelike rising while \(P_{\text{FA}}+P_{\text{MD}}\) becomes stable to 0 after its wavelike dropping. Then, we set a parameter \(f_{\rm a}\) in order to maximize the accuracy rate of the navigation algorithm, as follows:  
\[f_{\rm a}=\max(P_{\rm D}). \tag{31}\]  
At the same time, we set another parameter \(f_{\rm c}\) to minimize the sum of the missing alarm rate \(P_{\rm MD}\) and false alarm rate \(P_{\rm FA}\) to evaluate the sum of integrity and availability risks, as follows:  
\[f_{\rm c}=\min(P_{\rm FA}+P_{\rm MD}). \tag{32}\]  
The simulation results are shown in Figure 10. The impact of \(n_{\rm j}\) on \(f_{\rm a}\) and \(f_{\rm c}\) is analyzed in Figure 10a. It can be seen that as \(n_{\rm j}\) increases, \(f_{\rm a}\) is stable at 1, while \(f_{\rm c}\) is stable at 0. No matter what the value of \(n_{\rm j}\) is, the value of \(f_{\rm a}\) and \(f_{\rm c}\) remains the same. That is, the value of \(P_{\rm D}\) would become to 1 and \(P_{\rm FA}+P_{\rm MD}\) would become to 0 as the value of \(Th_{\rm j}\) increases no matter what the value of \(n_{\rm j}\) is. The difference is that \(Th_{\rm j}\) is different when \(P_{\rm D}\) increases to 1. Therefore, according to Equation (15), the jump parameter \(n_{\rm j}\) is set to 2. Additionally, according to the trend of \(P_{\rm D}\) and \(P_{\rm FA}+P_{\rm MD}\) when \(n_{\rm j}\) is set to 2, \(Th_{\rm j}\) is set to 1.5 m, as shown in Figure 10b. Then, we will have \(P_{\rm D}=100\%\) and \(P_{\rm FA}+P_{\rm MD}=0\%\). Figure 10: The impact of jump parameter settings: (**a**) the impact of jump verification window; (**b**) the impact of jump verification threshold (\(n_{\rm j}=2\)). Figure 9: The influence of jump parameters on accuracy rate, false alarm rate and missing alarm rate. Next, the effect of the slow-change parameter on \(P_{\text{D}}\) and \(P_{\text{FA}}+P_{\text{MD}}\) is analyzed based on the data set \(\widetilde{\Psi}_{3}\). In this experiment, the value range of the slow-change verification window \(n_{\text{s}}\) is set to the range 1 to 20, and the value range of \(Th_{\text{s}}\) is set as follows:  
\[ \begin{split}& Th_{\text{s}}=0.01t\text{ m}\\ &\text{s.t. }t=0\,\sim 1000\end{split}  \tag{33}\]  
The simulation results are shown in Figures 11 and 12. In Figure 11, it can be seen that when the value of \(n_{\text{s}}\) is constant, with the increase in \(Th_{\text{s}}\), \(P_{\text{D}}\) becomes stable after its wavelike rising and \(P_{\text{FA}}+P_{\text{MD}}\) becomes stable after its wavelike falling. According to Figure 12a, the influence of the slow-change verification window on \(f_{\text{a}}\) and \(f_{\text{c}}\) can be analyzed and reflected. In order to maximize the accuracy rate \(P_{\text{D}}\) and minimize \(P_{\text{FA}}+P_{\text{MD}}\), we pick the slow-change parameters as \(n_{\text{s}}\)\(=5\) and \(Th_{\text{s}}\)\(=1.28\) m. Then, we will have \(P_{\text{D}}=100\%\), and \(P_{\text{FA}}+P_{\text{MD}}=0\%\), as shown in Figure 12b. Figure 11: The influence of slow-change parameters on accuracy rate, false alarm rate and missing alarm rate.