# Student's t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter for GPS Navigation
## 4 Results and Discussion
### Environment 1: Effect of Outliers on Data with Changing Standard Deviation over Time  
In this section, the performance of various filters in suppressing outliers in a GPS multipath environment will be validated. The setup for this environment is the multipath effect noise \(\delta_{mp}\), which follows a Student's t-distribution with 3 degrees of freedom (\(v=3\)), as illustrated in Figure 3. The thermal noise \(\delta_{noise}\) is modeled as Gaussian noise with mean and standard deviations. To simulate realistic environments, the additional disturbances induced by outliers are injected into the pseudorange measurements at predetermined time steps. The outliers are incorporated into the multipath error by scaling according to predefined values. These configurations will allow for a comprehensive assessment of each filter, which handles and mitigates the impact of outliers and noise in the given environment. Figure 4 illustrates the resulting error sequence over the full simulation period, where 30 outliers are introduced based on varying standard deviations. These outliers are embedded in the Student's t-distributed multipath noise, producing heavy-tailed behavior and occasional sharp deviations. The induced pseudorange errors span a wide range, from tens up to more than a hundred meters, emulating the abrupt multipath disruptions commonly encountered in urban GPS environments. This setup enables a robust evaluation of each filter's ability to handle diverse noise characteristics and mitigate severe measurement faults. Figure 5 analyzes the performance differences among MCCEKF-1, MCCEKF-2, and STMCCEKF with fixed kernel bandwidth. In the experiments, both MCCEKF variants adopt a fixed kernel bandwidth of \(\sigma=50\) but differ in their kernel function or implementation settings, which leads to slight variations in robustness to outliers. Additionally, the STMCCEKF with a fixed kernel bandwidth is \(\sigma_{\text{ST}}=30\). These kernel bandwidths play a vital role in determining the performance of each filter by affecting the sensitivity to noise and outliers. Figure 6 compares the position errors of the EKF and STMCCEKF filters, with fixed and adaptive kernel bandwidths, under multipath interference. During periods without outliers, both filters exhibit similar performance. However, when significant anomalies are introduced, STMCCEKF demonstrates improved robustness by effectively suppressing most of the induced noise. Figure 3: Student’s t probability density function (**a**) without Gaussian outliers, (**b**) with Gaussian outliers. From Figures 5 and 6, it can be observed that, during periods without outliers, MC-CEKF and EKF show similar performance. However, when noises are introduced, STMC-CEKF suppresses most of the noise caused by these disturbances. Nonetheless, there are occasional instances where the kernel bandwidth is insufficient to handle the noises effectively. After using STMCCEKF, the performance improves in terms of noise suppression. It is important to point out that the MCCEKF exhibits greater sensitivity to measurement  
Figure 4: Outliers with varying standard deviations in Environment 1. Figure 5: Position errors of EKF, MCCEKF-1, MCCEKF-2, and STMCCEKF (with fixed kernel bandwidth) for Environment 1: (**a**) east; (**b**) north; (**c**) altitude. Figure 6: Position errors of EKF, STMCCEKF, and STMCCEKF-AKB for Environment 1: (**a**) east; (**b**) north; (**c**) altitude. noise, whereas the MCCEKF variant incorporating the Student's t kernel effectively suppresses noise, demonstrating its resilience against outliers. This contrast further validates the STMCCEKF's capability in attenuating measurement uncertainties, thereby improving state estimation accuracy in non-Gaussian noise environments. The STMCCEKF-AKB is based on the Student's t-distribution and incorporates an adaptive kernel bandwidth that adjusts according to the noise characteristics. Figure 7 shows the variation in kernel bandwidth over time for Environment 1.