# Student's t Kernel-Based Maximum Correntropy Criterion Extended Kalman Filter for GPS Navigation
## 4 Results and Discussion
### Environment 2: Impact of Outliers in a Time-Varying Noise Environment  
In this section, to evaluate the characteristics of various filters, an environment was designed that includes both time-varying observation noise and outliers. This setup aims to verify whether the proposed filters perform better in handling time-varying observation noise and to observe if they can also effectively suppress outliers within this context. The goal is to achieve simultaneous handling of time-varying observation noise and outliers within a single filtering framework, thereby improving overall filtering accuracy. In this experiment, the characteristics of pseudorange error illustrated in Figure 8 represent a time-varying, non-Gaussian environment constructed by injecting thermal noise of different intensities into two specific time intervals. In addition, ten outliers with varying magnitudes were randomly inserted throughout the signal to simulate abrupt disturbances. This setup reflects a more complex and realistic noise condition compared to static Gaussian assumptions. Traditional Gaussian-based filters like the EKF may struggle under such conditions due to their reliance on fixed noise statistics. In contrast, robust  
Figure 7: Variation in kernel bandwidth over time for Environment 1.  
filtering approaches, such as those incorporating the Student's t-distribution or adaptive noise covariance estimation, are more suitable for handling such non-stationary noise environments. These methods can better accommodate dynamic fluctuations and suppress outlier effects, thereby enhancing overall estimation robustness. Figure 9 shows the adaptive kernel bandwidth variation over time in Environment 2. The kernel bandwidth decreases sharply at several points, reflecting the filter's response to abrupt noise changes of outliers. This behavior highlights the filter's ability to adjust dynamically for enhanced robustness under non-Gaussian conditions. Based on Figure 10, it can be observed that the performance of MCCEKF and STMCCEKF is comparable to that of EKF during periods without anomalies. However, when anomalies occur, both MCCEKF and STMCCEKF demonstrate superior robustness, with STMCCEKF exhibiting a more effective suppression of large deviations in position errors. As illustrated in the test environment, time-varying noise combined with randomly injected outliers creates a challenging environment for state estimation. Although MCCEKF adopts a wider kernel bandwidth \(\sigma\), its ability to mitigate the effects of outliers is limited. In contrast, STMCCEKF employs a Student's t-distribution kernel with adaptively  
Figure 8: Outliers in time-varying noise in Environment 2. Figure 9: Variation in kernel bandwidth over time for Environment 2.  
adjusted bandwidth \(\sigma\) and degrees of freedom \(v\), where \(\sigma\) is capped at 12 to maintain numerical stability. This configuration enhances its adaptability to heavy-tailed noise and confirms the critical role of \(v\) in improving robustness against non-Gaussian disturbances. In STMCCEKF, both the process and measurement noise are modeled using the Student's t-distribution, which inherently reduces sensitivity to extreme values. As a result, the presence of outliers has minimal impact on the overall estimation accuracy. Additionally, STMCCEKF incorporates a generalized stochastic integration rule (SIR), enabling it to effectively combine the statistical properties of both Student's t and Gaussian distributions for enhanced performance under complex noise conditions. Although the proposed STMCCEKF introduces additional computations due to the use of the adaptive Student's \(t\) kernel and the fixed-point iteration, its computational complexity remains feasible for real-time implementation. The iteration typically converges within 3-7 steps per time update, and the associated operations primarily involve matrix/vector multiplications and scalar evaluations, which are computationally efficient and parallelizable.