## 4 Conclusions  
This paper introduces an extended Kalman filter based on the maximum mixture correntropy criterion with nonlinear regression (MCCEKF-NR) for GPS navigation and positioning applications. The mixture correntropy was extended with the original correntropy by combining the kernel functions with EKF, and integrating them into a nonlinear regression-based filtering algorithm technique. The resulting filter was designed from the corresponding cost function, which offers better robustness and filtering performance than typical correntropy-based filters and also other types of EKF-based filters. By repeatedly altering the Taylor expansion's center point, the iterated extended Kalman filter enhances the extended Kalman filter's linearization. Although more processing power is needed, the linearization error is decreased.  
Although there is no set standard for choosing the kernel width in the original correntropy, it has a big influence on the system as a whole. Thus, more flexibility is possible when employing the MCC, which integrates the kernel bandwidth, including distinct algorithm techniques. Using the fixed-point iteration for Gaussian and mixed-Gaussian noises ensures effective performance in the presence of outliers. This method avoids problems such as divergence resulting from extremely large or very small kernel bandwidth when the outliers are present. The effect of the ratio between the kernel bandwidth and the filter performance also plays a vital role in choosing an appropriate kernel bandwidth. The proposed MCCEKF-NR and Mixture-LRMCCEKF can achieve better performance as compared to the different MCCEKF techniques for a suitable kernel bandwidth; this has  
Figure 7: The root mean square error (RMSE) for different MCCEKF with respect to different positions: (**a**) east, (**b**) north, and (**c**) altitude.  
been verified from the simulation results. Additionally, we implemented the linear and nonlinear regression-based algorithms for the MCCEKF to improve the estimation accuracy by considering the iterated EKF (IEKF) for the GPS navigation.  
Further, the simulation tests for GPS navigation were included to demonstrate the efficiency. The findings indicate that the adaptive regression filtering algorithm based on the kernel entropy principle exhibits a discernible enhancement in navigation accuracy when interconnecting with the traditional techniques. Consequently, it shows promising potential as a substitute for the GPS navigation processor, particularly when dealing with observables that exhibit mixed-Gaussian errors. We demonstrate our findings in four different methods of error calculation for Gaussian and mixed-Gaussian noise environments. Performance comparisons were conducted for a number of techniques, such as EKF, MCCEKF-BC, MCCEKF-DS, MCCEKF-GL, MCCEKF-NR, and Mixture-LRMCCEKF. The kernel entropy-based MCCEKF with nonlinear regression algorithm showed encouraging results in terms of improving navigational accuracy. Moreover, this work can be extended to design MCCEKF-based filters for state-constrained processes, and systems with continuous networks.  
Conceptualization, A.B.; methodology, A.B.; software, A.B.; validation, A.B.; writing original draft preparation, A.B.; writing review and editing, A.B. and D.-J.J.; supervision, D.-J.J. All authors have read and agreed to the published version of the manuscript.  
The author gratefully acknowledges the support of the National Science and Technology Council, Taiwan under grant numbers NSTC 112-2221-E-019-030 and NSTC 112- 2811-E-019-004.  
Not applicable.  
Data available statementData are contained within the article.  
The authors declare no conflicts of interest.