## 2 Methodologies
### Maximum Correntropy Criterion-Based Extended Kalman Filter with Nonlinear Regression  
MCCEKF-NR is an approach for additional performance improvement. The flow diagram for a single cycle of the extended Kalman filter using the maximum correntropy criterion is shown in Figure 1 with nonlinear regression (MCCEKF-NR), which involves the nonlinear regression-based computation procedure for MCCEKF. In order to improve the shortcomings of the KF and estimate the noise covariance matrices, an extended KF can be employed as the filter with noise-adaptivity technique. The adaptive approach has the advantage of maintaining covariance in line with actual performance. The covariance-matching and correlation methods use the innovation sequences to estimate the noise covariances. Making the residual's real covariance value compatible with its theoretical value is the fundamental tenet of the covariance-matching strategy. It is assumed that the mean of the process and measurement noise terms is zero, at which point the anticipated state estimate and measurement residual are assessed. Other than that, the non-additive noise formulation is applied using the same methodology as the additive noise EKF. By repeatedly changing the Taylor expansion's center point, the iterated extended Kalman filter enhances the extended Kalman filter's linearization [35]. At the expense of higher processing demands, this lowers the linearization error. The designer can adjust the scalar that parameterizes the extra term to balance the mean-square-error and peak error performance criterion. The procedures involved in computing the EKF based on the maximum correntropy criterion with nonlinear regression (MCCEKF-NR) is presented below:  
Figure 1: Flow diagram for the MCCEKF-NR. 1. Let us first consider a kernel bandwidth \(\sigma\), and \(\varepsilon\) with small positive value for the error tolerance. Then initialize \(\hat{\mathbf{x}}_{0}\), \(\mathbf{P}_{0}\);
2. Then predict, \(\hat{\mathbf{x}}_{k}=f(\hat{\mathbf{x}}_{k-1})\), \(\widetilde{\mathbf{P}}_{k}=\mathbf{F}_{k-1}\hat{\mathbf{P}}_{k-1}\mathbf{F}^{ T}{}_{k-1}+\mathbf{Q}_{k-1}\)
3. Updating, \(\varphi_{k}=\mathbf{y}_{k}-\mathbf{h}(\widetilde{\mathbf{x}}_{k}),\delta_{k}= \mathbf{H}_{k}\widetilde{\mathbf{P}}_{k}\mathbf{H}_{k}^{T}+\mathbf{R}_{k}\), \(\beta_{k}=\varphi_{k}^{T}\delta_{k}^{-1}\varphi_{k}\). If \(f|\beta_{k}|>\theta\) then \(\hat{\mathbf{x}}_{k}=\widetilde{\mathbf{x}}_{k},\hat{\mathbf{P}}_{k}= \widetilde{\mathbf{P}}_{k}\). Otherwise
4. Iteration loop: Calculation of \(\hat{\mathbf{x}}_{k,t}\) using the following steps: i. \[\hat{\mathbf{x}}_{k}^{(t)}=\widetilde{\mathbf{x}}_{k}+\mathbf{K}_{k}^{(t-1) }(\mathbf{y}_{k}-h(\hat{\mathbf{x}}_{k}^{(t-1)})-\mathbf{H}(\hat{\mathbf{x}}_ {k}^{(t-1)})(\widetilde{\mathbf{x}}_{k}-\hat{\mathbf{x}}_{k}^{(t-1)}))\] ii. \[\mathbf{K}_{k}^{(t-1)}=\widetilde{\mathbf{P}}_{k}^{(t-1)}\mathbf{H}^{T}( \hat{\mathbf{x}}_{k}^{(t-1)})\times\left(\mathbf{H}(\hat{\mathbf{x}}_{k}^{(t -1)})\widetilde{\mathbf{P}}_{k}^{(t-1)}\mathbf{H}^{T}(\hat{\mathbf{x}}_{k}^{(t -1)})+\mathbf{R}_{k}^{(t-1)}\right)^{-1}\] iii. \[\widetilde{\mathbf{P}}_{k}^{(t-1)}=\mathbf{B}_{k,\mathbf{P}}( \mathbf{C}_{k,k}^{(t-1)})\mathbf{B}_{k,\mathbf{P}}^{T}\] iv. \[\mathbf{R}_{k}^{(t-1)}=\mathbf{B}_{k,\mathbf{R}}(\mathbf{C}_{k,k}^{(t-1) })\mathbf{B}_{k,\mathbf{P}}^{T}\] v. \[\mathbf{C}_{k,x}^{(t-1)}=diag(G_{\sigma}(\epsilon_{k,1}^{(t-1)}), \ldots,G_{\sigma}(\epsilon_{k,n}^{(t-1)}))\] vi. \[\mathbf{C}_{k,y}^{(t-1)}=diag(G_{\sigma}(\epsilon_{k,n+1}^{(t-1) }),\ldots,G_{\sigma}(\epsilon_{k,n+m}^{(t-1)}))\] vi. \[e_{k,i}^{(t-1)}=z_{k,i}-d_{i}(\mathbf{K}_{k}^{(t-1)})\] viii.