## 3 The Statistical Regression Adaptive Kalman Filtering Algorithm
### The Robustness Lake Trial  
The noise measurement is enlarged to verify SRAKF's robustness. At the time \(k\), the noise measurement is chosen to be one time and two times as much as the reference value of the noise measurement.  
\[\overline{\mathbf{R}}_{k}=\begin{cases}\mathbf{R}_{k}&\text{P}_{r}[0,0.95]\\ 2\mathbf{R}_{k}&\text{P}_{o}[0.95,1]\end{cases} \tag{41}\]  
where \(\text{P}_{r}\) and \(\text{P}_{o}\) are the probabilities of the reference value the noise measurement and outlier, respectively; where [0.0.95] denotes the reference value of the noise measurement, and [0.95,1] denotes the outlier at time \(k\), respectively.  
In the robustness lake trial, the trajectory comparison results for SRAKF, UKF, and CKF are shown in Figures 15 and 16. The latitude and longitude errors are shown in Figures 17 and 18, respectively. To obtain the quantization results, the RMSE of the position for SRAKF, UKF, and CKF are shown in Table 5.  
\begin{table}
\begin{tabular}{c c c c} \hline
**RMSE** & **UKF** & **CKF** & **SRAKF** \\ \hline RMSE\({}_{\text{La}}\)(deg) & 1.1179 \(\times\) 10\({}^{-5}\) & 4.6089 \(\times\) 10\({}^{-6}\) & 2.0964 \(\times\) 10\({}^{-6}\) \\ RMSE\({}_{\text{Lo}}\) (deg) & 1.6797 \(\times\) 10\({}^{-3}\) & 1.4708 \(\times\) 10\({}^{-3}\) & 6.6276 \(\times\) 10\({}^{-4}\) \\ \hline \end{tabular}
\end{table}
Table 4: RMSE for different algorithms.  
Figure 16: Longitude positions for the robustness lake trial.  
Figure 15: Latitude positions for the robustness lake trial.  
Figure 17: Latitude errors for the robustness lake trial.  
Figure 18: Longitude errors for the robustness lake trial.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**RMSEAlgorithms** & **UKF** & **CKF** & **SRAKF** \\ \hline RMSE\({}_{\text{La}}\)(deg) & 2.9388 \(\times\)\(10^{-5}\) & 2.9013 \(\times\)\(10^{-5}\) & 5.4470 \(\times\)\(10^{-6}\) \\ RMSE\({}_{\text{Lo}}\)(deg) & 2.3801 \(\times\)\(10^{-3}\) & 2.3713 \(\times\)\(10^{-3}\) & 1.4696 \(\times\)\(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: RMSE for different algorithms.  
performance of the three algorithms, the results of error analysis are also shown in Table 5. In Table 5, the SRAKF latitude error is less than the other two algorithms, where the former RMSE\({}_{\text{La}}\) is 5.4470 \(\times\) 10\({}^{-6}\) deg, and the UKF and CKF are 2.9388 \(\times\) 10\({}^{-5}\) deg and 2.9013 \(\times\) 10\({}^{-5}\) deg, respectively. The improvement in RMSE\({}_{\text{La}}\) is about 81.47% and 81.23% compared to UKF and CKF, respectively. In Figure 18, by employing the proposed SRAKF, the RMSE\({}_{\text{Lo}}\) for SRAKF is 1.4696 \(\times\) 10\({}^{-4}\) deg, which is lower than UKF and CKF. SRAKF reduces the RMSE\({}_{\text{Lo}}\) by about 93.83% and 93.80% compared to UKF and CKF, respectively, so the position estimation accuracy has greatly been improved by the proposed SRAKF.  
It can be seen that SRAKF performs better than UKF and CKF in handling with the changeable measurement noises. When the sensors are influenced by large noise, the SRAKF can still adaptively make an adjustment for \(\mathbf{R}_{k}\) and \(\mathbf{P}_{k|k-1}\), and it can make the algorithm more stable during the whole trial. However, the UKF cannot deal with the larger noise, because the large noise has a great impact on \(\mathbf{X}_{k|k-1}\) and \(\mathbf{P}_{k|k-1}\). In other words, SRAKF outperforms UKF and CKF because it can estimate \(\mathbf{P}_{k|k-1}\) and \(\mathbf{R}_{k}\) by using the EM method to choose the suboptimal solution of \(\Phi_{k}\). Meanwhile, the truncation errors are effectively eliminated by the unscented transformation, so the estimation accuracy of the algorithm is improved greatly. Hence, the proposed SRAKF has best performance among three algorithms when the noise measurement is time-varying.