## 4 Synthetic Time Series Analysis
Figure 9 shows the formal errors of the three components of YNDC were of the same magnitude. The means of the formal errors in Figure 9 were 1.81 mm, 1.53 mm and 1.93 mm, respectively, for Up, North and East components, and the correspondent standard deviations were 0.28 mm, 0.21 mm and 0.35 mm. Similarly, the standard deviations of the three weight factors derived from the three formal errors were 0.14, 0.13 and 0.16. By using the three formal errors in Figure 9, the same 500 simulations as Case 1 were carried out and the RMS values of the derived CMEs by IPCA and PCA are shown in Figure 10, from which we can see that the RMS values derived by IPCA were all smaller than those by PCA for three formal errors. The relative improvements of IPCA relative to PCA were 1.41%, 0.83%, 2.45%, respectively, for formal errors of Up, North and East components. The relative improvements of IPCA, with respect to PCA, for all eight stations are illustrated in Figure 11, where the improvement of East formal error component ranked first among the three components for all of the eight stations. The improvements corresponding to the three formal errors were proportional to their standard deviations or the standard  
Figure 8: Root mean square (RMS) with white and power law noise combination considered. deviations of their weight factors. In other words, the larger the standard deviation of the weight factor, the better the IPCA performed. Since the magnitude of East formal errors was also the largest, we simulated the largest Up formal errors by scaling the original Up formal errors with a factor of 0.4 (See Table 2). The corresponding improvement is also demonstrated in Figure 11 in red. Although the Up formal errors scaled by 0.4 were larger than those of East, the improvement was still less than that of East. The reason for this was that the standard deviation of the weight factor kept unchanged, regardless of whether the Up formal errors were scaled by 0.3 or 0.4. Thus, it is reasonable to conclude that the improvement of IPCA relative to PCA was mainly related to the standard deviation of the weight factor. Moreover, the improvement for the Up formal errors scaled by 0.4 was slightly better than that scaled by 0.3; therefore, the magnitude of formal errors also impacted the performance of IPCA. Figure 10: RMS values of extracted CME with different formal errors. Figure 9: Formal errors of three components (Up scaled by 0.3). Case 3: Impact of different initial CME signal  
The initial CME was extracted by PCA in Cases 1-2. In order to check the impact of different initial signals on the performance of IPCA, we plotted the extracted CME signals of YNDC using the initial CME signals derived by PCA and weighted Stacking [18] in Figure 12, where the two CME signals were exactly the same. Nevertheless, as shown in Table 3, the number of iterations of PCA was less than that of weighted Stacking. It is reasonable to conclude that although the CME derived by IPCA was not related to the initial signal, the CME derived by PCA was preferred as the initial signal. Case 4: Local effect and white, power law noise combination  
Due to different environments for different stations, local effects certainly exist and may be strong in some stations or some epochs [1]. The reconstructed time series derived from the second PC to the ninth PC were, respectively, employed to simulate the local effects for the selected eight stations. Specifically, the second PC was the local effects for the first selected station, the third PC for the second station, and so on. In this case, the simulated time series is expressed with:  
\[RMS=\sqrt{\frac{\sum\left(\mathbf{S}_{0i}-\hat{\mathbf{S}}_{0i}\right)^{2}}{m_{i}}} \tag{15}\]0  
where \(\mathbf{L}_{0}(t_{k},i)\) represents the local effects, and the other symbols denote the same meaning as Equation (14). With the local effects considered, the RMS values of the extracted CME signals of 500 simulations are shown in Figure 13 for the station YNDC, in which the performance of IPCA was also superior to PCA.