# Anomaly Detection and Identification in Satellite Telemetry Data Based on Pseudo-Period
## 5 Simulation Results  
To verify its effectiveness, the proposed method was tested using the measured data of the momentum wheel speed of a satellite for which the data length was equal to 1658. The measured data are shown in Figure 10.  
Figure 10: Measured data of the momentum wheel speed of a satellite. Figure 9: Flow chart of detection and identification of the satellite anomalous telemetry data based on the pseudo-period. First, the measured data were compressed by the SP method, and the salient points were extracted. The compressed data are shown in Figure 11.  
As shown in Figure 11, the number of salient points extracted from data was 67, and the corresponding data compression rate was 24.74. The data were further symbolized. The number of the symbols was determined as 5 by the CFDP method. Therefore, the corresponding characters {**a,b,c,d,e**} were selected. Then, the strings to be measured were obtained as:  
{ddddcdacdadddddacadddddcdabcaeddcadeacdeddeeaaeedddadadadaeeddcdadadadae}  
The standard pseudo-periodic sequence string was given as {ddcdacda}. The standard pseudo-periodic sequence string was used to segment the pseudo-periodic. The pseudo-periodic was divided as follows:  
To verify the accuracy of the pseudo-periodic partition, the standard pseudo-periodic sequence string was used as the reference, and the exact pseudo-periodic in the data was manually determined. The results were compared with the results obtained by the pseudo-periodic extraction algorithm proposed in this paper. Table 2 shows the analysis results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Extraction Method** & **Partition Points** & **1** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline Proposed method & 263 & 512 & 763 & 960 & 1169 & 1383 & 1645 \\ Manual method & 264 & 513 & 763 & 961 & 1169 & 1385 & 1644 \\ Error analysis & - & \(-1\) & 0 & \(-1\) & 0 & \(-2\) & \(+1\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis of the pseudo-period extraction for the data. Figure 11: Salient points of the measured data of the rotational speed of a momentum wheel. In Table 2, \"\(-\)\" indicates the advanced error, and \"+\" indicates the delayed error. As shown in the error analysis, the proposed pseudo-period extraction method based on the data shape showed high accuracy, making this method suitable for application to the actual telemetry data pseudo-period extraction. The anomaly data in each pseudo-period were further identified, and the phase-plane trajectory of the normal standard pseudo-period data of the measured data was given, as shown in Figure 12.  
The phase-plane was divided into six statistical regions. The data points of each region were counted as the normal data of the standard pseudo-period. The eigenvectors of the phase-plane trajectory were obtained, as shown in Table 3.  
The pseudo-periodic phase-plane trajectories of the measured data were obtained, and the phase-plane trajectory eigenvectors were obtained by statistical analysis in the same statistical region, as shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pseudo-Period** & **Statistical Interval** & **A1** & **A2** & **B1** & **B2** & **C1** & **C2** \\ \hline
1 & 0.116 & 0.116 & 0.100 & 0.064 & 0.428 & 0.176 \\
2 & 0.096 & 0.088 & 0.096 & 0.068 & 0.371 & 0.278 \\
3 & 0.068 & 0.096 & 0.088 & 0.104 & 0.400 & 0.244 \\
4 & 0.056 & 0.102 & 0.051 & 0.061 & 0.403 & 0.326 \\
5 & 0.058 & 0.058 & 0.033 & 0.082 & 0.391 & 0.377 \\
6 & 0.070 & 0.107 & 0.079 & 0.107 & 0.243 & 0.392 \\
7 & 0.100 & 0.103 & 0.100 & 0.096 & 0.257 & 0.342 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Eigenvectors of the plane trajectories of the pseudo-periodic phases for the measured data. Figure 12: Phase-plane trajectory of the standard pseudo-periodic reference data for the rotational speed of a momentum wheel. The cosine similarity distances between the pseudo-periodic vectors presented in Table 4 and the standard pseudo-periodic vectors presented in Table 3 were calculated.