# Anomaly Detection and Identification in Satellite Telemetry Data Based on Pseudo-Period
## 3 Anomaly Detection and Identification of Satellite Telemetry Data Based on Phase-Plane Trajectory
Figure 7: (**a**) Sinusoidal data with hopping interference; (**b**) Phase-plane trajectory of the sinusoidal data with hopping interference. on the phase-plane, the methods based on statistics could be applied to the analysis. The first-order moments between the center point and each phase trajectory point depicted the density of distribution of phase trajectory points; however, it lost the morphological information of data. Considering the accuracy and computational complexity of feature extraction of the phase-plane trajectory, and based on the distribution of the overall phase-plane trajectory of the data, the phase-plane was divided into six statistical regions [A1 A2 B1 B2 C1 C2], as shown in Figures 6b and 8b. The distribution points of the phase-plane trajectories of the different shape data were different in each statistical region. The number of the phase-plane trajectories in each statistical region was taken as the feature to quantitatively describe the phase-plane trajectories of the data, and then the eigenvectors were established as:  
\[V_{i}=\left[N_{i}^{A1},N_{i}^{A2},N_{i}^{B1},N_{i}^{B2},N_{i}^{C1},N_{i}^{C2}\right] \tag{15}\]  
where \(N_{i}^{A1},N_{i}^{A2},N_{i}^{B1},N_{i}^{B2},N_{i}^{C1},N_{i}^{C2}\) are the fractions of the number of the phase-plane trajectories in each statistical region out of the total number of points. The eigenvector \(V_{i}\) reserves the morphological information of data; therefore, eigenvectors of data with different morphological features are quite different from each other. As shown in Figures 6b and 8b, the distribution of the phase-plane trajectories in six statistical regions indicated that the main points of the phase-plane trajectories of the standard sinusoidal data were distributed in the A1 and A2 regions, the phase-plane trajectories of the sinusoidal data with the hopping disturbance were mainly distributed in the B1 and B2 regions, and the phase-plane trajectories of the impulse data with noise were mainly distributed in the C1 and C2 regions. Therefore, different types of data could be distinguished through the data distribution of each statistical interval. The similarity between the phase-plane eigenvector of the measured data and that of the sample data was further calculated. The threshold of the similarity judgment was set to identify the anomaly type of the measured data. To highlight the distribution characteristics of the data shape in each statistical interval, cosine similarity [27] was selected to calculate the similarity between the vectors. If the phase-plane trajectory eigenvectors of each pseudo-periodic data and the sample data were \(V_{i}\) and \(V_{i}\), the chord similarity would be expressed as:  
\[sim(V,Vs)=\frac{\sum\limits_{i=1}^{l}V_{i}Vs_{i}}{\sqrt{\sum\limits_{i=1}^{l}V _{i}^{2}}\times\sqrt{\sum\limits_{i=1}^{l}Vs_{i}^{2}}}=\frac{\sum\limits_{i=1} ^{l}V_{i}Vs_{i}}{\|V\|\cdot\|Vs\|} \tag{16}\]  
When the value of \(sim(V,Vs)\) approached more to 1, the angle between the two eigenvectors was more close to 0, indicating that the states, respectively, corresponding to \(V\) and vs. were more similar to each other. If vs. referred to the _i-th_ anomalous condition, the conclusion would be drawn that there existed the _i-th_ anomaly on the data within the corresponding pseudo period.