# Anomaly Detection and Identification in Satellite Telemetry Data Based on Pseudo-Period
## 2 Pseudo-Periodic Extraction of Satellite Telemetry Data
### Symbolization Method of Satellite Telemetry Data
However, the key points with a time interval that is too short may lead to a redundant number of salient points. Based on the KP method, this paper proposed a method to extract salient points of data shape (SP). By judging the interval between adjacent key points, which are extracted by the KP method, salient points can be filtered to obtain the final extracted salient points. The judgment function is given by:  
\[\varphi_{\lambda}\Big{(}\hat{x}_{j}\Big{)}= \begin{cases}\widetilde{x}_{i-1}& \tau_{i}\leq\lambda\\ LE\Big{(}\widetilde{x}_{i-1}:\widetilde{x}_{i}\Big{)}&\tau_{i}>\lambda\end{cases}  \tag{5}\]  
where \(\hat{x}_{j}\) is the final extracted salient point, \(\widetilde{x}_{i}\) is the first extracted salient point, \(\lambda\) is the time-length judgment threshold, and \(\tau_{i}\) is the time interval from \(\widetilde{x}_{i-1}\) to \(\widetilde{x}_{i}\). \(LE(\cdot)\) is a function that obtains the local extremum points between \(\widetilde{x}_{i-1}\) and \(\widetilde{x}_{i}\). The pseudo code of the salient points extraction algorithm for the data shape is given below:  
Figure 2: Filtering results of the measured current data of a solar panel. The algorithm of the salient points extraction for the data shape has two parameters: threshold of a turning point and time-length judgment threshold. When processing different numerical characteristics of telemetry data, the SP algorithm parameters need to be adjusted. The threshold of the turning point setting method is described in [23]. The time-length judgment threshold is set empirically, while its value in the real process of using can be determined by means of quantile method [24]. After the time differences of each group of adjacent salient points in the data sequence \(\left\{\tau_{j}\right\}\) are calculated, the \(\tau_{j}\) corresponding to the first quartile would be the reference value. Feature points were extracted by SP and KP from the filtered data of Figure 2, and the results are shown in Figure 3.  
As shown in Figure 3, both methods could extract the data shape. The KP method could extract 335 data feature points with the compression rate of 2.98, and the SP method could extract 107 data feature points with the compression rate of 9.34. Therefore, the SP method had a higher compression rate. Line segments were used to connect the extracted salient points in time order, and then, the compressed shape of the data could be obtained. The slope of the compressed data was used as the feature of the data shape, and the data were symbolized. For the simplicity of the symbolization, the slope values between the adjacent salient points were converted to the tilt angle values. \[\theta_{j}=atan\Big{(}k_{j}\Big{)}*180/\pi \tag{6}\]  
where \(k_{j}=\hat{x}_{j+1}-\hat{x}_{j}\) is the slope between the adjacent salient points. Based on the tilt angle, the number of intervals was determined, and a query table for the symbol conversion of the tilt angle in the range of \([-90^{\circ}\sim+90^{\circ}]\) was established. There was no overlap of the intervals in the table, and the corresponding symbols of each interval were usually expressed by letters {**a**, **b**,..., **z**}; for example, if there were two intervals, the symbols set was {**a**, **b**}, and the rest could be expressed in the same manner. In the symbolization of an angular data sequence, first, the cardinality of the symbol sets must be determined. If the symbol set is too large, the symbol sequence of the data will be too complex, whereas information will be lost if the symbol set is insufficient. The efficiency of data symbolization is directly related to the adoption of an appropriate symbol set in the process of data sequence symbolization. When the size of the symbol set is reasonably set, the ideal data compression effect can be achieved. To obtain a reasonable semiotic set, the sequence of the angle-value data is analyzed by the clustering method, and the number of symbol sets is determined by the number of clusters and each of the clusters corresponding to a symbol.