# Two-Line Element Outlier and Space Event Detection Method Based on Multi-Strategy Genetic Algorithm
## 2 Method
### Sample Data Generation  
TLE sets serve as a crucial data format, encapsulating the essential orbital parameters of Earth-orbiting objects at a specific epoch. For instance, TLE data typically comprise orbital information, including the NORAD ID, B* drag term, orbital inclination, eccentricity, and mean motion of the space target. The Simplified General Perturbations 4 (SGP4) model, developed in the 1980s, is an analytical method encapsulating orbital information to form Two-Line Element (TLE) sets. Unlike complex numerical integration methods, SGP4 simplifies the satellite perturbation model, significantly reducing computational requirements while providing a position accuracy of approximately 1 km. This capability enables the rapid cataloging of thousands of space objects. It is important to note that TLEs must be computed using the SGP4 model for a reliable determination of an object's position and velocity. Thus, we employed the SGP4 model to calculate the prediction errors of TLEs, as illustrated in Figure 1, depicting the sample generation process. The TLE values' track parameter \(a_{i}\) at time \(t_{i}\) is extrapolated to \(t_{i+1}\) to obtain \(a_{i}^{i+1}\), and the value at \(t_{i+1}\) is utilized to obtain track parameter \(a_{i+1}\). The predicted values of the orbital parameters are subtracted from the TLE values to obtain the prediction intervals \(dt_{i}^{i+1}=t_{i+1}-t_{i}\) and prediction errors \(da_{i}^{i+1}=a_{i}^{i+1}-a_{i}\). Dividing the two quantities yields the rate of change of the prediction errors of the orbital parameters \(\Delta da_{i}^{i+1}=da_{i}^{i+1}/dt_{i}^{i+1}\). Similarly, the change rate of the m prediction errors \(\Delta da_{i}^{i+1}\), \(\Delta da_{i}^{i+2}\), \(\cdots\Delta da_{i}^{i+m}\) for a single element is obtained by extrapolating the TLE value at time \(t_{i}\) to time \(t_{i+1},t_{i+2},\cdots,t_{i+m}\), to where the subsequent m values refer.  
It should be noted that the generation of patterns can be achieved not only using the semi-major axis, but also by selecting the orbital inclination, eccentricity, and orbital energy. This process is repeated for all catalog values, resulting in prediction errors at different times. They comprise the sample data of the prediction errors, i.e., the set \(\mathbf{S}\).  
\[\mathbf{S}=\begin{bmatrix}\begin{pmatrix}\Delta\text{da}_{1}^{2},\Delta di_{1}^ {2}\\ \Delta\text{da}_{2}^{3},\Delta di_{2}^{3}\end{pmatrix}&\begin{pmatrix}\Delta \text{da}_{1}^{3},\Delta di_{1}^{3}\\ \Delta\text{da}_{2}^{4},\Delta di_{2}^{4}\end{pmatrix}&\cdots&\begin{pmatrix} \Delta\text{da}_{1}^{m},\Delta di_{1}^{m}\\ \Delta\text{da}_{2}^{3},\Delta di_{2}^{3}\end{pmatrix}\\ \vdots&\vdots&\ddots&\vdots\\ \begin{pmatrix}\Delta\text{da}_{n}^{n+1},\Delta di_{n}^{n+1}\end{pmatrix}&\begin{pmatrix} \Delta\text{da}_{n}^{n+2},\Delta di_{n}^{n+2}\end{pmatrix}&\cdots&\begin{pmatrix} \Delta\text{da}_{n}^{n+m},\Delta di_{n}^{n+m}\end{pmatrix}\end{bmatrix} \tag{1}\]  
where \(\Delta\text{da}_{n}^{n+m}\) is the rate of change of the semi-major axis prediction error from the nth catalogued value to the \(n+m\)th catalogued value. \(\Delta\text{di}_{n}^{n+m}\) is the rate of change of the orbital inclination forecast error from the nth catalogued value to the \(n+m\)th catalogued value.  
Figure 1: Sample data generation.