# Fault Prediction for Rotating Mechanism of Satellite Based on SSA and Improved Informer
## 3 Methods  
### Ssa  
This paper employs the singular spectrum analysis (SSA) algorithm to decompose and reconstruct satellite turntable telemetry data. SSA is a time-series analysis method based on Singular Value Decomposition (SVD) [23]. The core idea of SSA is to decompose a time series into components of different frequencies. This allows for noise removal and helps identify periodic patterns in the signal, achieving effective denoising [24]. Through the analysis of eigenvectors, it is possible to determine the frequency, amplitude, and phase of the periodic components present in the signal, thereby revealing the periodic variations of the signal. Generally, SSA is divided into two parts: matrix decomposition and series reconstruction [25].  
The analysis object of SSA is one-dimensional time series with finite length. Firstly, we need to select the appropriate window length to arrange the original time series with delay to obtain the trajectory matrix:  
Step 1: Decomposition  
The original time series is mapped into a vector sequence of length \(L\) to form \(i+L-1\) vectors of the following length:  
\[\mathrm{X_{i}}=(x_{i},\ldots,x_{i+L-1})^{\mathrm{T}}\quad(1\leq\mathrm{i}\leq \mathrm{K}) \tag{1}\]  
These vectors form a trajectory matrix, as follows:  
\[\mathbf{X}=\begin{bmatrix}X_{1}&X_{2}&\cdots&X_{i-L+1}\\ X_{2}&X_{3}&\cdots&X_{i-L+2}\\ \vdots&\vdots&\ddots&\vdots\\ X_{L}&X_{L+1}&\cdots&X_{i}\end{bmatrix} \tag{2}\]  
The trajectory matrix \(\mathbf{X}\) is decomposed of singular value decomposition. By calculating \(\mathbf{X}\mathbf{X}^{\mathrm{T}}\) and performing Singular Value Decomposition (SVD), \(L\) eigenvalues are obtained, as follows:  
\[\mathbf{X}=\mathbf{X}_{1}+\ldots+\mathbf{X}_{d} \tag{3}\]  
\[\mathbf{X}_{\mathrm{i}}=\sqrt{\lambda_{\mathrm{i}}}\mathrm{U_{i}}\mathrm{V_{ i}^{\mathrm{T}}} \tag{4}\]  
where \(\lambda_{\mathrm{i}}\) is the eigenvalue of the matrix \(\mathbf{X}\mathbf{X}^{\mathrm{T}}\); \(\mathrm{U_{i}}\) and \(\mathrm{V_{i}}\) are orthogonal eigenvectors of the trajectory matrix \(\mathbf{X}\).  
Step 2: Reconstruction  
Reconstruction involves grouping the sub-matrices obtained from SVD to form several sub-matrices and calculating the contribution rate of each sub-matrix to determine its variability. The sub-matrices include high-frequency sub-matrices, low-frequency sub-matrices, and noise sub-matrices [26]. These sub-matrices are then restored to a reconstructed sequence with a shape similar to the original signal through diagonal averaging. The contribution rate can be expressed as follows:  
\[\eta=\sum_{i=1}^{r}\lambda_{i}/\sum_{i=1}^{d}\lambda_{i} \tag{5}\]  
The sub-matrices obtained by grouping are reconstructed into \(L\times K\) matrix \(\mathrm{X_{ij}}\) via the diagonal average formula, and the expression of reconstructed time series \(Y\) is as follows [24]:  
\[\hat{y}=\left\{\begin{array}{rl}\frac{1}{k}\sum_{p=-1}^{k}X_{p,k-p+1}^{*}(1 \leq k\leq L^{*})\\ \\ \frac{1}{k^{*}}\sum_{p=-1}^{L^{*}}X_{p,k-p+1}^{*}(L^{*}\leq k\leq K^{*})\\ \\ \frac{1}{N-k+1}\sum_{p=-K^{*}+1}^{-K^{*}+1}X_{p,k-p+1}^{*}(K^{*}\leq k\leq N) \end{array}\right. \tag{6}\]where \(L^{*}=\min\{L,K\}\), \(L^{*}=\min\{L,K\}\), \(N=L+k-1\), the sum of the sequences obtained after reconstruction is the total time series after singular spectrum analysis decomposition.