## 3 Program Design
### Classification of Health Status Classes Based on Spectral Clustering  
Due to the limitations of expert experience, which is subjective, limited, and difficult to update, it is difficult to determine reasonable thresholds to classify health states for different components. For satellite components in particular, the status of a component can often only be classified as \"qualified\" or \"failed\" by means of a fixed threshold. This inevitably leads to a waste of satellite resources. Spectral clustering is a nonparametric clustering method that groups data based on graph theory and linear algebraic methods [41]. Compared with traditional clustering methods, spectral clustering focuses more on the relationship and similarity between samples rather than relying only on the distance between samples. In the classification of component health status levels, there may be complex intrinsic connections and interactions between samples; spectral clustering can better capture these relationships to more accurately classify health status levels. Additionally, spectral clustering has the following unique advantages: (1) it is suitable for nonconvex data distributions and can accurately capture complex structures and correlations; (2) it has strong robustness and is effective in handling noise and outliers and the clustering results are not easily distorted [42; 43]; (3) the number of states can be divided artificially to adapt to different situations. When using spectral clustering for health status ranking, the WD-based HDDD is used as a feature. The main calculation steps are as follows [44]:  
1 Construct a similarity matrix \(S\), where \(S(i,j)\) denotes the similarity of the ith data point to the jth data point. 2 Construct the Laplace matrix. The Laplace matrix \(L\) is constructed from the similarity matrix \(S\). The commonly used Laplace matrices are the standard Laplace matrix and the symmetric normalized Laplace matrix. The standard Laplace matrix is defined as  
\[L=D\ -\ S \tag{6}\]  
Figure 5: Data standardization and mapping effects. where D is the degree matrix and the diagonal elements represent the degree of each data point. The symmetric normalized Laplace matrix is defined as  
\[\text{L}=\text{I}\ -\ \text{D}^{-1/2}\text{SD}^{-1/2} \tag{7}\]  
where I is the unit matrix. \(\text{\textregistered}\) Eigenvector decomposition. Eigenvector decomposition is performed on the Laplace matrix L to obtain the eigenvectors and the corresponding eigenvalues. \[\text{Lv}=\lambda\text{v} \tag{8}\]  
where L is the Laplace matrix, v is the eigenvector, and \(\lambda\) is the eigenvalue. The eigenvector corresponding to the first k eigenvalues is usually chosen as the new data representation, where k is the specified number of clusters. \(\text{\textregistered}\) Clustering. The data are clustered using a clustering algorithm (e.g., K means) based on the information in the feature vector. The data are classified into different classes of health status. We present an example using the solar sail shell temperature and bearing temperature data from a real satellite. As there are generally no \"failure\" states, except for severe failures and end of life, the HDDD of the ten-year data was therefore clustered into three states: \"Excellent, Good and Qualified\". The original data and the HDDD clustering effect are shown in Figure 6.  
Figure 6: The HDDD for the (**a**) shell temperature and (**b**) bearing temperature of a solar sail. (**c**) The spectral clustering effect of HDDD for shell temperature. (**d**) The spectral clustering effect of HDDD for bearing temperature. Figure 6 clearly demonstrates that, even for the same degree of degradation, different health characteristic parameters of the same component yield varying HDDD values. Therefore, the health state thresholds should be different for different components. It can be observed that the different state classes in Figure 6 are clearly delineated and clustered well. However, the use of expert experience only allows for delineation using fixed thresholds.