## 2 HAC-FCM Algorithm Establishment  
When dealing with the spectral polarization characteristics of space debris, traditional clustering algorithms can struggle due to the complexity and high dimensionality of the data. The FCM algorithm, which utilizes fuzzy theory-based clustering, can provide a degree of membership for each vector point to each category, making it a popular choice. However, the FCM algorithm has some limitations when applied to space debris spectral polarization data. Firstly, when new samples are introduced into the FCM algorithm, it can cause the existing initial sample density to become unbalanced, leading to further unbalanced clustering results within the same cluster. Secondly, the FCM algorithm only considers the point with the minimum distance to a single calculation, which can result in two extreme cases when the class contains too many initial points. The first case is when the points are too close to each other, which can lead to inaccurate clustering. The second case is when the points are too far apart, which can result in the clustering algorithm failing to converge. Additionally, the FCM algorithm cannot incorporate the number of samples into the final operation results, which can cause inaccuracies in clustering, especially for non-spherical data. Finally, the FCM algorithm has limited universality and is not suitable for situations where greater randomness is required. To overcome the limitations of the FCM algorithm, a combination of hierarchical clustering and FCM can be used for clustering complex and high-dimensional data such as spectral polarization data of space debris. Hierarchical clustering is used to determine the actual clusters and initial center points before applying the FCM algorithm, leading to more accurate and stable clustering results. As a result, a machine learning clustering analysis algorithm is developed for space debris targets based on spectral polarization information, which combines the HAC and FCM clustering algorithms. The algorithmic framework and clustering details of the verification process are illustrated in Figure 1. By combining these two algorithms, the accuracy and robustness of the clustering algorithm are significantly improved, making it more suitable for analyzing complex data such as spectral polarization data of space debris. The maximum value of a cluster of two points can only be established by HAC using a small-scale test set. In this paper, the number of initialized clusters was set to 1 to obtain a complete clustering tree with all distances of the clustering hierarchy. The standardized European Target Miles was selected as the measurement standard, and its target function is  
Figure 1: Schematic diagram of HAC-FCM clustering algorithm structure. shown in Equation (1). When the number of clusters in the test set is known, the distance _dist_ is calculated by obtaining the weighted mean of the normalized Euclidean distance between the minimum vector cluster and the normalized Euclidean distance within the maximum vector cluster for the specified number of clusters _dist_. Once the distances are obtained, the remaining data are trained and hierarchical cluster is used until there are no more clustering distances less than or equal to _dist_. The hierarchical tree is then cut into clusters to achieve different numbers of clusters. The number of clusters can be allocated from 1 to \(t\) (where \(t\) is the number of clusters of the final result). At this point, the number of cluster \(K\) is the initial cluster of the FCM algorithm, and the cluster \(c_{i}\) is the initial cluster of the FCM algorithm. \[Cost=\sum\limits_{i=1}^{N}\left(argmin_{j}\left(\left|\frac{x_{i}-c}{S_{k}} \right|^{2}\right)\right) \tag{1}\]  
The FCM algorithm is a widely used clustering algorithm that partitions \(N\) initial data points with no clear affiliation into \(K\) clusters, where each cluster center represents the representative value of that cluster.