## 3 Program Design  
### WD-Based Multiparameter Component Assessment Model  
#### 3.1.1 WD-Based Single Parameter Health Calculation  
Among the commonly used methods to characterize the differences in data distributions are total variation, JS dispersion, KL distance, etc. However, for satellite data with complex variations, the Wasserstein distance is more suitable than these methods. Total variation is insensitive to subtle variations in the distribution and shape differences, which makes it difficult to capture early anomalies in components in time [32]. JS dispersion does not accurately reflect the differences in dealing with nonoverlapping data distributions. The asymmetric characteristics of the KL distance make its results unstable [33]. To overcome these limitations, the Wasserstein distance (WD) is introduced as a mathematical tool for measuring differences in probability distributions. It has attracted a lot of attention and research in recent years with the rise of deep learning. The WD was proposed to address some of the limitations and shortcomings of the traditional distance metric when dealing with probability distributions [29; 34]. It provides more information when measuring differences between data, rather than being limited to statistical distributions of  
Figure 3: Overall program. values. In contrast to the MMD, the WD takes into account the differences in position and shape between the two distributions. It captures nonlinear relationships and local nuances, assessing the dissimilarity by determining the optimal mass transfer between distributions. The WD quantifies the average cost of transporting mass from one distribution to another in an optimal manner [34; 35; 36]. In summary, the superiority of the WD lies in its ability to take into account the structure and shape of probability distributions. The WD describes the differences between the state distributions of data by minimizing the transfer cost. This gives it better performance in dealing with practical problems such as multimodality, high-dimensional data, and noise and makes it more suitable compared with other distribution description methods [37]. The mathematical representation of the WD is as follows [30; 38]:  
\[\text{Wp (P,Q)}=\left(\inf_{\gamma\in\pi\left(\text{P,Q}\right)}\int_{\text{R \leavevmode\
obreak\ \times\leavevmode\
obreak\ R}}||\text{x \leavevmode\
obreak\ - \leavevmode\
obreak\ y}\ ||^{\text{P}}\text{d}\gamma\left(\text{x},\text{y}\right)\right)^{\frac{1}{ \rho}} \tag{1}\]  
where \(\pi\)(P,Q) denotes the set of all joint probability distributions with marginal distributions P and Q; \(\gamma\) is the transfer plan from P to Q, which describes how the mass is transferred from P to Q. In the component assessment model, the WD is used to describe the variability between the real-time operational data of the component and the benchmark data. By calculating the WD, a positive data distribution health indicator, HDDD, can be obtained to quantify the health status of the component. \[\text{HDDD}\leavevmode\
obreak\ =\leavevmode\
obreak\ 1\leavevmode\
obreak\ - \leavevmode\
obreak\ \text{W}_{\text{P}}(\text{P,Q}) \tag{2}\]  
A high HDDD value indicates a small difference between the real-time data and the benchmark data, suggesting that the component is in a good state of health. Conversely, a low HDDD value indicates a potential abnormality or deterioration in the component's health. The advantage of the new method is illustrated by the X-axis current data of a faulty gyroscope. The data are derived from telemetry data from a real satellite. From Figure 4, it can be seen that the current data are abnormally elevated around day 1500 and a fault occurs. At this time, both the SOC and HDDD based on the WD capture the anomaly timely and accurately. However, the HDDD based on the MMD is not sensitive to the anomaly and has a certain lag. This reflects the ability of the HDDD based on the WD to detect faults and anomalies in a timely manner, especially in the early stages of faults and anomalies.