## 1 Introduction  
Some faults may occur during the operation of satellites because of the uncertainties of the space environment, the limitations on pre-launch testing, and the imperfect design of satellites. It remains a challenge to detect and solve incipient faults and avoid further complications [1, 2]. At present, the detection of faults in satellites is done by comparing the telemetry parameters with the preset thresholds [3, 4]. However, this method is not suitable for detecting incipient faults because they may not change significantly during early manifestation of the fault. Compared to system signals, the amplitudes of incipient faults are small, typically from 1 to 10% [5, 6] and can easily be masked by the normal variations in systems [7]. Model-based fault detection methods have the advantage of being highly intelligent and interpretable. Examples of such systems include the Livingstone2 and the HyDE fault diagnosis models developed by National Aeronautics and Space Administration (NASA) [8] as well as the Testability Engineering and Maintenance System (TEAMS), developed by Qualtech Systems Inc (QSI) [9]. All of these model-based fault detection methods require manual fault modeling by experts in the field. In addition, the accuracy of these models has a determining impact on their final performance at detecting faults. However, due to the high cost yet low yield of components used in satellites, the lifetime tests or the acquisition of actual fault samples required to build accurate models may often be too costly. Moreover, model-based fault detection methods are only capable of detectinga narrow range of faults. This means that there may be faults that occur outside of the range detectable by the models and this may result in undetected incipient faults [10]. In recent years, data-driven fault detection methods have become a popular field of research [11; 12; 13]. These methods effectively utilize the extensive ground test data and operational data of satellites, with limited involvement of experts and do not require accurate mathematical models. [PERSON] et al. [14] proposed a fault detection method named the Inductive Monitoring System (IMS) and used it to detect faults in the Columbia Space Shuttle, and the TacSat-3 satellite. [PERSON] et al. [15] compared four data-driven fault detection methods containing IMS and One Class Support Vector Machine (OS-SVM). [PERSON] et al. [16] proposed a fault detection method based on Gaussian Process Regression (GPR) and implemented it to detect faults in the periodic telemetry parameters of satellites [16]. [PERSON] et al. [17] proposed a spacecraft fault detection method based on Long Short-Term Memory (LSTM) networks and dynamic thresholds. [PERSON] et al. [18] proposed a fault detection method using a combination of multivariate convolutional LSTM and probabilistic principal component analysis (PCA) and applied it to the Korea Multi-Purpose Satellite 2 (KOMPSAT-2). Among these data-driven fault detection methods, distance-based fault detection methods such as IMS and OS-SVM are suitable for detecting fault data that are significantly different from normal data. However, these methods are less effective when the fault data and normal data mostly overlap. The fault detection methods using GPR or LSTM are suitable for detecting faults in regular (e.g., periodic) and time-continuous systems, but are ineffective when the telemetry parameters include interruptions in time. Among the data-driven fault detection methods, methods based on PCA rely on the simplicity of the principle and on good real-time performance and have been used extensively in detecting faults in equipment [19; 20]. The Hotelling's \(T^{2}\) and squared prediction error (SPE) statistic are usually used to measure deviations in the principal subspace, and to measure deviations in the residual subspace, respectively. However, the conventional \(T^{2}\) and SPE statistics are not effective at detecting incipient faults that are easily drowned out by noise [21]. [PERSON] et al.