# Fault Detection Algorithm Based on Dynamic Global-Local Preserving Projection
## 1 Introduction  
The rapid development of big data, artificial intelligence, and other technologies and the integration of informatization and automation have made industrial processes increasingly complex and highly integrated. All links of a complex system are closely connected. Even a minor failure may cause a major security accident. Therefore, fault-detection technology is crucial in modern industrial processes. In recent years, deep-learning methods have achieved great success in fault diagnosis. However, their practical applications still have some challenges. Deep-learning methods typically require large amounts of labeled data for training. However, large-scale labeled data may be difficult to obtain in the field of fault diagnosis. In addition, deep-learning models are barely explanatory; thus, explaining why a particular fault diagnosis result is produced can be difficult. The artificial neural network has also made some progress in the field of fault diagnosis. However, the artificial neural network also has some limitations and shortcomings in fault diagnosis. The construction of the neural network model consumes a lot of time and cost. Secondly, the network model requires high-quality sample data, while the acquisition and processing of sample data may be limited in the actual situation. In addition, the complexity and size of the network model increases with the increase in the sample data and may not be suitable for large systems. The data-driven fault-detection method can analyze and mine the feature relationship between data to build a monitoring model. This method is often used for system fault detection because of its data-driven nature. Many industrial-process data can be collected and preserved by extensively applying distributed control systems. This scenario creates conditions for data-driven, fault-diagnosis methods [1; 2; 3; 4; 5]. Classic data-driven methods include PCA [6; 7; 8; 9], fisher discriminant analysis (FDA) [10], partial least squares [11; 12; 13], and independent PCA [14; 15; 16]. These methods have been widely used in process monitoring and faultdetection for industrial systems. However, these algorithms maintain only the data's global structure while ignoring the local features of the data. Stream-learning methods represented by LPP [17; 18; 19] and neighborhood preserving embedding (NPE) [20; 21; 22] have been developed and widely used in recent years. These methods can extract hidden intrinsic properties from high-dimensional data and maintain their local structural features. They preserve the data's local structure by constructing the neighborhood connection graph. However, these methods ignore the data's global structure. They can project the data only into a very narrow area; in particular, they cannot project the data with a non-neighbor relationship to a very long distance. Therefore, these methods have certain limitations. Neither PCA nor LPP can consider data's global and local structures. Therefore, the important features of the data cannot be fully represented in the low-dimensional space using these methods. In response to this problem, [PERSON] et al. proposed a fault-detection method for global-local structure analysis by constructing a double-objective function, combining the characteristics of PCA and LPP [23]. On the basis of similar ideas, [PERSON] et al. proposed local and global PCA [24]. In this method, a double-objective function based on LPP and PCA was constructed using the ratio for the simultaneous extraction of global and local features. [PERSON] et al. proposed an algorithm called GLPP [25]. This method successfully unifies global and local structures in the same framework. [PERSON] et al. proposed a nonlocal and local structure-preserving projection algorithm for fault detection [26]. Nonlocal and local similarity weight coefficients are utilized in this method to control the distance between the projected data points. Thus, the data's nonlocal and local structures are preserved.