# Intelligent Fault Diagnosis of Rotating Machinery Using Hierarchical Lempel-Ziv Complexity
## 1 Introduction  
Rotating machinery is commonly used in modern industries, such as the aero-engine, vehicle, ship, and railway industries [1]. In industrial applications, the strict working environment may result in localized damage on rotating machinery. If the damage cannot be timely diagnosed, it may cause disastrous failure and serious economic loss. Therefore, it is crucial to conduct the fault diagnosis of rotating machinery (FDRM) so as to ensure its safety operation [2]. Until now, many advanced techniques have been developed to accomplish the FDRM, such as the vibration-based fault diagnosis method [3, 4], acoustic emission-based fault diagnosis method [5, 6], and rotary encoder-based fault diagnosis method [7]. Among these techniques, the vibration signal method is most widely applied in industrial applications due to the its advantage of easy measurement and high scalability [8]. Generally, three main stages are included in the vibration-based method: data collection, fault feature extraction, and pattern identification. During the three stages, the fault feature extraction lays a good foundation for FDRM. Some advanced fault feature approaches have been proposed for the FDRM, such as blind source separation [9], wavelet-based method [10], and adaptive decomposition methods [11]. Unfortunately, the complex dynamical structure and complex operating conditions of rotating machinery often generate nonlinear and non-stationary characteristics in the measured vibrationsignals, resulting in the difficulty in extracting the weak fault characteristics from the vibration signals [12]. Thanks to the rapid development of complexity theory, some complexity indexes are used to extract fault features of rotating machinery, such as Lempel-Ziv complexity (LZC) [13], approximate entropy [14], fuzzy entropy [12; 15], permutation entropy [16; 17], symbolic dynamic entropy [18], and multi-scale entropy [19]. Related research have indicated that LZC is powerful in analyzing vibration signals of mechanical systems. [PERSON] et al. [13] used LZC to distinguish different bearing failure severity. [PERSON] et al. [20] calculated LZC after wavelet transformation to recognize the fault severity of bearing. [PERSON] et al. [21] proposed a signal decomposition and reconstruction method based on LZC and the double-dictionary matching pursuit. Meanwhile, [PERSON] et al. [22] proposed a fault diagnosis method that was based on Sparsogram and LZC. Moreover, [PERSON] et al. [23] utilized the permutation of the LZC to quantify the complexity of the signal, whereas [PERSON] et al. [24] proposed a novel symbolic aggregate approximation and LZC method for fault diagnosis of rolling bearings. However, the existing LZC methods have one common problem when analyzing the vibration signals. Since the fault information is usually embedded in vibration signals over different scale domains [25], the LZC-based methods only perform single-scale analysis, and thus the fault characteristics cannot be comprehensively described. In order to match the fault characteristics comprehensively, we proposed multi-scale Lempel-Ziv complexity (MLZC) to extract the fault features over multiple scales by multi-scale analysis [26]. However, the coarse-grained process is actually a linear smoothing process, which only considers the low-frequency components through the averaging process, and thus the fault information of the high-frequency component is discarded. In actual application, the fault information is embedded in both the low-high and high-frequency components of measured vibration signal. For example, a real vibration signal and its corresponding frequency spectrum are shown in Figure 1a,b, respectively. As can be seen, both the low-frequency (0-500 Hz) and high-frequency (2500-3500 Hz) components contain the main fault information of rotating machinery. Therefore, the fault extraction performance of MLZC still needs to be improved.