## 1 Introduction  
Nowadays, reciprocating compressors are extensively utilized in the industrial compressible fluids sector, particularly in the petrochemical industry. It is imperative to conduct equipment monitoring, maintenance, and fault research which ensure the smooth operation of these compressors [1]. The vibration signals of reciprocating compressors exhibit strong non-stationary characteristics and multi-component coupling characteristics due to their complex structures and multiple excitations [2]. The inherent structural characteristics of reciprocating compressors result in vibration signals that encapsulate the vibrational attributes of various components, rendering the identification of fault characteristics challenging [3]. The crucial issue in fault diagnosis of mechanical equipment is the ability to accurately extract the features that reflect the state of the machine. In recent years, multiple methodologies for feature extraction in mechanical fault diagnosis have been devised by researchers [4]. These methodologies include analysis methods based on signal processing in the time domain, frequency domain, and time-frequency domain [5]. Feature extraction methods are informed by information theory, including information entropy [6] and mutual information [7]. Additionally, techniques based on sparsity measurement [8] and deep learning have gained prominence [9; 10]. Nonetheless, sparsity analysis is often used to monitor the working trend and state changes of equipment, and the signals generated by reciprocating compressors may not have obvious sparsity, which makes it difficult to effectively use sparsity [11]. A significant amount of historical operating data is required to obtain features through deep learning [12]. In practical engineering applications, it may be difficult to obtain adequate operational data on reciprocating compressors. Hence, this article will not delve into the latter two methods. Fault diagnosis is essentially a process of recognizing fault patterns and its success relies on effectively extracting and utilizing information from vibration signals. Empirical evidence indicates that the characteristic distributions and statistical parameters of reciprocating compressors in varying conditions are discernible, thereby offering a foundation for the development of fault classification models [13]. Traditional approaches may be inadequate for complex datasets as they frequently presuppose linearity and stationarity within the data. Yet, in practical engineering contexts, the data commonly present non-linear and non-stationary behaviors. This renders traditional time-frequency methods in signal processing limited in effectively extracting features from non-stationary vibration signals. The emergence of non-linear technology has led to the widespread use of analysis methods based on various information theories in the manufacture of mechanical equipment [14]. Therefore, entropy-based fault detection methods are increasingly garnering attention because of their ability to capture the intricacy and irregularity of signals. Commonly utilized entropy techniques include approximate entropy (ApEn) [15], sample entropy (SE) [16], fuzzy entropy (FE) [17], permutation entropy (PE) [18], dispersion entropy (DE) [19], and others. The above methods are used to measure the complexity and irregularity of time series on a single time scale. In practical applications, [PERSON] [20] considered the multi-scale characteristics of complex signal and introduced multi-scale sample entropy (MSE) to assess time series complexity and irregularity across different scales. Addressing the issue of substantial fluctuations in MSE entropy values, [PERSON] [21] proposed refined composite multi-scale sample entropy (RCMSE). Among these methods, SE solves the issue of self-similarity that ApEn demonstrates and effectively measures the complexity and irregularity of time series.