## 1 Introduction  
Reciprocating compressors (RCs) are widely used in the petrochemical, refrigeration, and gas transportation industries [1; 2; 3]. The valve is the core part of RC, which determines the stable and efficient operation, and it is one of the most easily damaged parts in RC. Therefore, an effective and accurate valve fault detection method can improve the operation reliability of the RC, reduce the occurrence of accidents, and reduce the downtime and cost of the RC [4; 5]. The fault diagnosis method (FDM) of RC is mainly developed from the FDM of rotating machinery. For example, the wavelet transform, which is widely used in rotating machinery, has been well applied to the fault diagnosis of internal combustion engines [6; 7; 8]. As one of the most commonly used FDM,vibration-based pattern recognition method has attracted significant attention in the field of rotating equipment fault diagnosis. However, the vibration characteristics of RC have non-linear, non-stationary, and multi-component coupling factors [3, 9], which pose challenges for RC fault diagnosis. Generally, the FDM of RC can be divided into three categories: model-based, data-driven, and a combination of the two [2, 10, 11]. The model-based FDM creates a mathematical model of the RC system or fault characteristics. The thermodynamic model [12], FSI model [13], and hybrid numerical model [14, 15] of the RC have been studied, and successively established a variety of mechanism model of the RC system fault. However, the establishment of RC system model requires very complex calculation and profound theoretical knowledge, which is difficult for most researchers. The data-driven method is based on the response data of the sensors, and it uses various algorithms to obtain the hidden information in the data for fault diagnosis. The diagnosis effect of data-driven method mainly depends on the quantity and quality of data and the conditions of collecting data, and it has low requirements for experience knowledge and fault mechanism. Therefore, it has been actively studied in the field of RC fault diagnosis, among which local mean decomposition [1, 16], deep confidence network and back-propagation neural network [17, 18, 19], support vector machine (SVM) [9, 20], k approximate regression [18, 21], Bayesian estimation algorithm [10, 22], big data [23], and other technologies have been successfully applied. The method of combining model and data-driven is to diagnose the system fault by fusing the system operation data with the system fault model. [PERSON]. et al. [24, 25, 26] established the induction motor current model and combined with the modulation signal bispectrum method to realize the fault diagnosis of gear and RC. [PERSON] et al. [27] proposed the method of combining the kinematic model of valve with acoustic emission signal to diagnose the leakage fault of the RC valve. When compared with model-based and data-driven methods, the method of combining the two can achieve a better diagnosis effect. However, presently, this method has relatively few applications in the RC valve leakage fault. This paper will serve as a successful application to this method. The quality and type of data is the key to data-driven FDM. The scientific test-rig, reliable sensors, and reasonable experimental conditions can guarantee the quality of data. The data types of data-driven method can be divided into single signal and multi-signal fusion. Single signal, such as vibration signal [3, 18, 28, 29, 30], acoustic signal [31, 32, 33, 34], current signal [24, 25], temperature signal [21], pressure signal [35], and instantaneous phase signal [36], etc., have been successfully applied to RC fault diagnosis. Although good diagnosis results have been achieved, the difficulty of data processing is increased because of the singleness of signal and the few effective fault features. Therefore, the FDM of multi-signal fusion has attracted the attention of scholars. [PERSON] and [PERSON].