## 1 Introduction  
Rolling element bearings are commonly employed to support rotating shafts in a rotary machine. A bearing mainly includes an inner race, an outer race, and several rolling elements. A fault in any one of the above-mentioned components can result in the reduced performance of the whole system, fatal machine failure, or catastrophic accident [1; 2; 3]. Based on the statistical results in Ref. [4], more than 50% of mechanical faults are caused by various bearing defects. Therefore, it is important and necessary to design a effective solution for bearing fault diagnosis to reduce the loss caused by mechanical failure. Analyzing different features extracted from vibration signals has been testified to play an exceptionally effective role in addressing the issue of fault detection or fault diagnosis of rotary machinery, because these obtained vibration signals provide abundant information regarding the working conditions of the bearings in a rotary machine [5; 6]. Up to now, many signal processing methods have been employed to analyze the collected vibration signal for features, and they can be summed into three types: time domain, frequency domain, and time-frequency domain [7]. At the preliminary stage of mechanical fault diagnosis, time domain analysis is the simplest one. Frequency domain methods typically diagnose mechanical faults by revealing key frequency information regarding faults, for example, fast Fourier transform (FFT) [8] and interpolated discrete Fourier transform (IpDFT) [9]. However, the above two methods alone can only extract a portion of the useful information from the vibration signals, and information from each of the other domains is lost. Therefore, more and more researchers have introduced time-frequency processing tools for mechanical feature extraction, such as short-time Fourier transform (STFT), wavelet transform-based methods, empirical mode decomposition (EMD), and ensemble empirical mode decomposition (EEMD) [10; 11; 12; 13]. As a self-adaptive signal processing algorithm, EMD can analyze the overlap information in both time and frequency domains [14]. Therefore, many EMD-based methods have been proposed for mechanical fault diagnosis. A fault diagnosis method was employed using envelope spectrum analysis with the EMD algorithm in Ref. [15]. [PERSON] and [PERSON] [16] proposed an effective method to search the features of subharmonic faults of large rotating machinery based on EMD; this is an adaptive and unsupervised approach that does not require a shift to the frequency domain. In Ref. [17], a novel method combining the parameter estimate of Alpha stable distribution (ASD) and EMD-based signal processing was proposed to diagnose low-speed bearing faults. In this method, the trend and noise components of the vibration signal were filtered to give a clear signal. An accurate autoregressive (AR) model can reveal important information or characteristics of a dynamic system, so this model was combined with EMD to construct a new strategy to extract effective features for bearing fault diagnosis [18]. In [19], a new fault diagnosis method using EMD and wavelet denoising analysis was designed for bearing systems. However, traditional EMD can suffer from a mode-mixing problem when used to analyze complex signals. To settle the mode-mixing issue, an improved version, called EEMD, was designed by [PERSON] and [PERSON] [20]. A multi-fault diagnosis method for axle bearings was introduced using EEMD and Hilbert marginal spectrum analysis [21]. In this method, an intrinsic mode function (IMF) confidence index was designed to realize the aim of adaptive self-selection of the useful IMFs. In Ref. [22], EEMD and multi-scale fuzzy entropy were introduced for feature extraction from complex vibration signals, and this method was also used for motor bearing fault diagnosis. Another effective fault diagnosis method uses improved EEMD and Hilbert square demodulation (HSD) [23].