## 1 Introduction  
Rolling bearings are fundamental components that are widely used in rotating machinery, for example, in airplanes, machining centers, and wind turbines. As a vulnerable link in these machines, rolling bearing faults can result in breakdown of the whole machinery. According to reference statistics, about 40% (for large machines) to about 90% (for small machines) of failures can be attributed to bearing faults [1]. Therefore, rolling bearing fault diagnosis has been the research hot topic in machine health monitoring over the past few decades. The process of rolling bearing fault diagnosis basically comprises fault feature extraction and fault classification. To extract the fault features from vibration signals is the first and most essential step. Traditionally, bearing fault features are extracted with time-domain or frequency-domain analysis methods [2; 3]. However, due to variations such as friction, damping, and operational conditions, bearing vibration signals often are nonlinear and nonstationary. Thus, bearing fault feature extraction is now more frequently analyzed through time-frequency/scale methods, including short-time Fouriertransform (STFT) [4], wavelet transform (WT) [5], and empirical mode decomposition (EMD). Without requirement of basis functions, EMD [6] is a data-driven method that adaptively decomposes a nonstationary and nonlinear time series into several approximately stationary time series defined as the intrinsic mode function (IMF) plus a final monotonic trend. In this way, EMD is complete and has been widely applied for fault feature extraction [7]. Nevertheless, EMD suffers problems of mode mixing, false mode, and endpoint effect. In [8], ensemble empirical mode decomposition (EEMD) was proposed to alleviate the mode aliasing problem of EMD by taking advantage of the property of frequency uniform distribution of Gaussian white noise. When analyzed with the EEMD method, the reconstructed signal containing residual noise and different realizations of signal plus noise may produce different numbers of modes, bringing difficulties to the final average operation. Then, complementary ensemble empirical mode decomposition (CEEMD) was proposed in [9] to improve the reconstruction error with EEMD by adding complementary pairs of Gaussian white noise, but the final averaging problem remained unsolved. This promoted the emergence of the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) [10], achieving a negligible reconstruction error and solving the problem of different numbers of modes at the final average. Although CEEMDAN has been proven to be effective in fault feature extraction [11; 12], it still has some aspects to be improved: (i) the presence of residual noise in the modes and (ii) the existence of spurious modes. To address these two issues, an improved CEEMDAN (ICEEMDAN) was further proposed and modes with less noise as well as more physical meaning can be obtained [13]. Although ICEEMDAN was originally proposed for biomedical signal processing [13; 14], very recently few work [15] has researched its potential for fault diagnosis. With the development of nonlinear dynamics theory, many nonlinear parameter estimation methods provide good alternatives to extracting the fault-related features hidden in vibration signals stemming from nonlinear systems. Among these parameters, entropy-based ones quantify the complexity of a time series and have shown very good performances in analyzing nonlinear vibration signals of mechanical systems. Recently, entropy and its variants attracted increasing attention in fault diagnosis [16]. The commonly used entropies are sample entropy (SampEn) [17; 18] and permutation entropy (PerEn) [19; 20; 21]. Although SampEn is one improved version of approximate entropy, it still leads to unreliable estimations for short signals and is not fast enough for long signals in real-time applications.