This study presents a two-step fault diagnosis scheme combined with statistical classification and random forests-based classification for rolling element bearings. Considering the inequality of features sensitivity in different diagnosis steps, the proposed method utilizes permutation entropy and variational mode decomposition to depict vibration signals under single scale and multiscale. In the first step, the permutation entropy features on the single scale of original signals are extracted and the statistical classification model based on Chebyshev's inequality is constructed to detect the faults with a preliminary acquaintance of the bearing condition. In the second step, vibration signals with fault conditions are firstly decomposed into a collection of intrinsic mode functions by using variational mode decomposition and then multiscale permutation entropy features derived from each mono-component are extracted to identify the specific fault types. In order to improve the classification ability of the characteristic data, the out-of-bag estimation of random forests is firstly employed to reelect and refine the original multiscale permutation entropy features. Then the refined features are considered as the input data to train the random forests-based classification model. Finally, the condition data of bearings with different fault conditions are employed to evaluate the performance of the proposed method. The results indicate that the proposed method can effectively identify the working conditions and fault types of rolling element bearings. Entropy 2019, 21, 96 2 of 18 become the mainstream for diagnosing the faults of rolling element bearings, since they do not require any prior knowledge about the parameters and mathematical model of diagnosis objects [2] . In general, the data-driven approach can be divided into three groups: the method of statistical analysis, the method of signal analysis, and artificial intelligence technology [3] . Some successful applications of statistical analysis methods in fault diagnosis have been reported in [4, 5] . As an advanced signal processing technology, time-frequency analysis methods were widely used to extract important condition features from non-stationary signals for fault detection and diagnosis. Its developments and applications in fault diagnosis were reviewed in [6] . As for the first two methods, an expert's rich knowledge will be required for their implementation and they also show some limitations in dealing with multi-fault diagnosis problems [5, 6] . The machine learning method, as an important brand of artificial intelligence, has been widely applied in various fields, such as pattern recognition, fault diagnosis and data mining. Meanwhile, the related theories development provides great facility for the intelligent fault diagnosis technology [7] . It can be concluded that fault diagnosis based on artificial intelligence is a typical pattern recognition problem, where features extraction and classifier construction are the two crucial issues [8] . Due to the non-linear dynamic characteristics of bearings' fault vibration signals, the traditional feature extraction methods based on Fourier transform and statistical analysis are no longer adequate for the effective and accurate diagnosis of bearing faults. Recently, various complexity measurement methods derived from information theory are proposed to characterize the vibration signals [9] [10] [11] . The widely used methods include energy entropy (EnE) [10, 12] , sample entropy (SE) [13, 14] , approximate entropy (ApEn) [15, 16] and fuzzy entropy (FE) [17] . Although these methods have made certain achievements in the health monitoring and fault diagnosis of rotating machines, they still possess some deficiencies [8, 11, 18] . Permutation entropy (PE), as a non-linear dynamic parameter, was introduced by Bandit and Pompe to measure the randomness and dynamic mutation of time sequences [19] . Based on its advantages of intelligibility, low time consumption and strong robustness, PE has achieved many successful applications in the fault diagnosis of rotating machines [20] [21] [22] [23] . In practice, however, vibration signals collected from complex systems usually possess non-linear and non-stationary properties and contain different signal components with multiscales. Most studies indicate that the PE feature with a single scale has certain limitations when describing the dynamic properties of vibration signals and some useful information riding in other scales may be ignored. To overcome this problem, multiscale permutation entropy (MPE) was proposed based on the coarse-grained processing of time series by Aziz and Arif [24] . However, the determination of the scale factor has a direct effect on the quality of MPE and it is often very difficult. More notably, the representation ability of MPE features could be weakened by the mixing components distributed in different frequency bands. To obtain an accurate description about signals, the entropy features extraction integrated with time-frequency analysis methods has become a hot spot in the fault diagnosis field. Some representative time-frequency analysis methods include wavelet transform (WT) [25] , empirical mode decomposition (EMD) [26] , local mean decomposition (LMD) [27] and variational mode decomposition (VMD) [28] . By using these methods, a complicated signal can be decomposed into a series of mono-components, and then the entropy features of mono-components can be extracted as features to reflect the local characteristics of the signals. For instance, wavelet packet decomposition (WPD)-based PE features were extracted to identify faults appearing in bearings [29] . Similarly, PE features combined with ensemble empirical mode decomposition were used to reveal the local characteristics of original signals under intrinsic time scales [30] . Unlike WT, the other three methods have removed the dependency to the base functions and indicate remarkable self-adaptability. Although EMD has been widely applied to signal processing and fault diagnosis [31] , the sensitivity to noise and sampling frequency also limits its development. As a non-recursive signal decomposition method, VMD was proposed recently for adaptive signal decomposition [28] . In [32] , the performance of VMD compared with LMD and EMD was discussed and the analysis results indicate that VMD has the advantages of high computation efficiency, strong robustness to background noise and accurate
One of the analyzed existing solutions is based on both machine learning and entropy REF .
Fault Diagnosis of Rolling Element Bearings with a Two-Step Scheme Based on Permutation Entropy and Random Forests
Entropy
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Mathematics Computer Science