A new rolling bearing fault diagnosis approach based on multiscale permutation entropy (MPE), Laplacian score (LS), and support vector machines (SVMs) is proposed in this paper. Permutation entropy (PE) was recently proposed and defined to measure the randomicity and detect dynamical changes of time series. However, for the complexity of mechanical systems, the randomicity and dynamic changes of the vibration signal will exist in different scales. Thus, the definition of MPE is introduced and employed to extract the nonlinear fault characteristics from the bearing vibration signal in different scales. Besides, the SVM is utilized to accomplish the fault feature classification to fulfill diagnostic procedure automatically. Meanwhile, in order to avoid a high dimension of features, the Laplacian score (LS) is used to refine the feature vector by ranking the features according to their importance and correlations with the main fault information. Finally, the rolling bearing fault diagnosis method based on MPE, LS, and SVM is proposed and applied to the experimental data. The experimental data analysis results indicate that the proposed method could identify the fault categories effectively.
The vibration signals of mechanical systems, especially for ones with fault, often show mutation, nonlinearity, and nonstationarity because of the strike, velocity chopping, structure transmutation, loading, and friction. Hence, it is very crucial for mechanical fault diagnosis to extract the fault feature information from the nonlinear and nonstationary signal. A primary method for dealing with the nonlinear and nonstationary signal is timefrequency analysis [
With the development of nonlinear dynamic theories, especially in recent years, a number of nonlinear parameters and methods, such as chaos theory, fractal dimension, and information entropy, have been applied to machine condition monitoring and fault diagnosis. For instance, Logan and Mathew elaborated the application of the correlation dimension to vibration fault diagnosis of rolling element bearing [
Recently, permutation entropy (PE) was proposed by Bandt and Pompe [
However, like traditional single scale nonlinear dynamic parameters ApEn and SampEn, PE detects the dynamic changes and randomness of time series only in a single scale. Recently, multiscale permutation entropy (MPE) was introduced by Aziz and Arif in the literature [
As the vibration signals collected from normal rolling bearing are random and irregular, the randomness and the dynamic behavior of the vibration signal will change abruptly when the rolling bearing of equipment works under a bad condition. Due to the complexity of mechanical system, the vibration signal is much more complex and contains much more important information in different scales. Hence, MPE is employed to detect the dynamic changes and fault features from the rolling bearing vibration signal.
In the paper, firstly the PE values with different scales are served as initial feature parameters to extract fault feature information from the bearing vibration signal. Since the feature vector concludes MPE values in different scales, which will lead to a high dimension and information redundancy, and it is also difficult to find out the features containing the main fault information, in this paper the LS proposed by He et al. [
The rest of the paper is organized as follows. In the second section, the definitions of PE and MPE are introduced, respectively. In the third section, the Laplacian score (LS) is introduced firstly, and then a new rolling bearing fault diagnosis method based on MPE, LS, and SVM is proposed. In the fourth section, the proposed method is applied to rolling bearing experimental data and some comparisons are made. Finally, the fifth section concludes the paper.
Permutation entropy (PE) was introduced recently to detect dynamic changes of time series by Bandt and Pompe [
Consider a time series,
If there exist two elements in
Therefore, if we suppose that the probability distribution for the distinct symbols be as
It is noticed that
Obviously,
There are three parameters to be considered in the calculation of PE, namely, the length of time series
The PEs of white Gaussian noise signals with different lengths when
As the Gaussian white noise signal is random and it should have an estimated value close to 1, therefore when
In addition, the time delay
The PEs of white Gaussian noise with different time delays.
Multiscale permutation entropy (MPE) is defined as the PE set of time series in different scales and is calculated as follows [
Consider the time series,
Calculate PE of each coarsegrained time series
In order to select the best
The MPE of Gaussian white noise signal with different embedding dimensions. Here
Theoretically, the extracted MPE features in 12 scales are able to identify the fault categories. However, the feature vector with a high dimension will be timeconsuming and information inefficient for fault diagnosis. Therefore it is necessary to select the most important features which contain the main fault information from the 12 features, which could avoid the dimension disaster and improve the performance and efficiency of rolling bearing automatically fault diagnosis.
Laplacian score (LS) is a popular feature ranking based feature selection method and is mainly founded on Laplacian eigenmaps and locality preserving projection. Its basic idea is to estimate the features according their locality preserving power [
Based on the advantages of MPE, LS, and SVM, the proposed rolling bearing fault diagnosis method is described as follows.
Calculate MPE for each rolling bearing vibration signal with parameter selection
Then the obtained MPEs in all scales (i.e., 12 PEs) are viewed as the initial feature vector to represent the main fault information of vibration signal.
LS is employed to rank the 12 features from low to high score according to their importance and relationships with fault information.
The first several features with the least scores are selected as the new feature vector.
The new feature vectors are used to train and test the SVM based multifault classifier to fulfill fault diagnosis automatically.
The proposed method can be described briefly as in Figure
Flow chart of the proposed method.
In step (4) of the proposed method, as too many features will cost much training time and cause information redundancy while too few ones cannot completely reflect the fault information and get a lower accuracy, the novel feature vector in this paper is constructed by the first five features with the lowest LSs to achieve an effective fault diagnosis.
The rolling bearing experimental data analyzed in the paper are kindly provided by Bearing Data Center, Case Western Reserve University [
The vibration signals of normal (NORM) bearing and bearings with fault (ORF, IRF, REF) are depicted in Figure
The time domain waveforms of normal and fault bearing vibration signals.
It is unobvious to identify the normal and fault rolling bearings from each other especially differentiating NORM from REF and IRF from ORF. Therefore MPE is utilized to analyse above signals and their MPEs are plotted as a function of the scale factor in Figure
The MPE of normal and faulty bearing vibration signals. The results are the average of ten trials.
From Figure
If the extracted MPEs with 12 scales from the vibration signal are viewed as the feature vector, it will increase computational time and complexity, and the redundant information will decrease the classification accuracy. However, it is difficult for us to find out which feature contains the main fault information. In the literature [
In this paper, normal and three faults (REF, IRF, and ORF) types of rolling bearing are under our consideration. Each type has 30 samples and there are totally 120 samples. By extracting MPE from each vibration signal, correspondingly, 120 initial feature vectors with 12 PEs can be obtained. For each fault type, 10 samples are randomly chosen for training and the remaining 20 samples are used for testing. Hence, a training dataset (with dimension
Then, the LS is used to rank the 12 features according to their importance and the results are shown as follows:
LS_{1 }< LS_{2 }< LS_{9}< LS_{11 }< LS_{10 }< LS_{12 }< LS_{6 }< LS_{7 }< LS_{8 }< LS_{3 }< LS_{4 }< LS_{5},
where the subscript stands for scale factor number. Therefore, the MPEs withNext, a multifault classifier consisting of three SVMs, that is, SVM1, SVM2, and SVM3, is trained, where SVM1 is used to distinguish normal from the fault, SVM2 is used to discriminate IRF from REF and ORF, and SVM3 is used to discriminate REF from ORF. The structure diagram of the multifault classifier is depicted as in Figure
Structure diagram of SVM based multifault classifier.
After training the SVMclassifier with the 40 training feature vectors, the remaining 80 testing features are used to test the trained SVMclassifier and the outputs of the multiclassifier is shown in Table
The SVMclassifier outputs of testing data with feature vector refined by LS.
Testing data  Fault type  SVM1  SVM2  SVM3 


NORM  +1 (20)  —  — 

IRF  −1 (20)  +1 (20)  — 

REF  −1 (20)  −1 (20)  +1 (20) 

ORF  −1 (20)  −1 (20)  −1 (20) 
For comparison, a multiclassifier based on back propagation (BP) neural network [
The BP classifier outputs of all samples with the same feature vector as SVMclassifier. The first 40 outputs are training data and the remaining 80 outputs are testing data.
In addition, in order to verify the essentiality of multiscale analysis using MPE, the PE value of the original vibration signal, namely, the MPE with scale factor
The outputs of SVMclassifier with feature vector consisting of one PE.
Samples  Fault type  SVM1  SVM2  SVM3 


NORM  +1 (20)  —  — 

IRF  −1 (20)  +1 (20)  — 

REF  −1 (20)  −1 (20)  +1 (18), −1 (2) 

ORF  −1 (20)  −1 (20)  −1 (16), +1 (4) 
The outputs of BPclassifier with feature vector consisting of the PE when
To verify that it is necessary and superior to refine the feature vectors using LS, without loss of generality, the MPE with scales 1, 2, 3, 4, and 5 are taken as the feature vector to train and test the SVMclassifier. After training the classifier, the outputs of testing data are given in Table
The outputs of SVMclassifier with feature vector consisting of the first five PEs.
Samples  Fault type  SVM1  SVM2  SVM3 


NORM  +1 (20)  —  — 

IRF  −1 (20)  +1 (20)  — 

REF  −1 (20)  −1 (20)  +1 (20) 

ORF  −1 (20)  −1 (20)  −1 (18), +1 (2) 
In consideration of the nonlinearity and nonstationarity of rolling bearing vibration signal, a novel rolling bearing fault diagnosis method based on MPE, LS, and SVM is proposed. Permutation entropy (PE) is defined to detect the dynamic changes of time series. For the complexity of the mechanical system, the vibration signal always contains much more important failure information in different scales. Therefore, in this paper MPE is adopted to extract the nonlinear fault characteristics from vibration signal. Besides, in order to achieve the fault diagnosis automatically, the SVM is utilized to construct the multifault classifier. Meanwhile, to refine the feature vector and select the most important features, Laplacian score (LS) is employed for feature selection. Finally, the proposed method is applied to rolling bearing experimental data. Also, the SVMclassifier is compared with BPclassifier and the single scale based PE is compared with MPE by analyzing the experimental data, and the comparison result indicates that the proposed method could get much higher identifying accuracy and has verified the necessities of analyzing the vibration signal with MPE and selecting feature by LS as well. Finally, the proposed method is aiming to fault diagnosis of rolling bearing and has been verified as an effective way by experiment data. However, the proposed method also has some problems, such as the number selection of feature vector refined by LS, the construction of multiclassifier, and its generalization to other bearings or gear fault diagnosis, and they will be discussed and solved in the future work.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Natural Science Foundation of Hunan Province, China (Grant no. 11JJ2026), the National Natural Science Foundation of China (Grants no. 51075131 and no. 51175158), and Hunan Provincial Innovation Foundation for Postgraduate (Grant no. CX2013B144). The first author would like to appreciate the kind help of C. Bandt. The authors would like to thank anonymous peer reviewers for their valuable suggestions.