In recent years, many studies have been conducted in bearing fault diagnosis, which has attracted increasing attention due to its nonlinear and non-stationary characteristics. To solve this problem, this paper proposes, a fault diagnosis method based on Empirical Mode Decomposition (EMD), Kernel Principal Component Analysis (KPCA), and Extreme Learning Machines (ELM) neural network, which combines the existing self-adaptive time-frequency signal processing with the advantages of non-linear multivariate dimensionality reduction KPCA approach and ELM neural network. First, EMD is applied to decompose the vibration signals into a finite number of intrinsic mode functions, in which the corresponding energy values are selected as the initial feature vector. Second, KPCA is used to further reduce the dimensionality for a simplified low-dimension feature vector. Finally, ELM is introduced to classify the extracted fault feature vectors for lessening the human intervention and reducing the fault diagnosis time. Experimental results demonstrate that the proposed diagnostic can effectively identify and classify typical bearing faults

Hang Yuan

Zihan Chen

English

Vibroengineering PROCEDIA

 200 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
Bearing fault diagnosis based on EMD-KPCA and ELM 
Zihan Chen1, Hang Yuan2 
School of Reliability and Systems Engineering, Beihang University, Beijing 100191, China 
Science and Technology on Reliability & Environmental Engineering Laboratory, Beijing 100191, China 
2Corresponding author 
E-mail: 1cheenzihan@126.com, 2buaayuanhang@163.com 
(Accepted 13 September 2014) 
Abstract. In recent years, many studies have been conducted in bearing fault diagnosis, which 
has attracted increasing attention due to its nonlinear and non-stationary characteristics. To solve 
this problem, this paper proposes, a fault diagnosis method based on Empirical Mode 
Decomposition (EMD), Kernel Principal Component Analysis (KPCA), and Extreme Learning 
Machines (ELM) neural network, which combines the existing self-adaptive time-frequency 
signal processing with the advantages of non-linear multivariate dimensionality reduction KPCA 
approach and ELM neural network. First, EMD is applied to decompose the vibration signals into 
a finite number of intrinsic mode functions, in which the corresponding energy values are selected 
as the initial feature vector. Second, KPCA is used to further reduce the dimensionality for a 
simplified low-dimension feature vector. Finally, ELM is introduced to classify the extracted fault 
feature vectors for lessening the human intervention and reducing the fault diagnosis time. 
Experimental results demonstrate that the proposed diagnostic can effectively identify and classify 
typical bearing faults. 
Keywords: fault diagnosis, empirical mode decomposition, kernel principal component analysis, 
extreme learning machines. 
1. Introduction 
Rolling element bearings are widely used to support rotating components, whose failures may 
cause machinery breakdown and economic loss. As a kind of common rotary machinery 
components, bearings have been conducted much research in Prognostics and Health Management 
fields. It is crucial to diagnose bearing faults at their early developing stages to prevent severe 
machinery failures. 
Bearing faults are diagnosed mainly by vibration analysis. Vibration signals measured from 
bearings are complex multi-component signals, generated by tooth meshing, gear shaft rotation, 
gearbox resonance vibration signatures and a substantial amount of noise. Bearing fault features 
of vibration signals can be extracted by many methods, such as short-time Fourier transform and 
wavelet analysis. Wavelet analysis can provide the local features of vibration signals in both the 
time and frequency domain, so it has been widely used in the roller bearing fault diagnosis. 
Whereas, energy leakage will occur in wavelet transformation, due to the limitation of the length 
of the wavelet bases. When scales are determined, the results of wavelet transform would be under 
the certain scale, whose frequency components may lost a lot of information. And then, Empirical 
Mode Decomposition (EMD) developed by Huang et al. is demonstrated to be superior to wavelet 
analysis in many applications [1]. EMD is an adaptive signal processing method developed 
especially for dealing with nonlinear and non-stationary data. EMD is an approach to the 
decomposition of nonlinear signal, which can self-adaptive decompose signals into a series of 
Intrinsic Mode Functions (IMF). The features of vibration signals, in this paper, is extracted by 
calculating EMD energy entropies of each IMF. But the EMD energy feature vectors always are 
high-dimensional, which means those are redundancy and difficult to farther calculate. By 
calculating the eigenvectors of the covariance matrix of the feature vectors, Principal Component 
Analysis (PCA) can linearly transforms a high-dimensional vector into a low-dimensional vector. 
Whereas, when dealing with non-linear process, PCA-based feature extraction approaches are 
limited due to their underlying linearity assumption. Schölkopf [2] et al. described a method for 
performing a nonlinear form of Kernel Principal Component Analysis (KPCA). KPCA is the 
BEARING FAULT DIAGNOSIS BASED ON EMD-KPCA AND ELM.  
ZIHAN CHEN, HANG YUAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 201 
reformulation of traditional linear PCA in a high-dimensional space which is related to the input 
space by a possibly nonlinear map. By using integral operator kernel functions, we could valid 
employ principal component analysis in a high-dimensional feature space. Recently, the 
development of neural networks has added new dimensions to diagnose faults of bearings. Neural 
networks can be used for complex systems which involve non-linear behavior and unstable 
processes, but there is no standard method to determine their structure and parameters, which 
determined by researcher using data-driven or experience-based methods. Recently, Extreme 
Learning Machines (ELM) neural network has been proposed as a simple and efficient approach 
to build single-hidden-layer feed-forward networks (SLFN) sequentially. ELM use random hidden 
nodes and find the output weights to minimize the sum-of-squares error in the training set by 
solving a linear system of equations [3]. ELM learning speed can be much faster than traditional 
feed-forward network learning algorithms like back-propagation algorithm, with better 
generalization performance. Compared with BP neural network, ELM provides greater 
performance at a much faster learning speed and with less human intervention. 
In this study, to solve the aforementioned problems, an integrated method that combines 
EMD-KPCA and ELM is applied to rolling bearing fault diagnosis. Therefore, this paper is 
organized as follows. Section 2 introduces the details of EMD, KPCA, and ELM; Section 3 
describes the validation process of the proposed method; Section 4 concludes this paper. 
2. Methodology 
First, through the EMD progress the original vibration signals are decomposed to several IMF 
vectors. After calculating EMD energy entropy, we get energy feature vectors which can illustrate 
the energy of vibration signal in different frequency bands. Second, a KPCA progress is used to 
reduce the dimensionality of the energy feature vectors. Third, to identify the fault pattern of the 
roller bearings, in this paper, an ELM neural network is used to serve as the classifier. And then 
the feature vectors, calculated by the KPCA process, are taken as the training and testing input. 
 
Fig. 1. Process of fault diagnosis  
2.1. Empirical mode decomposition  
The main function of EMD is to decompose the original vibration signal into a series of IMF, 
from high frequencies to low frequencies, based on an iterative sifting process. Then the IMF can 
represent simple oscillatory mode in the signal, with the assumption that signal consists of 
different simple intrinsic modes of oscillation (i.e., different simple IMF).  
With the definition, the vibration signal (the ݔ(ݐ) in this paper) is then decomposed into ݊ 
intrinsic modes and a residue ݎ(ݐ) [1]: 
ݔ(ݐ) = ෍ ܫܯܨ௜(ݐ)
௡
௜ୀଵ
+ ݎ(ݐ). (1) 
Thus, the vibration signal can be decomposed into ݊-empirical modes and a residue ݎ(ݐ), 
which is the mean trend of ݔ(ݐ). 
EMD energy entropies of different vibration signals illustrate that the energy of vibration 
Energy entropy 
Calculation Raw 
vibration 
signals
IMF 
components
Energy 
feature 
vectors
EMD KPCA Reduced 
feature 
vectors
ELM network 
Trained with 
training sample
Normal
Inner race defect
Ball defect
Outer race defect
 Fault pattern
identification
BEARING FAULT DIAGNOSIS BASED ON EMD-KPCA AND ELM.  
ZIHAN CHEN, HANG YUAN 
202 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
signal in different frequency bands, which will change when bearing fault occurs. To illustrate this 
change case as mentioned above, the energy of each IMF is calculated. 
The energy of the ݊ IMF is ܧଵ, ܧଶ,…, ܧ௡ respectively, ܧ is the 2-norm of [ܧଵ, ܧଶ,…, ܧ௡]: 
ܧ௜ = න |ܫܯܨ௜(ݐ)|ଶ݀ݐ
ାஶ
ିஶ
, ݅ = 1, 2, . . . , ݊, (2)
ܧ = ටܧ௜ଵଶ + ܧ௜ଶଶ + ܧ௜ଵଶ +⋅⋅⋅ +ܧ௜ଶଶ . (3)
Then the EMD energy feature vector is defined as: 
ܨ௜ = [ܧ௜ଵ/ܧ, ܧ௜ଶ/ܧ,⋅⋅⋅, ܧ௜ଶ/ܧ]. (4)
2.2. Kernel principal component analysis 
In general, given a set of centered observations ܠ௞, ݇ = 1,…, ܯ, ܠ࢑ ∈ ࡾே, ∑ ܠ௞ெ௞ୀଵ = 0. In a 
dot product space ۴, which is related to the input space by a possibly nonlinear map: 
Φ: ܀ே → F,     ܠ → ܆. (5)
Then we get a set of new data ઴(ܠ௞), ݇ = 1,…, ܯ, ઴(ܠ௞) ∈ ۴. We assume that the new data 
are centered ∑ ઴(ܠ௞)ெ௞ୀଵ = 0, and use the covariance matrix in ۴: 
ܥ = 1ܯ ෍ Φ൫ܠ௝൯Φ൫ܠ௝൯
்.
ெ
௝ୀଵ
(6)
Defining an ܯ×ܯ matrix ۹ by: 
۹௜௝ = ቀΦ(ܠ௜) ⋅ Φ൫ܠ௝൯ቁ. (7)
Now we want to find eigenvalues ߣ ≥ 0 and eigenvectors ܄ ∈ ࡲ\ሼ0ሽ satisfying: 
ߣ܄ = ̅ܥ܄. (8)
And, there exist coefficients ߙ௜ (݅ = 1,…, ܯ) such that: 
܄ = ෍ ߙ௜Φ(ܠ௜)
ெ
௜ୀଵ
. (9)
Now we get: 
ߣ ෍ ߙ௜൫Φ(ܠ௞)Φ(ܠ௜)൯
ெ
௜ୀଵ
= 1ܯ ෍ ߙ௜ ቌΦ(ܠ௞) ෍ Φ൫ܠ௝൯
ெ
௝ୀଵ
ቍ
ெ
௜ୀଵ
ቀΦ൫ܠ௝൯Φ(ܠ௜)ቁ , ݇ = 1, . . . , ܯ. (10)
Combining Eqs. (7) and (10), we get: 
ܯߣ۹α = ۹ଶα, (11)
where ߙ denotes the column vector with entries ߙଵ,…, ߙெ. We can solve an eigenvalue problem 
BEARING FAULT DIAGNOSIS BASED ON EMD-KPCA AND ELM.  
ZIHAN CHEN, HANG YUAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 203 
to find the solution of Eq. (12): 
ܯߣα = ۹α. (12) 
For the purpose of principal component extraction, we need to compute projections onto the 
eigenvectors ܄௞ in ۴ . Let ܠ be a test point, with an image ઴(ܠ) in ۴; then: 
൫܄௞ ⋅ Φ(ܠ)൯ = ෍ ߙ௜௞൫Φ(ܠ௜)Φ(ܠ)൯
ெ
௜ୀଵ
. (13) 
In order to compute dot products of the form ൫Φ(ܠ) ∙ Φ(ܡ)൯, we use kernel representations: 
݇(ܠ, ܡ) = ൫Φ(ܠ)Φ(ܡ)൯. (14) 
2.3. Extreme learning machine 
ELM is an effective learning algorithms of SLFN. ELM has ݊ input nodes, ܰ hidden nodes, 
and, ݉ output nodes. The hidden biases and input weights and are randomly chosen, and the 
output weights are analytically determined using Moore–Penrose (MP) generalized inverse. 
Given a training set ܫ = ሼ(ܠ௜, ܜ௜)|ܠ௜ ∈ ܀௡, ܜ௜ ∈ ܀௠, ݅ = 1,…, ܰሽ, activation function ݃(ݔ), 
and hidden node number ෩ܰ: 
෍ ߚ௜ ௜݃൫ܠ௝൯
ே෩
௜ୀଵ
= ෍ ߚ௜݃൫ܟ௜ܠ௝ + ܾ௜൯
ே෩
௜ୀଵ
= ܗ௝, ݆ = 1, . . . , ܰ, (15) 
where ܟ௜ = [ݓ௜ଵ, ݓ௜ଶ, … , ݓ௜௡]் is the weight vector connecting the ݅th hidden node and the input 
nodes,  ઺௜ = [ߚ௜ଵ, ߚ௜ଶ, … , ߚ௜௠]் is the weight vector connecting the ݅th hidden node and the output 
nodes, and ܾ௜ is the threshold of the ݅th hidden node. ܟ௜ܠ௝ denotes the inner product of ࢝௜ and ࢞௝. 
Step 1: Randomly assign input weight ܟ௜ and bias ܾ௜, ݅ = 1,…, ෩ܰ: 
෍ ߚ௜݃൫ܟ௜ܠ௝ + ܾ௜൯
ே෩
௜ୀଵ
= ܜ௝, ݆ = 1, . . . , ܰ. (16) 
Step 2: Calculate the hidden layer output matrix ۶: 
۶ = ൥
݃(ܟଵܠଵ + ܾଵ) ⋯ ݃(ܟே෩ܠଵ + ܾே෩)
⋮ ⋯ ⋮
݃(ܟଵܠே + ܾଵ) ⋯ ݃(ܟே෩ܠே + ܾே෩)
൩
ே×ே෩
. (17) 
Step 3: Calculate the output weight ઺: 
ߚ = ۶ା܇. (18) 
3. Experiment results 
The data used in the experiment of this paper comes from the bearing data center, which 
provides access to ball bearing test data for normal and faulty bearings. The bearing data center 
conducted a series of experiments using a 2 hp Reliance Electric motor, and acceleration data was 
measured at locations near to and remote from the motor bearings. Three common types of fault 
of bearing (i.e. inner race defect, ball defect and outer race defect) were seed respectively using 
BEARING FAULT DIAGNOSIS BASED ON EMD-KPCA AND ELM.  
ZIHAN CHEN, HANG YUAN 
204 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533  
electro-discharge machining. 
Faults ranging from 0.007 inches in diameter to 0.040 inches in diameter were introduced 
separately at the inner raceway, rolling element (i.e. ball) and outer raceway. The motor speed is 
1750 r/min, and the sampling rate is 120 KHz. 
First, the original vibration signals were decomposed into some IMF by EMD. The IMF 
vectors of every signal had different dimensionality because EMD can self-adaptive decompose 
signals, as shown in Fig. 2. It is clearly show that only the first several IMF had dominant fault 
information. 
 
Fig. 2. IMF feature vectors 
Second, KPCA was used to reduce the dimensionality of the four IMF vectors to eliminate 
their redundant. The IMF of every samples was converted, separately, to three characteristics, 
which means each of the new four feature vectors had three rows.  
In this paper we use radial basis functions kernel (i.e. ݇(ܠ, ܡ) = exp(− ‖ܠ − ܡ‖ଶ 2ߪଶ⁄ )). This 
kernel is a measure of closeness, equal to 1 when the points coincide and equal to 0 at infinity. 
Through this process, not only reduced the data redundancy, also increased the data visibility. 
As shown in Fig. 3, the four pattern was clearly clustered. The feature vectors calculated by KPCA 
were separated, for further use, to train-set vectors and test-set vectors. 
Finally, an ELM neural network was adopted to identify the various fault patterns. The train 
feature vector in the four patterns respectively were taken as the ELM neural network inputs, and 
the hidden layer included 25 nodes. Each pattern was trained by 100 samples, and tested by 20 
samples. For the purposes of classification, the four pattern is marked as 1, 2, 3, and 4,  
respectively. 
By applying the ELM neural network that had been trained to the train-set, most of the test 
samples were identified successfully, as shown in Fig. 4(a). As can be seen from the figure, only 
the seventh sample was not properly detected, and the accuracy was 98.5 %, which meant the fault 
diagnosis and fault classification was successfully conducted. 
As indicated in Fig. 4, ELM, which only needs to set the node number in the training process, 
provides higher performance over BP in terms of both learning speed and human intervention. In 
this case study, the training and testing time was total 0.077838 sec, in which the training and 
testing samples are 100 and 80, respectively. However, on the same condition, training and testing 
time of fault diagnosis with BP was 0.870891 sec. The accuracy of bearing fault classification 
gradually decreases as the node number ranges from 25 to 500 with a step of 25. 
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
samples
IM
F E
ne
rgy
Normal Feature Vector
 
 
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
samples
IM
F E
ne
rgy
Inner Fault Feature Vector
 
 
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
IMF9
IMF10
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
samples
IM
F E
ne
rgy
Ball Fault Feature Vector
 
 
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
IMF9
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
samples
IM
F E
ne
rgy
Outer Fault Feature Vector
 
 
IMF1
IMF2
IMF3
IMF4
IMF5
IMF6
IMF7
IMF8
IMF9
IMF10
BEARING FAULT DIAGNOSIS BASED ON EMD-KPCA AND ELM.  
ZIHAN CHEN, HANG YUAN 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. NOVEMBER 2014. VOLUME 4. ISSN 2345-0533 205 
 
Fig. 3. Pattern clusters 
a) 
 
b) 
Fig. 4. Results of ELM performance verification 
4. Conclusions 
This study proposed an efficient algorithm for bearing fault diagnosis based on EMD, KPCA, 
and ELM. First, feature vectors of vibration signals from bearing are extracted and calculated with 
EMD, then, to make the diagnosis more accurate and effective, KPCA is introduced to reduce the 
feature vector dimensions, finally, an ELM network, which is trained with pre-determined samples, 
are used to classify the fault pattern. The proposed method is demonstrated with several tests, and 
the accuracy is 98.5 %, meanwhile, the time consuming of ELM is much less, compared with BP 
neural network. 
Future works are expected to focus on two aspects. First, to improve diagnosis accuracy, 
influence factors of diagnosis, such as the training node number, will be analyzed. Then, 
comparisons between the proposed methods with traditional diagnosis methods, such as support 
vector machine, or radial basis neural network, will be conducted. 
Acknowledgements 
This research was supported by the National Natural Science Foundation of China (Grant 
No. 61074083, 50705005, and 51105019), and by the Technology Foundation Program of 
National Defense (Grant No. Z132013B002). 
References 
[1] Huang N. E., et al. The empirical mode decomposition and the Hubert spectrum for nonlinear and 
non-stationary time series analysis. Proceedings of the Royal Society A: Mathematical, Physical and 
Engineering Sciences, Vol. 454, 1971, p. 903-995, 1998. 
[2] Schölkopf B., Smola A., Müller K. Nonlinear component analysis as a kernel eigenvalue problem. 
Neural Computation, Vol. 10, Issue 5, 1998, p. 1299-1319. 
[3] Romero E., Alquézar R. Comparing error minimized extreme learning machines and support vector 
sequential feed-forward neural networks. Neural Networks, Vol. 25, 2012, p. 122-129. 
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0.4
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Bearing fault diagnosis based on EMD-KPCA and ELM

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Bearing fault diagnosis based on EMD-KPCA and ELM

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