Engineering Mathematics

1. SPECIAL SECTION ON DATA-DRIVEN MONITORING, FAULT DIAGNOSIS AND CONTROL OF
CYBER-PHYSICAL SYSTEMS
Received October 6, 2017, accepted November 7, 2017, date of publication November 15, 2017,
date of current version February 14, 2018.
Digital Object Identifier 10.1109/ACCESS.2017.2773665
Blind Source Separation Method for
Bearing Vibration Signals
HE JUN 1, YONG CHEN1, QING-HUA ZHANG2, GUOXI SUN2, AND QIN HU2
1School of Automation, Foshan University, Foshan 528000, China
2Guangdong Provincial Key Laboratory of Petrochemical Equipment Fault Diagnosis, Guangdong University of Petrochemical Technology, Maoming 525000,
China
Corresponding author: Guoxi Sun (guoxi.sun@gdupt.edu.cn)
This work was supported in part by the National Natural Science Foundation of China under Grant 61301300 and Grant 61673400, in part
by the Guangdong Natural Science Foundation under Grant 2016A030313823, and in part by the Guangdong Characteristic Innovation
Project of colleges and universities under Grant 201463104.
ABSTRACT In underdetermined blind source separation (UBSS) of vibration signals, the estimation of the
mixing matrix is often affected by noise and by the type of the used clustering algorithm. A novel UBSS
method for the analysis of vibration signals, aiming to address the problem of the inaccurate estimation
of the mixing matrix owing to noise and choice of the clustering method, is proposed here. The proposed
algorithm is based on the modified k-means clustering algorithm and the Laplace potential function. First,
the largest distance between data points is used to initialize the cluster centroid locations, and then the mean
distance between clustering centroids average distance range of data points is used for updating the locations
of cluster centroids. Next, the Laplace potential function that uses a global similarity criterion is applied to
fine-tune the cluster centroid locations. Normalized mean squared error and deviation angle measures were
used to assess the accuracy of the estimation of the mixing matrix. Bearing vibration data from Case Western
Reserve University and our experimental platform were used to analyze the performance of the developed
algorithm. Results of this analysis suggest that this proposed method can estimate the mixing matrix more
effectively, compared with existing methods.
INDEX TERMS Signal underdetermined blind source separation, Laplace potential function, k-means,
bearing vibration signal.
I. INTRODUCTION
Rotating machinery is the most common mechanical equip-
ment in the petrochemical industry and rolling bearings are
the crucial components of rotating machinery that are prone
to failure. Therefore, early detection of the bearing running
status and early fault diagnosis help to ensure safety and
reliable operation of machinery equipment [1]. Even small
defects of rotating machinery will be manifested in the vibra-
tion signal; however, the collected bearing vibration signal
of rotating machinery may be mingled with that from other
vibration sources or with strong background noise, which can
make the detection of faults less efficient, affecting diagnosis.
To resolve this problem and thus improve fault forecasting
and diagnosis, the most critical step is to separate the fault
signal from the mixed signal.
Blind source separation (BSS), which aims to recover the
sources that contribute to the measured signal without any
knowledge of the mixing system, has been widely used in
speech recognition [2], fault diagnosis [3], [4], and image
processing [5]. However, in practical applications, the num-
ber of sensors is always smaller than the number of signal
sources. This situation is known as the problem of under-
determined blind source separation (UBSS) [6], [7]. When
signals can be represented with sufficient sparsity, the UBSS
approach consists of two steps: (1) estimation of the mixing
matrix and (2) recovery of the underlying signal sources. The
accuracy with which the mixing matrix is estimated directly
affects the performance of the UBSS algorithm; therefore,
estimation of the mixing matrix is critical in the UBBS
approach; consequently, much attention has been devoted to
solving this problem.
Under the assumption of sparse signal representation,
estimation of the mixing matrix can be described as a cluster-
ing problem. Hence, clustering methods, such as k-means,
have been used for estimating the mixing matrix for the
UBSS problem, resulting in many publications [6], [8], [9].
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VOLUME 6, 2018

2. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
To overcome the drawbacks of the k-means method
(sensitivity to the initialization of clusters and random initial
allocation of cluster centroids), a UBSS method based on the
k-means and AP clustering was proposed with the affinity
propagation clustering method used for estimating the exact
number of clusters [10], and the k-means method with the
AP algorithm for initialization was used for estimating the
mixing matrix. However, random assignment of initial clus-
ter centroids can still significantly affect the performance
of the clustering method. To overcome these drawbacks,
factorial k-means clustering was proposed, which aimed at
discovering the structure of clusters in a lower-dimensional
subspace [11]. However, in this method, the number of
clusters and the subspace dimensionality which have to be
provided, which may affect the estimation of the mixing
matrix. To adaptively determine the number of clusters, a self-
learning k-means clustering method was reported in [12], and
a global optimization method was used for minimizing the
cluster distortions, based on the current cluster configuration.
Unfortunately, the performance of the k-means clustering
method heavily relies on the initial distribution of cluster
centroids. To alleviate this dependence of k-means clustering
on initial centroid locations, a novel method for setting initial
centroid locations was introduced in [13], based on obtaining
Euclidean distances from the original dataset. The above-
mentioned methods are based on k-means, and they perform
very well on noiseless data; however, in practical engineering
applications, the signal typically features significant back-
ground noise, which can negatively affect the performance
of the k-means method. To alleviate these effects, Ahmed
used potential functions to adjust the method convergence
rate in rapidly-exploring random tree planning [14]. Cui used
a local potential function to build a novel quantum neural
network model [15]; local potential functions were the basic
components, allowing to better adapt the model to different
data structures. Although these methods introduced some
advantages, they did not affect the performance of clustering.
Note that all of the above-mentioned methods are based
on using the k-means clustering method for estimating the
mixing matrix in UBSS, which implies there is a solution
for addressing natural drawbacks of the k-means clustering
algorithm. But in fact, for the working conditions of rele-
vance to rotating machinery in the petrochemical industry,
collected vibration signals are always mixed with unknown
background noise. Thus, we are facing the challenge of
dealing with the signal that is mixed with some unknown
background noise, which makes it more difficult to ensure
robustness with respect to the initial locations of cluster
centroids. The proposed method focuses on overcoming the
natural drawbacks of the k-means approach, and on solving
the problem of volatile performance of the k-means clustering
algorithm, aiming at developing a better way to estimate
the mixing matrix for UBSS of vibration signals. In this
paper, a bearing vibration signal BSS method, combined with
improved k-means and Laplace potential function methods,
is proposed. In the proposed approach, the data points at
the maximal and minimal distances from the rest of the data
points in the sample are considered as initial cluster centroids.
During the iterative learning of clustering, the mean distance
of data for each class defines the new centroid location.
In the next step, these data points for which the distance
from a certain cluster centroid is smaller than the average
distance for the class described by that centroid, are labeled
as belonging to that class. Finally, to eliminate clustering
fluctuations owing to the presence of background noise, the
Laplace potential function is used for minor adjustments of
the k-means clustering results.
The remainder of this paper is organized as follows.
In Section 2, the UBSS approach is briefly described. The
proposed method and its application to bearing vibration
signals (case study) are described in detail in Section 3
and Section 4, respectively. Finally, some conclusions are
presented in Section 5.
II. BLIND SOURCE SEPARATION METHOD
In this section, the background knowledge of UBSS approach
was introduced, and then the improved k-means method was
proposed, at last, the measurement of mixing matrix estima-
tion was presented.
A. UBSS APPROACH
In the BSS approach, the measured signal x(t) can be acquired
without knowledge of mixing. Thus, sampling statistics is
commonly used for the measured signal. Therefore, when
pre-processing the measured signal, it is assumed that there
are two independent signals, one is the n dimensional source
signal, which is described as s(t) = [s1(t), s2(t), · · · , sn(t)]T ,
while the other one is the m dimensional measured signal,
expressed as x(t) = [x1(t), x2(t), · · · , xm(t)]T . Hence, the
mixing model that links the source signal and the measured
signal can be written as
x(t) = As(t) (1)
where A is called the mixing matrix, which is expressed as
A = [a1, a2, · · · , an] with m × n dimensions. Meanwhile,
if A is the noise matrix, the measured signal x(t) can be
obtained as the source s(t) mixed with the mixing matrix in
the transfer process. The mixed model is given by





x1(t)
x2(t)
.
.
.
xm(t)





=





a11 a11 · · · a1n
a21 a22 · · · a2n
.
.
.
.
.
.
.
.
.
.
.
.
am1 am2 · · · amn





×





s1(t)
s2(t)
.
.
.
sm(t)





(2)
Based on the above analysis, the principal objective of
BSS is to estimate the source signal s(t) with assuming
s(t) and x(t) are independent each other. In this paper, we used
the UBSS method for vibration signal analysis of rotating
machinery; that is, for actual application industrial vibration
signals, the dimensionality m of the measured signal is lower
than the source signal dimensionality n. However, the UBSS
approach requires the signal to be very sparse; consequently,
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3. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
FIGURE 1. Time domain and frequency domain signals.
many approaches were attempted to ensure signal sparseness,
such as the Fourier transform. Here, we use the sparse signal
representation as in [16], and estimate the column vectors of
the mixing matrix. Combining equations (1) and (2), the sig-
nal x(t) can re-expressed as
x(t) = a1s1(t) + a2s2(t) + · · · + ansn(t), t = 1, 2, · · · , T
(3)
To make the signal sparse, the Fourier transform is com-
monly used for transforming signals from the time domain
to the frequency domain. The time domain and the frequency
domain signals are shown in Fig. 1 [17].
Fig.1 (b) shows that, in the frequency domain, the data
are approximately linear, implying that the measured signal
is sufficiently sparse. But for the source signal s(t) there is
no threshold for determining sufficient signal sparsity, and
there is only once si(t)(i = 1, 2, · · · , n) that satisfies with the
following equation:
xi(t)/xi+1(t) = ai,j/ai,j+1 (4)
That is, when there is only one source signal, the slope of
the observed signal xi and the mixing matrix Ai fall onto the
same straight line.
B. IMPROVED K-MEANS
Usually, the UBSS approach consists of two steps: 1) esti-
mation of the mixing matrix, which is the most important
step, and 2) separation of the source signals. Based on the
above mentioned two-step approach, the mixing matrix is
estimated in the first step of the proposed novel approach
that combines modified k-means clustering with the Laplace
potential function. Then, in the second step, the shortest path
method is used for recovering the source signals.
To overcome the drawbacks of k-means clustering, a mod-
ified k-means clustering method with two improvements is
proposed here. First, in the commonly used k-means clus-
tering approach, the initial cluster centroid locations are ran-
domly assigned, which often negatively affects the estimation
of the mixing matrix by increasing the estimation uncertainty.
To easier distinguish between different clusters in the process
of clustering, we propose to use the maxima and minima of
the entire dataset as initial centroids.
The initial centroid z is given as
z = arg max dist(xi, xj), (i 6= j, i, j ∈ n) (5)
xh = arg min
n
X
h=1
(dist(xi, xh) + dist(xj, xh)) (6)
where the centroid z denotes the initial centroid, which can
described as z = [xi, xj, xh], where xi, xj, xh are the centroids
for different classes, respectively, which are defined as the
maximal distances of each centroid.
Then, in the process of iterative learning of clustering, after
setting, for each class, its center of mass as a cluster centroid,
all data points within a distance d from a certain cluster
centroid are assigned to that class. The average distance of
data points in the i-th class from the class’s centroid zi is
di = mean
q
X
j=1
dist(zi, xij), i = 1, 2, · · · C, j = 1, 2, · · · q
(7)
where di is the distance of the i -th class datum from its cluster
centroid zi, and xij denotes j-th datum of i-th class. After
the average distance for i-th class is obtained, adjusting the
i-th class data proceeds according to the following condition
that should be satisfied.
(xi, xij) ← dist(zi, xij) < di,
(i = 1, 2, · · · , C; j = 1, 2, · · · q) (8)
After adjusting the data labels according to the distance
of data points from cluster centroids, centroid locations are
updated as follows:
znew
i = mean(dist(zi, xij)),
(i = 1, 2, · · · , C; j = 1, 2, · · · q) (9)
where znew
i is the new i-th class cluster centroid, obtained by
applying the above updating equation. Throughout the pro-
cess of clustering, the above steps are performed iteratively
until the locations of the cluster centroids no longer change.
Algorithm 1 The Modified k-Means Method
Step 1: Initialize k centroids using Equations (5) and (6);
Step 2: For each class, calculate the mean distance of the
class data from the class centroid, using Equation (7);
Step 3: Adjust the samples for each class according to the
condition in Equation (8), where the distance of the data
point from the centroid is smaller than the class average
distance di;
Step 4: Update centroid locations using the new data
for each class, using Equation (9). If centroid locations
changed in this iteration, go to Step 2; otherwise, clustering
is complete.
With regard to initialization and updating of cluster cen-
troid locations, this modified k-means algorithm performs
better than conventional k-means clustering algorithms.
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4. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
However, the unknown background noise and outliers are the
most important factors that can affect the performance and
the stability of the proposed clustering algorithm. At the same
time, the clustering process is a critical step in the estimation
of the mixing matrix; therefore, the method’s stability and
accuracy during this step can strongly affect the estimation
of the mixing matrix.
To address this problem, the Laplace potential func-
tion [14] was used for fine-tuning and trimming corrections
of cluster centroid locations. Owing to the assumptions of
sparsity, the source signal at most points is zero or near
zero, and its probability distribution follows the Laplace
distribution. In practical applications, many source signals
have sparse characteristics. If the source signal is not sparse,
it can be transformed to a sparse representation using the
sparse signal transform tool.
The model of the Laplace potential function can be
assumed as
f (x(t)) = exp(−(
m
X
i=1

6. xi(t) − zj

8. )/b), j = 1, 2, · · · , k
(10)
Generally speaking, when the Laplace potential function is
used for estimations, the sampled data need to be pretreated.
After regularization, the sample data are projected onto the
unit hyper-sphere.
x̃(t) = x(t) ×
x(t)
kx(t)k2
, t = 1, 2, · · · , T (11)
Thus, according to the given Laplace potential function
model, the estimation potential function of a cluster center
is given as
8(zj) =
T
X
t=1
exp(−(
m
X
i=1

10. x̃i(t) − zij

12. )/b), j = 1, 2, · · · , k
(12)
where b is the scale factor parameter, calculated as
b =
1
N
N
X
i=1
(
1
C
C
X
i=1

14. x̃i(t) − ux̃i

16. (13)
Here, ux̃i is the mean value of i-th x̃i(t), x̃i(t) denotes the
regularization of x(t), and C is the class number. The scale
factor parameter can be used to evaluate the influence of noise
on signal. For large b, the peak value of noise can be smoothed
out. It helps to reduce the influence of noise and to make the
estimation results more accurate. From the definition of the
potential function, there are some local maxima that are the
cluster centroids of the observed signal. However, the noise
hidden in the observed signal that makes the local maximum
of the function cannot be obtained accurately. To reduce the
influence of noise, a global similarity test function was intro-
duced to solve the Laplace potential function local maximum,
which is given as
J(z) =
k
X
j=1
T
X
t=1
(exp(−(
m
X
i=1

18. x̃i(t) − zij

20. )/b))γ
(14)
where the elimination factor, γ , is used to alleviate the
impact of the scale factor b on the probability distribution
of observed signal, obtained by the method of correlation
comparisons [18].
The local maximum of J(z) can be obtained as follows
equation with its value is zero.
dJ(z)
dzik
=
T
X
t=1
γ
b
(exp(−(
m
X
i=1
|x̃i(t)−zik|)/b))γ
×sign(x̃i(t)−zik)
(15)
In addition, the new cluster centroid zij can be obtained in con-
tinuous iterations, with the updating equation (16), as shown
at the bottom of this page, where zu+1
ij is the new cluster
centroid for i-th class after (u + 1)th iteration, and ε is a very
small positive integer for avoiding the error arising when the
denominator becomes zero. Usually, ε = 10−9.
C. ESTIMATION OF THE MIXING MATRIX
Estimating the underdetermined mixing matrix is the most
important step. After clustering iterations, the cluster centroid
z is obtained, following which the mixing matrix of the
observed signal needs to be calculated. However, using the
cluster centroid z to calculate the mixing matrix amounts to a
simple linear programming problem. According to the sparse
signal characteristics, when the observed signal is sufficiently
sparse, at a certain point there is only one si(i = 1, 2, · · · , n)
in the source signal s(t) that satisfies the condition that the
i-th source signal si and i-th column vector of the mixing
matrix A are distributed in the same direction. In other words,
the observed signal x(t) can gather on a line with the rake
ratio, and the column normalization condition of the mixing
matrix A defined as

22. a·j

24. 2
= 1 (17)
where a·j is the j-th column of the mixing matrix, which,
when combined with Equation (4), constitutes the following
zu+1
ij =
N
P
i=1
(exp(−(
m
P
i=1

26. xi(t) − zj

28. )/b))γ · xi(t)(

30. xi(t) − zj

32. + ε)
N
P
i=1
(exp(−(
m
P
i=1

34. xi(t) − zj

36. )/b))γ (

38. xi(t) − zj

40. + ε)
, j = 1, 2, · · · , k (16)
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41. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
FIGURE 2. Experimental machinery platform.
FIGURE 3. Experimental bearing elements.
set of equations.
(
ai,j/ai,j+1 = xi(t)/xi+1(t)
a2
i,j + a2
i,j+1 = 1
(18)
By solving this set of equations, the column vectors of the
estimated mixing matrix Ā for the mixing matrix A can be
obtained.
III. SIMULATIONS AND CASE STUDY
In this section, the experiment platform and the performance
evaluate criterion were introduced, and then the results analy-
sis was carried out. Two kinds of bearing vibration data used
to validate the effectiveness of mentioned methods and two
evaluation criterions introduced to evaluate the performance
of mentioned methods.
A. EXPERIMENTAL DATA AND EVALUATION CRITERION
To evaluate the estimation accuracy of the mixing matrix
using this proposed method, the normalized mean square
error (NMSE) and deviation angle measures were used
as the evaluation criteria. Two datasets of bearing vibra-
tion were used as the experimental data. One dataset of
bearing vibration was from Case Western Reserve Uni-
versity (CWRU) [19], while the data in the other bearing
vibration dataset were collected using our experimental plat-
form (Guangdong Province Key Laboratory of Petrochemical
Equipment Fault Diagnosis, GPKLPEFD) [20]. The exper-
imental platform and experimental failure parts are shown
in Fig 2 and 3, respectively.
The experimental data from the CWRU and GPKLPEFD
datasets are described in Tables 1 and 2, respectively.
NMSE has been widely used for evaluating the estima-
tion accuracy of the mixing matrix, and is given in the
TABLE 1. Experimental data from CWRU.
TABLE 2. Experimental data from GPKLPEFD.
following expression:
NMSE = 10 log10





m
P
i=1
n
P
j=1
(α̂ij − αij)2
m
P
i=1
n
P
j=1
α2
ij





(19)
where m and n are the number of rows and columns of
the original mixing matrix A, respectively, while α̂ij and αij
denote the i-th row and j-th column elements of the esti-
mated mixing matrix Ā and the original mixing matrix A,
respectively. The smaller the NMSE value, the higher is the
estimation accuracy of the mixing matrix.
Another widely used criterion, the deviation angle, is def-
ined as follows:
ang(a, â) =
180
π
arccos
a, â
kak ·
â
!
(20)
where ang(a, â) captures the extent of angular similarity
between the column vectors of Ā and A, while a is the column
vector of A and ā is the column vector of Ā corresponding
to a. Generally speaking, a smaller deviation angle indicates
a higher estimation accuracy of Ā.
B. CASE STUDY AND ANALYSIS OF RESULTS
First, in our experiment, the classical k-means clustering,
the improved k-means clustering method proposed in this
paper, and the Laplace potential function were abbreviated
as kmeans, impkmeans, and LPH, respectively. The number
of signal sources was assumed to be n = 4, and the number
of observed signals was set to m = 2. Then, the stochastic
mixing matrix was given as follows:
Astochastic =
0.4566 0.7435 0.6118 0.6571
0.8896 0.3312 0.9981 0.7538
The four source signals s(t) and the above mentioned
mixing matrix Astochastic were used for mixing, yielding
the two observed signals x(t). In this experiment, with the
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42. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
TABLE 3. Deviation angles between the estimated mixing matrices and their corresponding source matrices.
TABLE 4. NMSE values that were obtained for the tested methods.
two observed signals x(t) assumed to have been obtained,
we sought to estimate the four source signals s(t). However,
to make the observed signals sparser, the signals were pro-
cessed by applying the Fourier transform, and the parame-
ter γ was set to γ = 2.
Two observed signals collected from CWRU and
GPKLEFD were used for estimating the respective mixing
matrices. The mixing matrices were obtained by using the
classical k-means clustering algorithm and the improved
k-means clustering algorithm proposed in this paper, and the
obtained matrices were as follows:
ĀCWRU
kmeans =
0.5243 0.4156 0.7838 0.5189
0.8515 0.9095 0.6210 0.8548
ĀGPKLPEFD
kmeans =
0.6477 0.5866 0.5779 0.5826
0.7619 0.8098 0.8161 0.8128
ĀCWRU
impkmeans =
0.5160 0.6318 0.6917 0.6409
0.8566 0.7751 0.7221 0.7676
ĀGPKLPEFD
impkmeans =
0.5921 0.5840 0.5664 0.5890
0.8056 0.8118 0.8242 0.8081
Here, ĀCWRU
kmeans and ĀGPKLPEFD
kmeans are the estimated mixing
matrices obtained using the classical k-means clustering algo-
rithm applied to the CWRU dataset and GPKLPEFD dataset,
respectively. The matrices ĀCWRU
impkmeans and ĀGPKLPEFD
impkmeans are the
mixing matrices that were estimated using the presently pro-
posed improved k-means clustering algorithm applied to the
CWRU dataset and GPKLPEFD dataset, respectively.
To increase the robustness of k-means clustering, the LPH
was combined with the classical k-means clustering algo-
rithm and the presently proposed improved k-means cluster-
ing algorithm to estimate the mixing matrices for the CWRU
and GPKLPEFD datasets, respectively. The obtained esti-
mated mixing matrices were
ĀCWRU
kmeans+LPH =
0.5160 0.4156 0.5442 0.6008
0.8566 0.9095 0.8389 0.7994
ĀGPKLPEFD
kmeans+LPH =
0.6542 0.5861 0.5824 0.5926
0.7564 0.8103 0.8129 0.8055
ĀCWRU
impkmeans+LPH =
0.5099 0.6223 0.6541 0.6429
0.8602 0.7827 0.7564 0.7659
ĀGPKLPEFD
impkmeans+LPH =
0.5857 0.5907 0.5856 0.5784
0.8105 0.8069 0.8106 0.8157
Next, Equation (20) was used to calculate the deviation
angles for all columns of the estimated mixing matrices and
stochastic mixing matrices; the deviation angles obtained for
the different methods are listed in Table 3.
As Table 3 illustrates, for the CWRU dataset, the four
deviation angle indices of impkmeans are better than that
of kmeans. Only three angle indices obtained using the
impkmeans+LPH method are better than that obtained using
the kmeans+LPH method. The four deviation angle indices
for the kmeans+LPH method are better than that obtained
using the kmeans method, which indicates that the Laplace
function achieves some fine-tuning and improve the results
of clustering.
The obtained NMSE criterion values are listed in Table 4
As Table 4 shows, the NMSE index obtained using the
impkmeans+LPH method is better than those obtained using
the other tested methods. The NMSE obtained using the
impkmeans method is better than that obtained using the
kmeans method.
IV. CONCLUSION
In this paper, a UBSS method based on k-means clustering
and the Laplace potential function was introduced for analysis
of bearing vibration signals. A combined k-means clustering
and potential function method has been developed to improve
the estimation accuracy of the mixing matrix. A novel adjust-
ment scheme for cluster centroids was proposed to improve
the locations of k-means cluster centroids, and a small cor-
rection was introduced into the k-means clustering algorithm
to weaken the effect of noise. Two bearing vibration datasets
were used for conducting experiments. NMSE and deviation
angle were used as criteria to estimate the performance of
the tested methods. In simulations, estimation of the contrast
between the mixing matrices can reflect the effectiveness
and accuracy of the underlying signal processing algorithm.
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43. H. JUN et al.: Blind Source Separation Method for Bearing Vibration Signals
Overall, we conclude that using a novel k-means clustering
algorithm in combination with the potential function method
yields more accurate estimations compared with classical
k-means clustering algorithms.
ACKNOWLEDGMENT
Thanks the Case Western Reserve University, Rockwell Sci-
ence Office of Naval Research and CVX for the bearing fault
data sets used in this research. Thanks for the valuable com-
ments from the anonymous reviewers who helped to improve
this paper very much. Thanks editors for their hard work for
this paper
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HE JUN received the Ph.D. degree from the South
China University of Technology, Guangzhou,
China, in 2012.
He is currently an Associate Professor of con-
trol science and engineering. His current research
interests include mechanical system and signal
processing, fault diagnosis and prognosis for
petrochemical equipment, and speech signal pro-
cessing and speaker recognition.
YONG CHEN received the Ph.D. degree from
Central South University, Changsha, China,
in 2009.
He is currently an Associate Professor with
the School of Automation, Foshan University.
His research interests include complex system
control and optimization, application of integrated
automation technology, and fault diagnosis and
prognosis for petrochemical equipment.
QING-HUA ZHANG received the Ph.D. degree
from the South China University of Technology,
Guangzhou, China, in 2004.
He is currently a Professor and the Dean of the
Guangdong Province Petrochemical Equipment
Fault Diagnosis Key Laboratory. His research
interests include: condition monitoring and fault
diagnosis of rotating machinery, intelligence
control, and applications of intelligent algorithms.
GUOXI SUN received the Ph.D. degree in signals
and systems from the South China University of
Technology, Guangzhou, China, in 2006.
His current interests include rotating machinery
fault diagnosis, pattern recognition, and machine
learning.
QIN HU received the B.Sc. degree in electrical
engineering and automation from Binzhou Univer-
sity, Binzhou, China, in 2010.
He received the M.Sc. degree from the
Guangdong University of Technology in 2013. His
research direction is artificial immune algorithm,
genetic programming, and fault diagnosis.
664 VOLUME 6, 2018