The correlation methods involved in feature extrac-
tion are as follows: firstly, rolling bearing vibration
signals are obtained as initialize signals. And then,
rolling bearing signals are decomposed into a variety
of sub-signals and wavelet coefficients through time-
frequency transform, DWT. Thereafter, in order to
show the hidden information, reconstructed signals
are plotted in a three-dimensional phase space dia-
gram using PSR. Finally, the minimum features
with the highest accuracy are selected using SVD.
Fault classification aims at diagnosing different fault
types using an improved ELM classifier, which will be
described particularly in fourth part. With these meth-
ods, fault types of rolling bearing can be diagnosed
accurately.
Feature extraction
DWT
DWT is a crucial time-frequency analysis algorithm
for processing non-stationary and non-linear signals.
It decomposes signals on a multi-scale basis, where
each signal is decomposed into several sub-bands.
21
One of the greatest strengths of DWT is well
time-frequency localization, due to using long-time
windows at low frequencies and short-time windows
at high frequencies. As a consequence, it can decom-
pose signal at different scales in accordance with dif-
ferent targets. The details of DWT refer to Saravanan
and Ramachandran
22
and Mori et al.
23
The rolling bearing signals obtained by acceler-
ation sensor can be expressed as t
i
, where
i ¼ 1, 2, ..., N and N is the length of signals.
a,b
tðÞ¼ a
jj
1=2
t b
a
ð1Þ
where a is the scale factor, b is the shift factor, and
is the mother wavelet.
The function of continue wavelet transform is
defined as below
W
f
a, bðÞ¼f,
a,b
¼ a
jj
1=2
Z
þ1
1
ftðÞ
t b
a
dt
ð2Þ
The continuous wavelet should be discrete in prac-
tical application by computer. The function of DWT
can be obtained by discretizing the scale factor and
shift factor of continuous wavelet transform. Usually,
the parameter a and b can be defined as below
a ¼ a
m
0
, b ¼ nb
0
a
m
0
, m, n 2 Z ð3Þ
By replacing a and b in equation (1) with a
0
and b
0
in equation (3), the discrete wavelet sequence is
defined as
m,n
tðÞ¼ a
0
jj
m=2
a
m
0
t nb
0
m, n 2 Z
ð4Þ
where
m,n
ðtÞ is discrete wavelet sequence. Hence, the
corresponding discrete wavelet coefficient can be writ-
ten as
W
f
a, bðÞ¼f,
a,b
¼ a
jj
m=2
Z
þ1
1
f ðtÞ a
m
0
t nb
0
dt
ð5Þ
By using DWT, each signal is decomposed into
several sub-bands, and then a series of detail coeffi-
cients D
i
and approximation coefficients A
i
can be
obtained through calculate each sub-band. The
detail coefficients D
i
is coefficient of high frequency
and the approximation coefficients A
i
is the coefficient
of low frequency. The appropriate coefficients are
used for signal reconstruction. The reconstructed
information X
i
is used for the following processing.
PSR
The wavelet coefficients including detail and approxi-
mate coefficients show non-linear and non-stationary
characteristics in one-dimensional space. To reveal the
Discrete Wavelet Transfrom(DWT)
Phase Space Reconstruction(PSR)
Sigular Value Decomposition(SVD)
Improved ELM Classifier
Wavelet coefficients
High dimensional
feature vectors
Feature vectors
Classification accuracy
Feature extraction
Classification
Original signals
Experiments to get bearing data
Data collection
Figure 1. Block diagram of proposed method.
DWT: discrete wavelet transform; PSR: phase space recon-
struction; SVD: singular value decomposition; ELM: extreme
learning machine.
Li et al. 3