LEAST SQUARES SUPPORT VECTOR MACHINE BASED CONDITION PREDICTION FOR BEARING HEALTH

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1
15
th
International Congress on Sound and Vibration

6-10 July 2008, Daejeon, Korea

LEAST SQUARES SUPPORT VECTOR MACHINE BASED
CONDITION PREDICTION FOR BEARING HEALTH
Fagang Zhao
1
, Jin Chen
1
and Lei Guo
1
1
State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University
Shanghai 200240, China
fagang@sjtu.edu.cn
Abstract
Due to the importance of condition maintenance, it is urgent to predict future condition in order to avoid
unexpected failure. So this paper presents a new scheme for the condition prediction of ball bearings
health based on least squares support vector machine (LS-SVM). Simulation and the practical
application have been carried out to validate the method. In the practical application, vibration data
which was collected from equipment is used to predict the future condition
1. INTRODUCTION
The manufacturing and industrial sectors are increasingly required to produce more and higher
quality products but avoid accidents as less as possible. As manufacturing equipments become
more complex and sophisticated, machine breakdowns are common. However, failure
conditions are difficult to identify and localize in a timely manner, scheduled maintenance
practices tend to reduce machine lifetime and increase down-time, resulting in loss of
productivity. So in order to prevent unexpected failures from shutdown, and reduce the
economic loss, the abnormal condition should be found as early as possible. Therefore,
condition monitoring and trend prediction is important for condition maintenance [1-3]. It uses
the features extracted form the raw data to make sure of the machine condition, and to predict
the trend. Trend prediction and residual life prediction is meaningful to maintenance decision.
To fulfil prognostics, there are three steps. At first, the defect or abnormality should be able
to be detected at its early stage and it is better to know which part causes the fault. Secondly, the
part or machine should be monitored continuously, so that we can get the trend data to predict
the machine state in future. At last, a prediction needs to be generated estimating the trend or the
residual useful life (RUL). Above the three steps the third step is the most difficult.
There are many indicators to detect the fault of equipments, and these indicators can also be
used to track the trend and predict the future condition. But how to select useful indicators as
the prediction parameters is difficult for researchers. There are many indicators, such as
time-domain statistical indicators: Peak-Peak (P-P), Root Mean Squares (RMS), Crest Factor,
Skew and Kurtosis; wavelet index, energy factor etc, there are so many indicators that we can
not use any indicator to predict the condition or residual life. According to [4] [5], we choose
RMS as the indicator in this research. Furthermore, to select the proper model is difficult in the
prediction of residual life, some researchers constructed the prediction models based on crack
propagation models: namely Paris Law [6], [7]. Ref. [8] uses neural network to predict bearing
life, and compares with the real life. Wang et al. [9] compared the results of applying recurrent
neural networks and neural–fuzzy inference systems to predict the fault damage propagation
trend. Yan et al. [10] employed a logistic regression model to calculate the probability of failure
ICSV15 • 6-10 July 2008 • Daejeon • Korea

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for given condition variables and an ARMA time series model to trend the condition variables
for failure prediction. Wang and Vachtsevanos [11] applied dynamic wavelet neural networks
to predict the fault propagation process and estimate the RUL as the time left before the fault
reaches a given value. Yam et al. [12] applied a recurrent neural network for predicting the
machine condition trend. Wang and Lee proposed a wavelet-neural network prediction
algorithm for performance evaluation, and evaluated and predicted the wearing condition of
machine spindle and cutting tools [13], and so on. In recent years, because the industry is in
urgent need of condition prediction and residual life prediction, the research in the field of fault
diagnosis is transferred to condition monitoring and prediction. So now researchers focus on
how to predict the future condition of machines intelligently and accurately, and reduce the
frequency of sudden accidents.
In this paper, we propose a new scheme for the condition prediction of a ball bearing health
based on least squares support vector machine (LS-SVM). This scheme can effectively research
equipments’ whole life cycle from the first time it comes into use to the final failure. In order to
validate the model, we carry out an experiment to test the new method. Fig. 1 is the whole flow
chart of this research.

Figure
.1.
the overall flow diagram of this research
2. THEORETICAL BACKGROUND OF LS-SVM [14]
Vapnik proposed support vector machines (SVM) method based on statistical learning [15].
Traditional support vector machine gets the solutions with optimal quadratic function. In the
process of optimal solution, the dimension of the matrix is related with the number of training
samples directly, and it is feasible to use inner product to solve the medium-scale optimal
solution. But to large-scale, the matrix should be decomposed or trimmed to reduce the
complexity. Much research has been done in the large-scale optimal solution. However, they
still use quadratic inequality constraints, which cost much time and can not process real-time
data. So it usually has to be used to process off-line data, which constricts the application of
SVM. Suykens [14] introduced variance term in the optimal function of SVM, and changed
constraints from inequality to equality, and then proposed SVM based on the equality
constraints, which is called Least Squares Support Vector Machine (LS-SVM). Since the
variance term was introduced into LS-SVM, optimal function of traditional SVM changed into
equality constraints, which the solution has changed from optimal quadratic function to linear
function, simplified the complexity of solution.
ICSV15 • 6-10 July 2008 • Daejeon • Korea
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The LS-SVM algorithm is as follows. Suppose the training set:

( )
{
}
,| 1,2,....,
k k
D x y k N= =
,
n
k
x
R∈
,
m
k
y R∈

Where
k
x
is the input data,
k
y
is the output data, in the primal space (
w
space), the
optimization problem can be describe as:

( )
2
,,1
1 1
,,
2 2
min
M
T
L
S i
w B i
L w B w w
ε
ε
γ ε
=
= +

(1)
Subject to the equality constraints:

( )
,1,...,
T
i i i
y w x B i M
ϕ ε= + + =
(2)
Where the nonlinear mapping
ϕ
:
n m
→ 
maps the input data into a high dimensional feature
space, which can be infinite dimensional. In the high dimensional feature space, the super
classification face is defined by
n
w R∈
,
B


.
w
is weight vector in the high dimensional
feature space,
B
is the bias term, and
i
ε
is the error variable,
γ
is the adjusting factor, and Eqn.
(1) is formula of the least squares support vector machine, which has been investigated by
Saunders et al [16] and Suykens & Vandewalle [17].
According to the optimal function Eqn. (1), we can define the Lagrange function:

( ) ( ) ( )
( )
1
,,;,,
M
T
LS LS i i i i
i
L w B J w B w x B yε α ε α ϕ ε
=
= − + + −

(3)
Where
i
α
denotes Lagrange multiplier, and the KTT optimality function is

( )
( )
1
1
0
0 0
0,1,...,
0 0,1,...,
M
i i
i
M
i
i
i i
i
T
i i i
i
LS
w x
w
LS
B
LS
i M
LS
w x B y i M
αϕ
α
α γε
ε
ϕ ε
α
=
=
∂⎧
= → =





= → =





⎪ = → = =





= → + + − = =





(4)
After eliminating of
w
and
i
ε
, we can get the following set of linear equations.

1
0
0 1
1
T
B
y
K I
α
γ

⎡ ⎤

⎤ ⎡ ⎤
=
⎢ ⎥

⎥ ⎢ ⎥
+

⎦ ⎣ ⎦⎢ ⎥
⎣ ⎦

(5)
Where
[ ]
1
...
M
x
x x=
,
[
]
1
;...;
M
y y y=
,
[
]
1 1;...;1
=

,
[
]
1
;...;
M
α
α α=

,1,...,
i j N
=
. As in the SVM
theory, according to Mercer’s condition, the matrix
K
can be written as

( )
( )
(
)
,
T
ij i j i j
K K x x x xϕ ϕ= =
(6)
Then the function estimation of LS-SVM is

( )
1
N
i ij
i
y
x K b
α
=
=
+

(7)
Where
i
α
and
b
can be computed by Eq. (5). RBF kernels one can take [15]

(
)
2
2
exp
ij i j
K x xη= − −
(8)
We can see from above that all the constraints have changed to be the equations, and we can
solve the linear equations to get the results. Obviously, linear equations can solved by least
ICSV15 • 6-10 July 2008 • Daejeon • Korea

4
square, which makes the computation easy and reduces computation time, So LS-SVM has
strong adaptability.
Furthermore, we choose normalized rooted mean squares error (NRMSE) as the index to
decide whether the prediction result is good or not. The expression is:

( )
2
,,
1
1
1
N
i pre i obs
i
obs
O O
N
NRMSE
S
=


=

(9)
Where
N
is the number of prediction data;
obs
S
is the standard deviation of samples;
,
i pre
O
is
the predicted value; and
,
i obs
O
is the true value at the time of
i
.
3. METHOD
In this research, we propose the method of LS-SVM to predict machines condition. The flow
chart of the proposed method is given in Fig.2.

Figure
.2. Flowchart of predict method with LS-SVM

There are many basis functions for LS-SVM, such as radial basis function (RBF) kernel,
linear function, polynomial function, wavelet function. RBF based LS-SVM has a good
adaptability to vibration signals, its robustness is better than the other basis functions based
LS-SVM, and its prediction preciseness is better than traditional SVM and neural network,
computation time is very small, but high efficiency. So In this Paper, RBF kernel will be used,
which is defined as:

2
2
(,) exp
2
K
σ
⎛ ⎞

=
⎜ − ⎟
⎜ ⎟
⎝ ⎠
x
y
x y
(10)
In order to get the more precise result, we utilize the leave-one-out cross validation approach.
The kernel width and the regularization parameter must be decided when we use the RBF
kernel. In this paper, we adopt a method to determine these parameters based on the
cross-validation idea. We define two data sets, namely the training set and the validation set
from the observed time series, respectively. The prediction error is estimated via cross
validation and when the model provides the lowest estimated error,
σ
is chosen. It can be
shown that for large data sets, cross validation is asymptotically equivalent to analytical model
selection. In this case, the computational cost of cross validation in terms of computational time
and training time is high.
ICSV15 • 6-10 July 2008 • Daejeon • Korea
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4. SIMULATION
In this section we present the results of the simulations and compare with the traditional
LS-SVM method. In the research of time series prediction, sunspot series and Mackey-Glass
time series are often used to test the algorithm. Here we use sunspot series. The sunspot data
extracted from Matlab toolbox. It is a sample of size
280m
=
. The first 200 values of the
sunspot data is used to train the model and the remaining values are used to predict. The
NRMSE yielded by LS-SVM is 1.758. For the sunspot dataset (normalized) we see that the
LS-SVM model provides us a good result. This method is significant when compare with
traditional LS-SVM. To illustrate the performance of the LS-SVM, the predicted time series are
shown in Fig. 3.
5
10
15
20
25
30
35
40

0.4

0.2
0
0.2
0.4
0.6
0.8
1
Predicted Serial
Real Serial

Figure
. 3.
The predicted result of the sunspot
5. EXPERIMENT
An experiment of condition monitoring is set up for validate the model. Fig. 4 is the position
where the sensor is installed.

Figure
. 4. Photo of equipment installed sensors
Then through data acquisition, preprocessing, feature extraction and feature reduction,
training samples and test samples are obtained. After that, train LS-SVM with the training
samples which are the time series. At last the model is employed to predict future condition and
compared with the result of traditional LS-SVM. Fig. 5 is the sketch of data acquisition system,
which includes sensors and signal conditioner, anti-aliasing filter, data acquisition computer,
oscilloscope and
dynamic analyzer
. Signals are probed by sensors. Then after signal conditioned
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and anti-aliasing filtered, the information is collected by computer. Oscilloscope and on-line
monitoring system are employed to analyze the validity of the signals. Fig. 6 is the result of the
predicted experiment series (normalized) used LS-SVM.
0
5
10
15
20
25
30
35
40
4
5
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
RMS Value
Predicted Serial
Real Serial

Figure 5. Sketch of data acquisition system Figure
. 6.
The predicted result of the experiment
6. CONCLUSIONS AND FUTURE WORK
According to the nonlinearity of bearings vibration, LS-SVM model is introduced into times
series prediction of vibration in this paper to predict bearing condition.
To provide a more reliable and real-time prognostic tool for the bearing condition, we
developed LS-SVM prediction approach to predict the behaviour of dynamic systems in this
paper. According to the example given above, we can see that it is useful to implement for both
bearing residual life prediction and condition prediction. Test results of this study showed that
the LS-SVM model is a reliable forecasting tool. It can capture the system’s dynamic behaviour
quickly and track the system’s features accurately. It is also a robust forecasting tool in terms of
its capabilities to accommodate different system operation conditions and variations in
system’s dynamic characteristics.
There are two aspects for the further research: one is to implement the predictor in other
complex industrial facilities and to develop new strategies for multi-step predictions, the other
one to find out if there is a better method to predict the time series or not.
7. ACKNOWLEDGEMENTS
The research was supported by the National Natural Science Foundation of China (Approved
Grant: 50675140) and the National High Technology Research and Development Program of
China (863 Program, NO. 2006AA04Z175).
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