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Time Series Prediction of Earthquake Input by using Soft Computing
Hitoshi FURUTA, Yasutoshi NOMURA
Department of Informatics, Kansai University, Takatsuki, Osaka569-1095, Japan
nomura@sc.kutc.kansai-u.ac.jp
Abstract
Time series analysis is one of important issues in science,
engineering, and so on. Up to the present statistical
methods[1] such as AR model[2] and Kalman filter[3]
have been successfully applied, however, those statistical
methods may have problems for solving highly nonlinear
problems. In this paper, an attempt is made to develop
practical methods of nonlinear time series by introducing
such Soft Computing techniques[4][5][6] as Chaos
theory[7], Neural Network[8][9], GMDH[10][11] and
fuzzy modeling[12][13]. Using the earthquake input
record obtained in Hyogo, the applicability and accuracy of
the proposed methods are discussed with a comparison of
those results.
1.Introduction
In this study, the prediction of external force such as
earthquakes and wind loads is employed to discuss the
accuracy and efficiency of the prediction methods, because
of the importance of its prediction from the engineering
point of view. In these days, monitoring and controlling
play important roles to reduce the vibration of high-rise
buildings due to earthquakes and wind loads. As
buildings are getting higher, the vibration of high-rise
buildings due to earthquakes and winds becomes a subject
of discussion. At present, many high-rise buildings have
vibration control systems on their own. However, the
vibration control system works using measured earthquake
input and acceleration.
On the other hand, traffic flows on such bridges as
Amarube Bridge, Kansai International Airport Bridge and
Akashi Strait Bridge under strong winds are controlled
with the intensity of earthquake input through measuring.
Then, unsuitable control may be done due to the effects of
time lag between real and measured wind velocities. In
order to solve these problems, it is desirable not only to
achieve wind-resistant structure of buildings and bridges
but also to establish a practical method of predicting the
earthquake input and wind loads.
In this paper, an attempt is made to develop practical
prediction methods of earthquake input, which behaves
irregularly time to time, by introducing such so-called soft
computing techniques as Chaos theory (Ito,1993[7],
Takens, 1981[14][15][16], Iokibe, 1994[17], Sakawa at all,
1998[18]) Neural Network (Chen at all, 1989[8],
Funabashi, 1992[9]) and GMDH (Group Method of Data
Handling)(Ivakhenemko, 1968[10], Hayashi, 1985[11]).
Many researches have revealed that the Chaos theory is
useful in dealing with complex systems, Neural Network is
applicable to various problems like pattern recognition and
function approximation, and GMDH can analyze highly
nonlinear systems which have a few input and output
variables. Numerical examples are presented to illustrate
the applicability of the proposed methods, and to compare
the characteristics of those methods.
2.Time Series Prediction by Chaos Theory
The definition of Chaos is done by several researchers,
and generally speaking, Chaos is the phenomenon which is
“non-periodic vibration governed by a deterministic
system”. The deterministic system means the system
governed by a definite constant rule. And the
non-periodic vibration means the movement which entirely
acts randomly. Thus, deterministic Chaos is considered as
a phenomenon which behaves irregularly at a glance, but is
governed by a definite deterministic rule.
Orbital instability is a characteristic caused by the
sensitivity to the initial state that two very near points in the
state space become a long way off when the steps proceed.
Lyapunov exponent is used to distinguish whether the time
series are chaotic or not. Lyapunov exponent expresses
the leaving velocity of the two orbits from near points time
to time. If Lyapunov exponent is positive, then the
behavior may be chaotic, else if it is negative, then it is not
chaotic. Due to the characteristic, the Chaos theory can
not be applied to long-term prediction, but is suitable for
short-term prediction.
There are some cases which behave chaotic; one is such
a behavior as logistic mapping problem which is governed
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
by a clear recursive formula, and the others are the
demands of water or traffic flows which are social
problems. In order to provide us with useful information,
it is desirable to predict it, and some researches have been
made so far.
It is necessary to discretize time series data at fixed
sampling interval. If the data sampled is chaotic, the
behavior is regard to be governed by a deterministic rule.
Then, if a non-linear deterministic rule is estimated, it is
able to predict the data of near future until the deterministic
rule does not work due to the sensitivity to the initial state.
This process is the most important to estimate the
dynamics. It is needed to reconstruct time series data in n
dimension state space with the embedding theorem of
Takens (Takens, 1981[14][15][16]) and make an orbit.
This orbit is called attractor, and the attractor which
appears when the behavior is chaotic is especially called
strange attractor. Then, the objective data vectors are
predicted by a local reconstruction method with the nearest
data vector plotted and the later data vectors. There are
some methods of reconstruction such as the tethelation
method, the local fuzzy reconstruction method and so on.
And the objective time series data is searched with the
transformation of the estimated data vector.
When a time series shows a chaotic behavior, it may
follow a deterministic rule. Then, it is possible to predict
the near future behavior of the time series with the aid of
Chaos theory. Here, it is attempted to predict the wind
velocity given by a time series record, using Local Fuzzy
Reconstruction Method (LFRM) (Iokibe at all, 1994[17])
and Deterministic Nonlinear Prediction Method using
Neighborhood’s Difference (NDM) (Sakawa at all,
1998[18])
The prediction process is performed as follows:
At first, the embedding of the time series data is done by
using the Takens’ theorem.
The time lag is calculated according to the embedding
results.
The dimension of the state space is determined as the
dimension showing the most strong chaotic behavior.
Figure 1 shows the earthquake acceleration. Figure 2
shows the attractors obtained by changing the time lag.
Figure 3 shows the variation of the Lyapunov exponent
analysis according to the change of embedding dimension.
Based upon those results, the optimal parameters can be
obtained as shown in Table 1.
In this study, prediction parameter is established in Table
1.
Figure 4 and 5 present the results calculated by NDM
and LFRM respectively. Although the both methods
provide satisfactory results with good accuracy, NDM can
present slightly better agreement with the observed data in
this example. Prediction start point is 100 step and
Prediction end point is 5000 step.
Figure 1. Earthquake acceleration
Figure 2. Results of Embedding by Takens’ Theorem
Figure 3. Results of Lyapunov exponent analysis
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
Table 1. Prediction parameters of chaotic analysis
Dimension 2
Neighborhood’s vector 8
Time lag 1
Figure 4. Prediction results by NDM
Table 2. Prediction Results by NDM
Correlation Coeff Initialized MSE
0.97645 0.21644
Figure 5. Prediction results by LFRM
Table 3. Prediction Results by LFRM
Correlation Coeff Initialized MSE
0.96729 0.25382
3.Time Series Prediction by Neural Network
Neural Network can be applied to various problems such as
pattern recognition, approximation of function,
classification, etc. Here, it is attempted to apply a
multi-layer Neural Network methods for the time series
prediction. As a learning method, the back-propagation
method (Chen at all 1989, Funabashi, 1992) is employed,
whose parameters are given in Table 2. Figureureure. 6
shows the convergence of the learning process. From
Figureureure. 7, it can be seen that the Neural Network can
provide a rather good prediction. However, comparing
with the Chaos theory, the prediction of the Neural
Network method still has such problems that the accuracy
depends on the number of layer and node and the learning
process.
Table 4. Parameter of Neural Network
Input Unit 4
Hidden Unit 5
Output Unit 1
Learning Coefficient 0.4
Momentum Coefficient 0.6
Slope of Sigmoid Function 1
Figure 6. Convergence of Learning Process
Figure 7. Prediction results of Neural Network
Table 5. Prediction Results by Neural Network
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
Correlation Coeff Initialized MSE
0.96925 0.30292
4. Time Series Prediction by GMDH
GMDH (Group Method of Data Handling) theory was
developed by Ivakhnemko to analyze a nonlinear problem
by introducing quadratic functions with the principle of
Heuristic Self-Organization (Ivakhnemko, 1968, Hayashi,
1995) GMDH is regarded as one of the methods of Neural
Network, but GMDH can deal with highly nonlinear
complicated problems by a layer model of quadratic
functions. In GMDH, multi-layer model is used, which is
the most distinctive characteristic. The success of the
model is dependent on the structure of multi-layer.
At first, it is necessary to divide input and output variables
into training data and checking data. Training data are
used to analyze the model, and checking data are used to
check the analyzed model. When there are N input and
output data, Nt training and Nc checking data, it must be
that the total of Nt and N is N.
A quadratic expression is constructed with any two of n
input data and output data, for variables of xp, xq and y (Eq.
1).
qpqpqp xfxexdxcxbxay 22 (1)
where a, b, c, d ,e and f are constant coefficients.
The six coefficientsa through f are estimated with training
data by a linear recurrent analysis. Intermediate
parameter Z is calculated with the substitution of estimated
coefficients and input data of checking data into Eq. 2.
qpqpqp xfxexdxcxbxaz 22 (2)
Then, square errors E are calculated as
2)( zyE (3)
Square errors calculated for n checking data are sorted
from small to large, and expressed as E1 to En(n-1)/2.
If E1 and E* satisfy Inequality 4, the algorithm was
terrminated. E* is the E1 obtained at the previous layer.
E1>E* (4)
If Inequality 4 is not satisfied, it is needed to calculate z
with substitution input data for Eq. 2. Then, z is regarded
as the input data for the next layer. And the operation is
repeated until the termination condition is satisfied.
If the algorithm is stopped, intermidiate parameter of
previous layer is substituted into the expression at the layer
where E* is gained. The estimated model is established
with repeating this operation to the first layer.
Figureureure. 8 presents the prediction results obtained by
GMDH. This shows a good agreement. In this example,
the calculation terminated at the first layer, because it
satisfies the termination condition. Eq. 5 presents the
quadratic function obtained through the method.
Figureureureure 8. Prediction results by GMDH
Table 6. Prediction Results by GMDH
Correlation Coeff Initialized MSE
0.96875 0.29738
12
2
1
2
2
12
000074.0001386.0001537.0
584467.1849673.0637718.0
nnnn
nn
xxxx
xxy (5)
5.Comparison of Chaos, Neural Network,
GMDH and Fuzzy modeling for Time Series
Prediction
Table 8 presents the correlation coefficients and mean
square errors obtained for the five methods. NDM Chaos
method can provide the best prediction, namely, its errors
of 1 % occurred 1321 cases in 4900 cases. Successively,
LFRM Chaos method provides satisfactory results with 1%
error for 1129 cases. Fuzzy modeling provides
satisfactory results with 1% error for 1098 cases in 4900
Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE
cases.
On the other hand, GMDH and Neural Network provide
satisfactory results with 1 % error for 36, 79 cases in 4900
cases. As a result, it is concluded that the prediction
methods based on chaotic time series analysis can provide
enough results for the prediction of earthquake inputs.
6.Conclusions
In this paper, the applicability and efficiency of the
Chaos Prediction methods, Neural Network and GMDH
were discussed and examined using the time series data of
earthquake acceleration. Neural Network and GMDH
methods could not present good results for such data with
highly chaotic characteristics such as Logistic mapping.
However, it requires the shortest time to predict the time
series data, once the learning could be done. This is one
of advantages of Neural Network from a practical point of
view.
In the development of predicting method of earthquake
acceleration, the Chaotic Prediction methods and Fuzzy
modeling are useful in short-term prediction. If a more
long-term prediction is required, other methods may be
superior, though the Chaotic Prediction methods can
provide more accurate solutions for the short-term
prediction.
In this study, Chaotic Prediction methods, Neural
Network, GMDH and Fuzzy modeling were compared and
discussed with emphasis on accuracy and computing time.
However, instead of using them independently, it is
desirable to combine those methods to compensate their
own defects.
Table 7. Prediction error
NDM LFRM NN GMDH Fuzzy
Err<-5% 777 1035 2979 4418 1136
-5%<Err<-4% 97 140 30 25 124
-4%<Err<-3% 121 174 37 17 150
-3%<Err<-2% 163 238 26 13 217
-2%<Err<-1% 272 403 27 19 304
-1%<Err<0% 497 586 41 26 510
0%<Err<1% 824 543 38 10 588
1%<Err<2% 556 400 31 6 329
2%<Err<3% 345 263 32 19 215
3%<Err<4% 205 198 25 12 184
4%<Err<5% 184 114 27 11 166
5%<Err 859 806 1607 324 977
-1%<Err<1% 1321 1129 79 36 1098
Table 8. Comparison of prediction results by All methods
Correlation Coeff Initialized MSE
NDM 0.97645 0.21644
LFRM 0.96729 0.25382
Neural Network 0.96925 0.30292
GMDH 0.96875 0.29738
Fuzzy Modeling 0.97291 0.23123
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Proceedings of the Fourth International Symposium on Uncertainty Modeling and Analysis (ISUMA’03) 0-7695-1997-0/03 $17.00 © 2003 IEEE