Time Series Prediction of Earthquake Input by Using Soft Computing-III

Time Series Prediction of Earthquake Input by using Soft Computing

Hitoshi FURUTA, Yasutoshi NOMURA

Department of Informatics, Kansai University, Takatsuki, Osaka569-1095, Japan

[email protected]

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


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


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



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


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


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


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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Time Series Prediction of Earthquake Input by Using Soft Computing-III