# Time Series Earthquake

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.

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.

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.

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,...

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