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Kalman Filtering and Neural Networks P4

CHAOTIC DYNAMICS Gaurav S. Patel Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario, Canada Simon Haykin Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada (haykin@mcmaster.ca) 4.1 INTRODUCTION In this chapter, we consider another application of the extended Kalman filter recurrent multilayer perceptron (EKF-RMLP) scheme: the modeling of a chaotic time series or one that could be potentially chaotic. The generation of a chaotic process is governed by a coupled set of nonlinear differential or difference equations. The hallmark of a chaotic process is sensitivity to initial conditions, which means that if the starting point of motion is perturbed by a very small increment,. | Kalman Filtering and Neural Networks Edited by Simon Haykin Copyright 2001 John Wiley Sons Inc. ISBNs 0-471-36998-5 Hardback 0-471-22154-6 Electronic 4 CHAOTIC DYNAMICS Gaurav S. Patel Department of Electrical and Computer Engineering McMaster University Hamilton Ontario Canada Simon Haykin Communications Research Laboratory McMaster University Hamilton Ontario Canada haykin@mcmaster.ca 4.1 INTRODUCTION In this chapter we consider another application of the extended Kalman filter recurrent multilayer perceptron EKF-RMLP scheme the modeling of a chaotic time series or one that could be potentially chaotic. The generation of a chaotic process is governed by a coupled set of nonlinear differential or difference equations. The hallmark of a chaotic process is sensitivity to initial conditions which means that if the starting point of motion is perturbed by a very small increment the deviation in 83 84 4 CHAOTIC DYNAMICS Table 4.1 Summary of data sets used in the study Network size Training length Testing length Sampling frequency f Hz Largest Lyapunov exponent lmax nats sample Correlation dimension DML Logistic 6-4R-2R-1 5 000 25 000 1 0.69 1.04 Ikeda 6-6R-5R-1 5 000 25 000 1 0.354 1.51 Lorenz 3-8R-7R-1 5 000 25 000 40 0.040 2.09 NH3 laser 9-10R-8R-1 1 000 9 000 1a 0.147 2.01 Sea clutter 6-8R-7R-1 40 000 10 000 1000 0.228 4.69 The sampling frequency for the laser data was not known. It was assumed to be 1 Hz for the Lyapunov exponent calculations. the resulting waveform compared to the original waveform increases exponentially with time. Consequently unlike an ordinary deterministic process a chaotic process is predictable only in the short term. Specifically we consider five data sets categorized as follows The logistic map Ikeda map and Lorenz attractor whose dynamics are governed by known equations the corresponding time series can therefore be numerically generated by using the known equations of motion. Laser intensity pulsations and sea clutter i.e. radar .