Neural Networks for Chaotic Time Series Prediction

There are many systems that can be described as chaotic: The readings from seismic monitoring stations in mines which describe the rock dynamics, from EKG which describe the fibrillation of a cardiac patient’s heart, and the share prices in financial markets which describe the optimism about the earning potential of companies are examples of observations of deterministic, non−linear, dynamical systems. Clearly, chaotic systems are an important class of dynamical systems. Often, the only information we have about such systems is in the form of a time series, i.e. a time−ordered sequence of system state observations; such time series do not necessarily contain observations for all states, i.e some system states may be hidden. The objective of time series analysis is then to build a model for the unknown dynamical system that generated the time series. Linear modeling is a common time series analysis technique, based on the fact that the signal produced by a finite dimensional linear system comprises a finite number of frequencies, a necessary condition for successful prediction using linear models. Time series produced by chaotic dynamical systems do not have a finite number of frequencies, but rather a continuous Fourier spectrum. Thus, linear time series prediction techniques lack predictive power in this case. Commonly, dynamical systems such as neural networks with feedback structures are used to model chaotic systems. The objective of this study is a systematic survey of the capability of different classes of neural network architectures to model chaotic time series. Given the significance of chaotic systems in our daily lives, we hope to make a contribution toward a better understanding of a class of modelling tools which are enjoying an ever increasing popularity.