A Mutual Information-based Method for the Estimation of the Dimension of Chaotic Dynamical Systems Using Neural Networks

In this paper, a method of estimating the dimension of dynamical systems from a time series, using neural networks, is examined. It is based (a) on the hypothesis that a member of a time series can be optimally expressed as a deterministic function of the d past series values (where d is the dimension of the system), and (b) on the observation that neural networks’learning ability is improved rapidly when the appropriate amount of information is provided to a neural structure which is as complex as needed. To estimate the dimension of a dynamical system, neural networks are trained to learn the component of the attractor expressed by a reconstructed vector in a suitable phase space whose embedding dimension m, has been estimated using the mutual information method. More specifically, the information supplied to the networks is represented by vectors consisting of the m past values of the time series, where m varies from 1 to D + 2, D being a pre-estimation for the maximum value of the embedding dimension of the system. The current method proposes that when m meets the dimension d of the dynamical system, the neural model of the attractor remarkably improves its learning ability, minimizing locally the RMS error of the training set. The logistic and the Henon map as well as the Lorenz and the Rosler attractors expressed as systems of difference equations, were examined to test the validity of the method.