Chaotic time series analysis: Prediction and noise reduction

Time series generated by chaotic dynamical systems confront the analyst with a new challenge. A chaotic system does not allow for long-term predictability, but the underlying determinism of its dynamics can be exploited to augment traditional statistical analysis. This dissertation describes new modelling procedures for chaotic time series, motivated by the geometrical perspective that comes from analyzing the evolution of state space trajectories of nonlinear dynamical systems.; Techniques for learning an approximate form of the system dynamics are discussed and demonstrated. Data is embedded in a state space using delay coordinates, and an ad hoc nonlinear representation of the dynamics is constructed. It is shown that local approximation schemes produce effective short-term predictions, and that the iteration of short forecasts is generally superior to computing a single estimate for long predictions. Error estimates are derived for the accuracy of approximation in terms of attractor dimension and Lyapunov exponents, the number of data points, and the extrapolation time. The techniques are demonstrated on both computer generated and real world data.; Several procedures for reducing noise in chaotic signals are presented. One method is a nonlinear averaging scheme based on the maximum likelihood principle, and another method is based on an optimal least squares solution of the shadowing problem. If the dynamical system is known exactly the noise can be reduced to the level of machine precision; if the dynamics must be learned from the data the noise reduction is limited by the accuracy of the learned representation. These noise reduction algorithms are demonstrated on several model dynamical systems, and our methods are compared to several other techniques that have been introduced by other groups.