Exploiting Chaos to Predict the Future and Reduce Noise

We discuss new approaches to forecasting, noise reduction, and the analysis of experimental data. The basic idea is to embed the data in a state space and then use straightforward numerical techniques to build a nonlinear dynamical model. We pick an ad hoc nonlinear representation, and fit it to the data. For higher dimensional problems we find that breaking the domain into neighborhoods using local approximation is usually better than using an arbitrary global representation. When random behavior is caused by low dimensional chaos our short term forecasts can be several orders of magnitude better than those of standard linear methods. We derive error estimates for the accuracy of approximation in terms of attractor dimension and Lyapunov exponents, the number of data points, and the extrapolation time. We demonstrate that for a given extrapolation time T iterating a short-term estimate is superior to computing an estimate for T directly.