(in Theory and Methods)

Estimating the Lyapunov Exponent of a Chaotic System With Nonparametric Regression

Daniel F. McCaffrey, Stephen Ellner, A. Ronald Gallant, Douglas W. Nychka
Journal of the American Statistical Association, Vol. 87, No. 419. (Sep., 1992), pp. 682-695.


We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov Exponent l1 from time series data generated by a nonlinear autoregressive system with additive noise. For systems with bounded fluctuations, l1 > 0 is the defining feature of chaos. Thus our procedures can be used to examine time series data for evidence of chaotic dynamics. We show that a consistent estimator of the partial derivatives of the autoregression function can be used to obtain a consistent estimator of l1. The rate of convergence we establish is quite slow; a better rate of convergence is derived heuristically and supported by simulations. Simulation results from several implementations-one "local" (thin-plate splines) and three "global" (neural nets, radial basis functions, and projection pursuit)-are presented for two deterministic chaotic systems. Local splines and neural nets yield accurate estimates of the Lyapunov exponent; however, the spline method is sensitive to the choice of the embedding dimension. Limited results for a noisy system suggest that the thin-plate spline and neural net regression methods also provide reliable values of the Lyapunov exponent in this case.

Keywords: Dynamical systems, Neural networks, Nonlinear dynamics, Nonlinear time series models, Projection pursuit regression, Thin plate smoothing splines