Estimating the Lyapunov Exponent of a Chaotic System with Nonparametric Regression

We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov Exponent λ1 from time series data generated by a nonlinear autoregressive system with additive noise. For systems with bounded fluctuations, λ1 > 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 λ1. 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.