ESTIMATING LYAPUNOV EXPONE0iTS WITH NONPARAMETRIC REGRESSION

vVe discuss procedures based on non parametric regression for estimating the dominant Lyapunov exponent Al from time-series data generated by a system xt =f(xt _l' x t _2' ... xt_d)+o-et , where XtEIR, and {et } is an iid sequence of random variables. For systems with bounded fluctuations in x t ' '\1 >0 is the defining feature of chaos. vVe show that any consistent estimator of the partial of/ax. can be used to obtain a consistent estimator J of .\. The rate of convergence we establish is quite slow. A better rate of convergence is derived heuristically. and supported by sim ulations. Sim ulation results from several implementations, one "local" (thin-plate splines) and three "global" (neural nets. radial basis functions. projection pursuit) are presented for two deterministic (0=0) chaotic systems. Local splines and the neural nets yield accurate estimates of the Lyapunov exponent. IImvc\·er. the spline method is sensitive to the choice of the embedding dimension. Limited results for a noisy (0->0) Henon system suggest that the thin-plate spline and neural net regression methods also provide reliable values of the Lyapunov exponent. Key \Vords and Phrases: Thin plate smoothing splines. neural networks. projection pursuit regression, chaotic dynamic systems, chaos, nonlinear dynamical systems. 1 Daniel ~IcCaffrey is a graduate student, Stephen Ellner is an Associate Professor..-\. ll. Gallant is a Professor and Douglas Nychka is an Associate Professor. All are members uf 11lt" Department of Statistics, North Carolina State University, Box 8230. Raleigh. :\C ~'G8.5-·"~0:3. The authors would like to acknowledge the support of the National Science Foundation and the North Carolina Experiment Station.