(in Theory and Methods)
Estimating the Lyapunov Exponent of a Chaotic System With
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
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
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