Estimating Lyapunov Exponents in Chaotic Time Series with Locally Weighted Regression

Nonlinear dynamical systems often exhibit chaos, which is characterized by sensitive dependence on initial values or more precisely by a positive Lyapunov exponent. Recognizing and quantifying chaos in time series represents an important step toward understanding the nature of random behavior and revealing the extent to which short-term forecasts may be improved. We will focus on the statistical problem of quantifying chaos and nonlinearity via Lyapunov exponents. Predicting the future or determining Lyapunov exponents requires estimation of an autoregressive function or its partial derivatives from time series. The multivariate locally weighted polynomial fit is studied for this purpose. In the nonparametric regression context, explicit asymptotic expansions for the conditional bias and conditional covariance matrix of the regression and partial derivative estimators are derived for both the local linear fit and the local quadratic fit. These results are then generalized to the time series context. The joint asymptotic normality of the estimators is established under general short-range dependence conditions, where the asymptotic bias and asymptotic covariance matrix are explicitly given. We also discuss extension to fractal time series, where the finite-dimensional probability measure is only assumed to possess a fractal dimension and can be singular with respect to the Lebesgue measure. The results on partial derivative estimation are subsequently applied to estimation of Lyapunovexponents. Using the asymptotic theory of the eigenvalues from a random matrix, we are able to characterize the asymptotic distribution for the estimators of local Lyapunov exponents. The local Lyapunov exponents may provide a way of quantifying short-term predictability in a system. The results may shed some light on the estimators of Lyapunov exponents.