*J. R. Statist. Soc.* B (1992)

54, No. 2, pp. 399-426

**Finding Chaos in Noisy Systems**

By

DOUGLAS NYCHKA, STEPHEN ELLNER and A. RONALD GALLANT

North Carolina State University, Raleigh, USA

and

DANIEL McCAFFREY

Rand Corporation, Santa Monica, USA

[*Read before* The Royal Statistical Society *at a meeting on Chaos organized by the* Research Section

*on Wednesday, October 16th, 1991*]

SUMMARY

In the past 20 years there has been much interest in the physical and biological sciences in non-linear dynamical systems that appear to have random, unpredictable behaviour. One important parameter of a dynamical system is the dominant Lyapunov exponent (LE). When the behaviour of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behaviour. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (neural networks and thin plate splines) it is possible to estimate the LE consistently. The properties of these methods have been studied with simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (1820-1900). On the basis of a nonparametric analysis there is little evidence for low dimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.

*Keywords*: CROSS-VALIDATION; DYNAMICAL SYSTEMS; LYAPUNOV EXPONENT; MARTEN FUR
RETURNS; NEURAL NETWORKS; NONPARAMETRIC REGRESSION; THIN PLATE SPLINES