Assessing Chaos in Time Series: Statistical Aspects and Perspectives

Chaos theory offers to time series analysis new perspectives as well as concepts and ideas that have a through contribution to statistics. On the other hand, statistical methodology has shown to play a crucial role for the comprehension of nonlinear and chaotic phenomena. One peculiar feature of chaotic systems is sensitivity to initial conditions, which is responsible of the unpredictability we experience in such phenomena. One of the most popular quantity that measures this property is the maximum Lyapunov characteristic exponent (MLCE). In this paper we discuss from a statistical perspective the problems arising in estimating both the MLCE and its generalizations in time series, issues that have recently deserved attention in the community of time series analysts. We also present a method based on resampling in order to assign confidence interval to the estimates of the MLCE. It is shown that in addition to answering the question of the presence of chaos, these methods give relevant contributions to the characterization of many other aspects of nonlinear time series.