A Personal Overview of Non-linear Time Series Analysis from a Chaos Perspective
The contents of this paper were presented as a Special Invited Lecture at the 15th Nordic Conference on Mathematical Statistics, Lund, Sweden, August 1994, and at the DMV (German Mathematical Society) Seminar programme in July 1994.
A personal overview of non-linear time series from a chaos perspective is given in an informal but, it is hoped, informative style. Recent developments which, in a radically new way, formulate the notion of initial-value sensitivity with special reference to stochastic dynamical systems are surveyed. Its practical importance in prediction is highlighted and its statistical estimation included by appealing to the modern technique of locally linear non-parametric regression. The related notions of an embedding dimension and correlation dimension are also surveyed from the statistical stand-point. It is shown that deterministic dynamical systems theory, including chaos, has much to offer to the subject. In return, some current results in the subject are summarized, which suggest that some of the standard practice in the former may have to be revised when dealing with real noisy data. Several open problems are identified.
