Embedding Theorems, Scaling Structures, and Determinism in Time Series

In this talk we discuss specific definitions of deterministic and stochastic for stationary time series. Our main purpose in doing so is to create a convenient rigorous framework in which to examine the interplay between state-space reconstruction (embedding theorems), scaling or fractal structures (the Grassberger-Procaccia algorithm), and the predictability properties of time series. Thus the definitions themselves are not as important as the clarity and precision they provide within the above context. In spite of the various pitfalls and limitations involved, choosing and adhering to a specific appropriate definition of determinism provides a firm foundation for proving theorems and constructing examples in those areas of chaos theory and time series concerned with reconstruction of the underlying source (or generating mechanism) of a time series. In this paper we provide some examples where our approach enables us to show that such reconstruction cannot be done.