Nonlinear Forecasting, Chaos and Statistics

Many natural and experimental time series are generated by a combination of coherent, low-dimensional dynamics and stochastic, high-dimensional dynamics. A famous example is the sunspot time series. In the first part of this paper a nonlinear forecasting algorithm is used to attempt to identify how much of the irregularity in an aperiodic time series is due to low-dimensional chaos, as opposed to high-dimensional noise. The algorithm is applied to experimentally generated time series from coupled diodes, fluid turbulence, and flame dynamics, and compared to dimension calculations. Theoretical results concerning the limitations of such forecasting algorithms in the presence of noise are reviewed in the second part of the paper. These results use a combination of ideas from dynamical systems and statistics. In particular, the state space reconstruction problem is investigated using properties of likelihood functions at low noise levels.