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Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements.They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the efFects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. %hile much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

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Exponential smoothing is a formalization of the familiar learning process, which is a practical basis for statistical forecasting. Higher orders of smoothing are defined by the operator Snt(x) = αSn−1t(x) + (1 − α) Snt−1(x), where S0t(x) = xt, 0 < α < 1. If one assumes that the time series of observations {xt} is of the form xt = nt + ∑ı=Nı=0 aıtı where nt is a sample from some error population, then least squares estimates of the coefficients a, can be obtained from linear combinations of the operators S, S2, …, SN+1. Explicit forms of the forecasting equations are given for N = 0, 1, and 2. This result makes it practical to use higher order polynomials as forecasting models, since the smoothing computations are very simple, and only a minimum of historical statistics need be retained in the file from one forecast to the next.