Nonlinear Forecasting for the Classification of Natural Time Series

There is a growing trend in the natural sciences to view time series as products of dynamical systems. This viewpoint has proven to be particularly useful in stimulating debate and insight into the nature of the underlying generating mechanisms. Here I review some of the issues concerning the use of forecasting in the detection of nonlinearities and possible chaos, particularly with regard to stochastic chaos. Moreover, it is shown how recent attempts to measure meaningful Lyapunov exponents for ecological data are fundamentally flawed, and that when observational noise is convolved with process noise, computing Lyapunov exponents for the real system will be difficult. Such problems pave the way for more operational definitions of dynamic complexity (cf. Yao & Tong, this volume). Aside from its use in the characterization of chaos, nonlinear forecasting can be used more broadly in pragmatic classification problems. Here I review a recent example of nonlinear forecasting as it is applied to classify human heart rhythms. In particular, it is shown how forecast nonlinearity can be a good discriminator of the physiological effects of age, and how prediction-decay may discriminate heart-disease. In so doing, I introduce a method for characterizing nonlinearity using `S-maps' and a method for analysing multiple short time series with composite attractors.