Discriminating low-dimensional chaos from randomness: A parametric time series modelling approach

Summary

A procedure for forecasting and, more in general, for making statistical inference about the degree of predictability of dynamical systems given by experimental time series is presented. A parametric time series modelling approach is taken, yielding a powerful technique for analysing the structure of a given signal. The approach we propose consists in fitting autoregressive processes to the data and forecasting future values of the system on the basis of the model selected. We distinguish between two possible forecasting techniques of a dynamical system given by experimental series of observations. The «global autoregressive approximation» views the observations as a realization of a stochastic process, the autoregression coefficients are estimated by best-fitting the model to all the data at once. The «local autoregressive approximation» views the observations as realizations of a truly deterministic process and the autoregression coefficients are continuously updated by using the nearest neighbours of the current state. Then, a proper comparison between the predictive skills of the two techniques allows us to gain insight into distinguishing low-dimensional chaos from randomness. The global procedure also gives adaptive spectral filters (all-poles filters) which are able to pick the dominant oscillations of the system. As an example of the application of the procedure the author considers the Henon map and stochastic processes as well (autoregressive moving average processes). The effects of additional noise (e.g. measurement errors) are also discussed.