Likelihood and Bayesian Prediction of Chaotic Systems

原文摘要

There has recently been considerable interest in both applied disciplines and in mathematics, as well as in the popular science literature, in the areas of nonlinear dynamical systems and chaotic processes. By a nonlinear, deterministic dynamical system, we mean a time series in which, starting at some initial condition, the values of the series are some fixed, nonlinear function of the previous states. One of the more intriguing aspects of these models is their propensity for displaying very complex, apparently random behavior, even when simple models are analyzed. A consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system. The difficulty in prediction stems from the fact that even the tiniest of errors, including computer roundoff, in either the specification of the function or the initial condition, can lead to huge errors in prediction. After a brief review of dynamical systems and the role of probability in dealing with uncertainty, a common example of a simple dynamical system, the logistic map, is introduced. In Section 2 statistical prediction for a dynamical system, based on observing a portion of a realization of the system with error, is considered. The emphasis is on likelihood and Bayesian approaches to the problem of prediction. Example computations, based on simulated data, suggest some novel characteristics of chaotic data analysis. Section 3 is an excursus into the relationships between ergodic theory and chaos, with specific application to Bayesian forecasting. In Section 4 a common numerical approach, which generates discrete-time dynamical systems, to the solution of differential equations is described briefly. An example involving the so-called Duffing equation is presented.

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