Likelihood and Bayesian Prediction of Chaotic Systems

Abstract

Abstract 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 com...