One important property of a system that changes over time, such as an economy or an ecosystem, is the extent to which its future behavior can be predicted. This work will develop statistical methods for quantifying predictability, and apply these methods to address some open questions in ecology, epidemiology and macroeconomics. The result of this work will be an understanding of how a complex system's dynamics can be divided in two parts: a part that is a possibly complex function of the previous history of the system and a random component that is unrelated to the system's past behavior. The ability to make predictions depends on both of these components; moreover, knowing the relative contributions of the components is necessary for understanding how the system responds to external shocks. In the past 20 years there has been much interest in the use of nonlinear models to explain seemingly unpredictable or random phenomena. Most techniques for analyzing data from a nonlinear dynamic system have been based on large data sets and properties of deterministic models. Such methods are not useful for biological and economic systems that are subject to random perturbations and observed over a limited amount of time. Statistical methods will be developed that address these situations by combining techniques of nonparametric regression and time series analysis. These statistical techniques will enable researchers to reliably estimate the rules governing a system's evolution over time (law of motion) and the average response of the system to small perturbations (Lyapunov exponent). Also, at a more theoretical level, the properties of artificial neural networks for approximating systems of many variables will be studied. The methods will be applied to empirical, substantive time series in ecology and epidemiology in order to quantify the predictability of the systems from past history and to identify the system's response to exogenous (e.g. environmental) shocks. These methods, coupled with general equilibrium economic models, will provide new evidence for resolving a longstanding controversy in macroeconomics: Are extreme fluctuations in financial markets natural phenomena or are they aberrations requiring government regulation?
