Mathematical Sciences: Estimation and Inference for Noisy Nonlinear Systems

随着时间的推移而变化的系统，如经济或生态系统，其一个重要特性是其未来行为可以预测的程度。这项工作将发展量化可预测性的统计方法，并将这些方法应用于解决生态学、流行病学和宏观经济学中的一些开放性问题。这项工作的结果将是理解如何将复杂系统的动力学分为两部分：一部分可能是系统以前历史的复杂函数，另一部分是与系统过去行为无关的随机组件。做出预测的能力取决于这两个组成部分；此外，了解各组成部分的相对贡献对于理解系统如何应对外部冲击是必要的。在过去的20年里，人们对使用非线性模型来解释看似不可预测或随机的现象产生了浓厚的兴趣。大多数分析非线性动态系统数据的技术都是基于大数据集和确定性模型的特性。这种方法对于受随机扰动影响的生物和经济系统，以及在有限时间内观察到的系统，是没有用的。统计方法将通过结合非参数回归和时间序列分析技术来解决这些情况。这些统计技术将使研究人员能够可靠地估计控制系统随时间演变的规则（运动定律）和系统对小扰动的平均响应（李雅普诺夫指数）。此外，在更理论化的层面上，我们将研究人工神经网络在多变量近似系统中的特性。这些方法将应用于生态学和流行病学的经验、实质性时间序列，以便从过去的历史中量化系统的可预测性，并确定系统对外源（例如环境）冲击的反应。这些方法与一般均衡经济模型相结合，将为解决宏观经济学中一个长期存在的争议提供新的证据：金融市场的极端波动是自然现象，还是需要政府监管的异常现象？

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?