Estimation from Dynamical Systems and Individual Sequences

摘要：现代计算技术的出现和近年来对混沌的兴趣使得人们越来越关注表现出随机行为的确定性系统。这类系统的统计分析常常因为对其行为的测量可能表现出非常长期的依赖性而变得复杂。首席研究员正在研究从遍历过程和单个序列的非参数估计，特别强调从动态系统的测量产生的过程和序列。他正在开发和证明以下问题的一致性方案：(i)估计生成给定离散时间动力系统的映射；（ii）估计动力系统的平稳密度和Lyapunov指数；(3)单个序列的密度和回归估计。此外，pi正在寻求确定序列的特征，并估计它们的诱导变换。现代计算的出现和最近科学对混沌的兴趣越来越多地集中在受确定性定律支配的物理系统上，但表现出不稳定或不可预测的行为，这是随机现象的特征。这种动力系统已经在诸如医疗诊断和天气预报等不同领域得到了应用。首席研究员（P.I.）正在开发统计方法，可以通过对系统随时间演变的观察来估计动力系统的潜在特性。如果希望预测或控制系统的未来行为，这一点很重要。私家侦探正在开发方法来估计在一个单位时间内决定系统演变的规则。他还在开发评估系统对初始条件敏感性的方法：如果系统开始时处于两个非常相似的状态，那么之后会有多大的不同？P.I.正在寻找对各种各样的系统都有效的方法，包括那些测量在很长时间尺度上表现出依赖性的系统。

DMS 9971964 Estimation from Dynamical Systems and Individual SequencesAndrew B. Nobel, University of North Carolina, Chapel HillABSTRACT:The advent of modern computing and the recent interest in chaos have focussed increasing attention on deterministic systems that exhibit random behavior. Statistical analysis of such systems is often complicated by the fact that measurements of their behavior can exhibit very long-range dependence. The Principal Investigator is studying non-parametric estimation from ergodic processes and individual sequences, with particular emphasis on processes and sequences that arise from measurement of a dynamical system. He is developing and proving the consistency of schemes for the following problems: (i) estimating the map generating a given discrete time dynamical system; (ii) estimating the stationary density and Lyapunov exponents of a dynamical system; and (iii) density and regression estimation from individual sequences. In addition, the P.I. is seeking to characterize deterministic sequences and to estimate their induced transformations.The advent of modern computing and recent scientific interest in chaos have focussed increasing attention on physical systems that are governed by deterministic laws, but exhibit erratic or unpredictable behavior that is characteristic of random phenomena. Dynamical systems of this sort have found application in such diverse areas as medical diagnostics and weather prediction.The Principal Investigator (P.I.) is developing statistical methods that can be used estimate the underlying properties of a dynamical system from observations of the system as it evolves in time. This is important if one wishes to predict or control the future behavior of the system. The P.I. is developing methods to estimate the rule that dictates the evolution of the system over a single unit of time. He is also developing methods to assess the sensitivity of the system to its initial conditions: if the system is started off in two very similar states, how different will it look later? The P.I. is seeking methods that will be effective for a wide variety of systems, including those whose measurements exhibit dependence over very long time scales.