Estimation from Dynamical Systems and Individual Sequences

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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.