Beneath the noise, chaos

• 成果类型: Article
• 署名作者: Lalley, SP
• 署名单位: Purdue University System; Purdue University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1018031203
• 发表日期: 1999
• 页码: 461-479
• 关键词: time-series; reduction

摘要

The problem of extracting a signal x(n) from a noise-corrupted time series y(n) = x(n) + e(n) is considered. The signal x(n) is assumed to be generated by a discrete-time, deterministic, chaotic dynamical system F, in particular, x(n) = F-n(x(0)), where the initial point x(0) is assumed to lie in a compact hyperbolic F-invariant set. It is shown that (1) if the noise sequence e(n) is Gaussian then it is impossible to consistently recover the signal x(n), but (2) if the noise sequence consists of i.i.d, random vectors uniformly bounded by a constant delta > 0, then it is possible to recover the signal x(n) provided delta < 5 Delta, where Delta is a separation threshold for F. A filtering algorithm for the latter situation is presented.