Estimating in Chaos: Cramér-Rao Bounds

Abstract

Estimating parameters of the dynamics from observations of a noise contaminated time series or estimating the clean time series itself is a classical challenge in the analysis of signals [Tre68]. This chapter contributes to the analysis of one’s ability to estimate a clean signal from observations of a noisy signal with the study of the case when the clean signal arises from a nonlinear dynamical system which exhibits chaotic evolution. We will argue that aspects of the very property of chaotic systems which makes them only slightly predictable and causes them to be nonperiodic, namely, the presence of positive Lyapunov exponents, also is connected with a Cramér-Rao bound which suggests an ability to estimate the clean time series with exponential accuracy. The bounds presented here give a rationale for the striking results on how well one can estimate clean data from noisy observations associated with manifold decomposition [Ham90, FS89, ABST93] as discussed in Chapter 7.