Chaotic time series data are observed routinely in experiments on physical systems and in observations in the field. The authors review developments in the extraction of information of physical importance from such measurements. They discuss methods for (1) separating the signal of physical interest from contamination ("noise reduction"), (2) constructing an appropriate state space or phase space for the data in which the full structure of the strange attractor associated with the chaotic observations is unfolded, (3) evaluating invariant properties of the dynamics such as dimensions, Lyapunov exponents, and topological characteristics, and (4) model making, local and global, for prediction and other goals. They briefly touch on the effects of linearly filtering data before analyzing it as a chaotic time series. Controlling chaotic physical systems and using them to synchronize and possibly communicate between source and receiver is considered. Finally, chaos in space-time systems, that is, the dynamics of fields, is briefly considered. While much is now known about the analysis of observed temporal chaos, spatio-temporal chaotic systems pose new challenges. The emphasis throughout the review is on the tools one now has for the realistic study of measured data in laboratory and field settings. It is the goal of this review to bring these tools into general use among physicists who study classical and semiclassical systems. Much of the progress in studying chaotic systems has rested on computational tools with some underlying rigorous mathematics. Heuristic and intuitive analysis tools guided by this mathematics and realizable on existing computers constitute the core of this review.

The analysis of observed chaotic data in physical systems

Recently proposed noise reduction methods for nonlinear chaotic time sequences with additive noise are analyzed and generalized. All these methods have in common that they work iteratively, and that in each step of the iteration the noise is suppressed by requiring locally linear relations among the delay coordinates, i.e., by moving the delay vectors towards some smooth manifold. The different methods can be compared unambiguously in the case of strictly hyperbolic systems corrupted by measurement noise of infinitesimally low level. It was found that all proposed methods converge in this ideal case, but not equally fast. Different problems arise if the system is not hyperbolic, and at higher noise levels. A new scheme which seems to avoid most of these problems is proposed and tested, and seems to give the best noise reduction so far. Moreover, large improvements are possible within the new scheme and the previous schemes if their parameters are not kept fixed during the iteration, and if corrections are included which take into account the curvature of the attracting manifold. Finally, the fact that comparison with simple low‐pass filters tends to overestimate the relative achievements of these nonlinear noise reduction schemes is stressed, and it is suggested that they should be compared to Wiener‐type filters.

On noise reduction methods for chaotic data.

Within the field of chaos theory several methods for the analysis of complex dynamical systems have recently been proposed. In light of these ideas we study the dynamics which control the behavior over time of river flow, investigating the existence of a low-dimension deterministic component. The present article follows the research undertaken in the work of Porporato and Ridolfi [1996a] in which some clues as to the existence of chaos were collected. Particular emphasis is given here to the problem of noise and to nonlinear prediction. With regard to the latter, the benefits obtainable by means of the interpolation of the available time series are reported and the remarkable predictive results attained with this nonlinear method are shown.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/96WR03535

Nonlinear analysis of river flow time sequences

A noise reduction method for signals from nonlinear systems

Noise reduction and prediction of hydrometeorological time series: dynamical systems approach vs. stochastic approach

We describe a method for noise reduction in chaotic systems that is based on projection of the set of points comprising an embedded noisy orbit in R d toward a finite patchwork of best-fit local approximations to an m-dimensional surface M'⊂R d , m≤d

Local-geometric-projection method for noise reduction in chaotic maps and flows.

SNR perfomance of a noise reduction algorithm applied to coarsely sampled chaotic data

We describe a family of algorithms for chaotic noise reduction when a ‘‘reference’’ time series is assumed to be given. These algorithms are variations of the local‐geometric‐projection (LGP) algorithm first introduced in. Signal‐to‐noise ratio (SNR) improvements achieved depend on the quality of the reference time series, the degree to which information available from knowledge of the reference data is used, and on the SNR initially present in the data. We report results for cases where the reference time series is the original noise‐free time series, and where it is taken to be a noise reduced version of the given noisy time series.

Chaotic noise reduction by local‐geometric‐projection with a reference time series

andGuan-Hsong Hsu*Department of MathematicsUniversity of MissouriColumbia, MO 65211ABSTRACTWe present a method for noise reduction that does not depend on detailed prior knowledge of system dynamics.The method has performed reasonably well for known maps and flows. Also we present an empirically basedtechnique to estimate the initial signal—to—noise ratio for time series whose dynamical origin may be unknown.1. INTRODUCTIONThere has been a remarkable growth of interest over the last ten years in the analysis of chaotic data, and ofnoisy chaotic data in particular.17 The mere kinematic phenomenon of chaos allows for the possibility of exper-imental detection of the existence of low—dimensional dynamics in physical systems exhibiting complex behavior.In specific circumstances this information might aid the construction of a simple physical model embodying theessential dynamics. Noise caused by instrumentation or by effects associated with unwanted background contami-nation, either statistical or dynamical, can cloud the analysis of chaotic data and even the diagnosis of the presenceof of low—dimensional dynamics. Correspondingly, there has been considerable interest in developing methods toreduce the level of noise in experimental data, especially in recent years.814In this paper, we present new results based on a local geometric projection scheme for noise reduction. Thescheme possesses considerable structure and flexibility in how it can be applied. Our method : (1) does not requireprior knowledge of an underlying dynamics, (2) in extensive numerical studies has performed reasonably well invery high noise environments (noise amplitude up to 100%, or 0db signal—to—noise ratio (SNR)) as well as low; (3)can be used for both map and flow data, and (4) has performed reasonably well for both low and high samplingrate flow data. Finally, (5) from systematic studies of performance of the method we have been able to develop anempirically based technique to estimate in some cases the noise level actually present in an experimental chaotictime series.

Noise reduction for chaotic data by geometric projection

We describe a general, systematic method for assessing the presence or absence of determinism in time series. Our method is rooted in the standard engineering paradigm of hypothesis testing. Our application of this procedure is novel, however, for we test given data sets against the class of data sets that produce smoothness. That is, our null hypothesis is that of determinism. We highlight two inherently interactive key features of our approach which conspire to make this treatment promising, the use of a smoothness detector and of chaotic noise reduction.

Guan-Hsong Hsu

Papers

Local-geometric-projection method for noise reduction in chaotic maps and flows.

SNR perfomance of a noise reduction algorithm applied to coarsely sampled chaotic data

Chaotic noise reduction by local‐geometric‐projection with a reference time series

Noise reduction for chaotic data by geometric projection

Method to Discriminate Against Determinism in Time Series Data