In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data-typically univariate-via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems. PACS numbers: 05.45.Tp Keywords: time series analysis, other things Nonlinear time-series analysis comprises a set of methods that extract dynamical information about the succession of values in a data set. This framework relies critically on the concept of reconstruction of the state space of the system from which the data are sampled. The foundations for this approach were laid around 1980, when deterministic chaos became a popular field of research and scientists were looking for evidence of chaos in natural and laboratory systems. One of the first-and still most spectacularapplications was the prediction of the path of a ball on a roulette wheel, which nicely demonstrated the power of these methods. Since then, nonlinear time-series analysis has left this narrow niche and moved into much broader use across all branches of science and engineering, as well as social science, the humanities, and beyond. The goal of time-series analysis is to learn about the dynamics behind some observed time-ordered data. Early approaches to this employed linear stochastic models-more precisely, autoregressive (AR) and moving average (MA) models 1 . These stationary Gaussian a) EB thanks the Max-Planck-Institut fur Physik komplexer Systeme for hosting the visit during which this paper was written. b) Electronic mail: lizb@cs.colorado.edu c) Electronic mail: kantz@pks.mpg.de stochastic processes are fully characterized by their twopoint auto-correlation function or by their power spectrum, respectively. There are many data sets where this type of analysis leads to a good characterization, such as temperature anomalies: differences between the daily (maximum, mean, minimum) temperature at a given place and the many-year average of that quantity for the corresponding calendar day. Data of this type possess an almost Gaussian distribution with an almost exponentially decaying auto-correlation function; typically the null hypothesis that they are generated by an AR(1) process cannot be rejected easily on the basis of observed data. Of course, we know that temperatures can be predicted much more accurately by high-dimensional physics-based models of the atmosphere than by AR(1) models. That scalar temperature data look like AR data comes from the projection of dynamics in a high-dimensional state space onto a single quantity. This illustrates that, depending on one's point of view and one's access to a system's variables, the very same system might appear to have very different complexity. As in any other analysis, the choice of a specific timeseries analysis method requires justification by some hypothesis about the appropriate data model. Time-series analysis is essentially data compression: we compute a few characteristic numbers from a large sample of data. This reduced information can only enhance our knowledge about the underlying system if we can interpret it, and it becomes interpretable through the fact that the arXiv:1503.07493v1 [nlin.CD]
Nonlinear time series analysis revolves around reconstructing a high dimensional dynamical system from an univariate time series, and studying the properties of the reconstructed dynamical system to derive knowledge on the original, univariate time series REF .
Nonlinear time-series analysis revisited
Chaos
Chaos
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