Nonlinear time-series analysis revisited.

By E. Bradley, Holger Kantz. Chaos, 2015

331 citations recorded by the source corpus.

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.

Subfields: Computer Networks and Communications, Economics and Econometrics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/4f590bcf973f91e5727b80ebe621195ab2ad5292

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Citations in the corpus

Backpropagation algorithms and Reservoir Computing in Recurrent Neural Networks for the forecasting of complex spatiotemporal dynamics (2019, methodology): “The identification of a useful embedding and the construction of a forecasting model have been the subject of life-long research efforts (Bradley & Kantz, 2015).”
Complex network approaches to nonlinear time series analysis (2019, background): “Most existing methods originating from nonlinear dynamics are state-space models, which build local models in ‘‘patches’’ of a reconstructed state space and then use these models to predict the next point on the system’s trajectory, which remains an active area of research [17].”
Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction (2016, methodology): “i li i li i i i i i , i ll t - ce reconstruction through delay-c ordinat embe ding [35].”
Higher Order Dynamic Mode Decomposition (2017, methodology): “These methods have been recently revisited, combined with diffusion maps [62, 9] and the Koopman operator [24], and generalized to a probabilistic framework for stochastic systems [42, 7, 8].”