Phase space reconstruction for non-uniformly sampled noisy time series

Jaqueline  Lekscha; Reik V.  Donner

doi:10.1063/1.5023860

ABSTRACT

Analyzing data from paleoclimate archives such as tree rings or lake sediments offers the opportunity of inferring information on past climate variability. Often, such data sets are univariate and a proper reconstruction of the system’s higher-dimensional phase space can be crucial for further analyses. In this study, we systematically compare the methods of time delay embedding and differential embedding for phase space reconstruction. Differential embedding relates the system’s higher-dimensional coordinates to the derivatives of the measured time series. For implementation, this requires robust and efficient algorithms to estimate derivatives from noisy and possibly non-uniformly sampled data. For this purpose, we consider several approaches: (i) central differences adapted to irregular sampling, (ii) a generalized version of discrete Legendre coordinates, and (iii) the concept of Moving Taylor Bayesian Regression. We evaluate the performance of differential and time delay embedding by studying two paradigmatic model systems—the Lorenz and the Rössler system. More precisely, we compare geometric properties of the reconstructed attractors to those of the original attractors by applying recurrence network analysis. Finally, we demonstrate the potential and the limitations of using the different phase space reconstruction methods in combination with windowed recurrence network analysis for inferring information about past climate variability. This is done by analyzing two well-studied paleoclimate data sets from Ecuador and Mexico. We find that studying the robustness of the results when varying the analysis parameters is an unavoidable step in order to make well-grounded statements on climate variability and to judge whether a data set is suitable for this kind of analysis.

ACKNOWLEDGMENTS

This work has been financially supported by the German Federal Ministry for Education and Research (BMBF) via the BMBF Young Investigators Group CoSy-CC²—Complex Systems Approaches to Understanding Causes and Consequences of Past, Present and Future Climate Change (Grant No. 01LN1306A) and the Studienstiftung des deutschen Volkes. Calculations have been performed with the help of the Python packages pyunicorn⁴³43. J. F. Donges, J. Heitzig, B. Beronov, M. Wiedermann, J. Runge, Q. Y. Feng, L. Tupikina, V. Stolbova, R. V. Donner, N. Marwan, et al., “Unified functional network and nonlinear time series analysis for complex systems science: The pyunicorn package,” Chaos 25, 113101 (2015). https://doi.org/10.1063/1.4934554 and MoTaBaR.²⁴24. J. Heitzig, “Moving Taylor Bayesian Regression for nonparametric multidimensional function estimation with possibly correlated errors,” SIAM J. Sci. Comput. 35, A1928–A1950 (2013). https://doi.org/10.1137/12087846X

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Phase space reconstruction for non-uniformly sampled noisy time series

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ReferencesSection:ChooseTop of pageABSTRACTI. INTRODUCTIONII. METHODSIII. MODEL SYSTEMSIV. APPLICATION TO PALEOC...V. CONCLUSIONSReferences <<CITING ARTICLES

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