Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience and economics, and to understand some of their underlying dynamical features. Several methods have been proposed for the analysis of multivariate time series, yet most of them neglect the effect of non-pairwise interactions on the emerging dynamics. Here, we propose a framework to characterize the temporal evolution of higher-order dependencies within multivariate time series. Using network analysis and topology, we show that our framework robustly differentiates various spatiotemporal regimes of coupled chaotic maps. This includes chaotic dynamical phases and various types of synchronization. Hence, using the higher-order co-fluctuation patterns in simulated dynamical processes as a guide, we highlight and quantify signatures of higher-order patterns in data from brain functional activity, financial markets and epidemics. Overall, our approach sheds light on the higher-order organization of multivariate time series, allowing a better characterization of dynamical group dependencies inherent to real-world data.
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All data needed to evaluate the conclusions of this paper are available on Zenodo (10.5281/zenodo.7210075).
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We thank L. Lacasa and J. Goñi for feedback on an earlier version of the manuscript. Data were provided (in part) by the HCP, the WU-Minn Consortium (principal investigators D. V. Essen and K. Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research and by the McDonnell Center for Systems Neuroscience at Washington University. E.A. acknowledges financial support from the SNSF Ambizione project ‘Fingerprinting the brain: network science to extract features of cognition, behaviour and dysfunction’ (grant no. PZ00P2_185716). A.S. and E.A. acknowledge support from the SNSF COST project ‘Mapping the higher-order dynamics of neurodegeneration in human brain networks’ (grant no. IZCOZ0_198144). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
The authors declare no competing interests.
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Santoro, A., Battiston, F., Petri, G. et al. Higher-order organization of multivariate time series. Nat. Phys. (2023). https://doi.org/10.1038/s41567-022-01852-0