Unveiling the higher-order structure of multivariate time series

Time series analysis has proven to be a powerful method to characterize several phenomena in biology, neuroscience, economics, and to understand some of their underlying dynamical features. Despite several methods currently exists for the analysis of multivariate time series, most of them do not investigate whether the signals stem from either independent, joint, or group interactions. Here, we propose a new framework to investigate the higher-order dependencies within a multivariate time series. We distinguish instantaneous co-fluctuation patterns at different group levels (pairs, triplets, etc), and then characterize the additional coherence of higher-order co-fluctuation patterns using TDA tools. We test our framework on coupled chaotic maps, demonstrating that it robustly differentiates various spatiotemporal regimes, including chaotic dynamical phases and various types of synchronization. By analysing fMRI signals, we find that, during rest, the human brain mainly oscillates between chaotic and partially intermittent states, with higher-order structures reflecting Default Mode Network and somatomotor regions, respectively. In financial time series, instead, the presence of higher-order structures can efficiently discriminate crises from periods of financial stability.