A topological approach for multivariate time series characterization: the epileptic brain

In this paper we propose a methodology based on Topogical Data Analysis (TDA) for capturing when a complex system, represented by a multivariate time series, changes its internal organization. The modification of the inner organization among the entities belonging to a complex system can induce a phase transition of the entire system. In order to identify these reorganizations, we designed a new methodology that is based on the representation of time series by simplicial complexes. The topologization of multivariate time series successfully pinpoints out when a complex system evolves. Simplicial complexes are characterized by persistent homology techniques, such as the clique weight rank persistent homology and the topological invariants are used for computing a new entropy measure, the so-called weighted persistent entropy. With respect to the global invariants, e.g. the Betti numbers, the entropy takes into account also the topological noise and then it captures when a phase transition happens in a system. In order to verify the reliability of the methodology, we have analyzed the EEG signals of PhysioNet database and we have found numerical evidences that the methodology is able to detect the transition between the pre-ictal and ictal states.