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 homo-logy 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 Phy-sioNet database and we have found numerical evidences that the methodology is able to detect the transition between the pre-ictal and ictal states.