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

Abstract

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

A new entropy based summary function for topological data analysis

“…It was defined in its current form in [26] but a precursor of this definition appears in [9]. Some successful applications of persistent entropy have been developed for pattern recognition of signals [21], [27]; complex systems [3] and biological images [1]. A more theoretical approach allows to use persistent entropy to distinguish topological features from noise [2].…”

Tracing patterns and shapes in remittance and migration networks via persistent homology

“…This allows differentiation between cycles of the same dimension according to the variation in the edge weights, and is most useful for distinguishing cycles that are born at a later stage in the filtration as illustrated in an example in the following section. The introduction of this coloring scheme supplements the information encoded within barcodes, and could potentially be combined with other tools, such as persistence landscapes [36,37], homological scaffolds [38], and persistent entropy [39,40], that enrich the information extrapolated from barcodes.…”

Information Topological Characterization of Periodically Correlated Processes by Dilation Operators

“…Having now a better idea on the mathematical concepts of homology, we turn the discussion on the persistent homology. persistent homology has been applied for time series classification, [35], image pattern recognition [36], biology [37], phylogenetic [38], science of language [39] and various other fields like [40] or for brain monitoring [46], [47]. As seen in the previous sections, the group homology is obtained thanks to the comparison of simplices at different dimensions.…”