Time topological analysis of EEG using signature theory

Anomaly detection in multivariate signals is a task of paramount importance in many disciplines (epidemiology, finance, cognitive sciences and neurosciences, oncology, etc.). In this perspective, Topological Data Analysis (TDA) offers a battery of "shape" invariants that can be exploited for the implementation of an effective detection scheme. Our contribution consists of extending the constructions presented in citechretienleveraging on the construction of simplicial complexes from the Signatures of signals and their predictive capacities, rather than the use of a generic distance as in citepetri2014homological. Signature theory is a new theme in Machine Learning arXiv:1603.03788 stemming from recent work on the notions of Rough Paths developed by Terry Lyons and his team citelyons2002system based on the formalism introduced by Chen citechen1957integration. We explore in particular the detection of changes in topology, based on tracking the evolution of homological persistence and the Betti numbers associated with the complex introduced in citechretienleveraging. We apply our tools for the analysis of brain signals such as EEG to detect precursor phenomena to epileptic seizures.