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Time topological analysis of EEG using signature theory

Stéphane Chrétien, Ben Gao, Astrid Thebault-Guiochon, Rémi Vaucher

TL;DR

The paper tackles anomaly detection in multivariate signals by developing a time-resolved topological framework built on signal signatures. It extends signature-based simplicial complexes to capture dynamic topology through Betti numbers $b_k$ and persistence entropy $PE$, using a LASSO-driven edge selection to form a filtration that evolves over time. The approach is applied to EEG data to identify precursor phenomena to epileptic seizures, showing that $b_1(t)$ and $PE(t)$ exhibit detectable changes around seizure events and during precritical periods. This topology-driven method offers a novel way to monitor brain dynamics and potentially anticipate seizures with a rigorously defined, time-varying geometric perspective.

Abstract

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 \cite{chretienleveraging} 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 \cite{petri2014homological}. 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 \cite{lyons2002system} based on the formalism introduced by Chen \cite{chen1957integration}. 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 \cite{chretienleveraging}. We apply our tools for the analysis of brain signals such as EEG to detect precursor phenomena to epileptic seizures.

Time topological analysis of EEG using signature theory

TL;DR

The paper tackles anomaly detection in multivariate signals by developing a time-resolved topological framework built on signal signatures. It extends signature-based simplicial complexes to capture dynamic topology through Betti numbers and persistence entropy , using a LASSO-driven edge selection to form a filtration that evolves over time. The approach is applied to EEG data to identify precursor phenomena to epileptic seizures, showing that and exhibit detectable changes around seizure events and during precritical periods. This topology-driven method offers a novel way to monitor brain dynamics and potentially anticipate seizures with a rigorously defined, time-varying geometric perspective.

Abstract

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 \cite{chretienleveraging} 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 \cite{petri2014homological}. 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 \cite{lyons2002system} based on the formalism introduced by Chen \cite{chen1957integration}. 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 \cite{chretienleveraging}. We apply our tools for the analysis of brain signals such as EEG to detect precursor phenomena to epileptic seizures.
Paper Structure (18 sections, 2 theorems, 6 equations, 20 figures, 1 algorithm)

This paper contains 18 sections, 2 theorems, 6 equations, 20 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Signature of rough paths (in a nutshell)
  3. Simplicial complex
  4. Some tools for simplicial complex analysis.
  5. Our simplicial complex construction algorithm
  6. The algorithm
  7. The induced filtration:
  8. Empirical study
  9. Importance of hyperparameter values
  10. Experiments with multiple EEG trajectories.
  11. Discussion
  12. Appendix
  13. Impact of the hyperparameters for $b_1 (t)$
  14. $\lambda_i$:
  15. $L$:
  16. ...and 3 more sections

Key Result

Theorem 1.1

Consider $X:[a,b]\rightarrow\mathbb{R}^d$ and $Y:[b,c]\rightarrow\mathbb{R}^d$. Define: as the concatenation of $X$ and $Y$. Then:

Figures (20)

  • Figure 1: $b_1(t)$ on 3 trajectories
  • Figure 2: $PE(t)$ on 3 trajectories
  • Figure : a) $\lambda_1=\lambda_2=1$.
  • Figure : a) $L=20s$.
  • Figure : a) $\lambda_1=\lambda_2=1$.
  • ...and 15 more figures

Theorems & Definitions (7)

  • Theorem 1.1: Chen's identity
  • Theorem 1.2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5