Topological Data Analysis for Directed Dependence Networks of Multivariate Time Series Data

TL;DR

This paper introduces a novel topological data analysis method for directed dependence networks in multivariate time series, especially brain networks, by decomposing asymmetric dependencies to better understand epileptic seizures.

Contribution

It proposes a new approach to handle directed, asymmetric dependence measures in TDA, overcoming limitations of symmetric-only methods for brain network analysis.

Findings

Decomposition of brain networks into symmetric and anti-symmetric parts.

Analysis of EEG data reveals differences in anti-symmetric components pre- and post-seizure.

Method shows promise for studying complex directed dependence in neural data.

Abstract

Topological data analysis (TDA) approaches are becoming increasingly popular for studying the dependence patterns in multivariate time series data. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes, which can be altered by various neurological impairments such as epileptic seizures. Existing TDA approaches rely on the notion of distance between data points that is symmetric by definition for building graph filtrations. For brain dependence networks, this is a major limitation that constrains practitioners to using only symmetric dependence measures, such as correlations or coherence. However, it is known that the brain dependence network may be very complex and can contain a directed flow of information from one brain region to another. Such dependence networks are usually captured by more advanced measures of dependence such as partial directed coherence, which is a Granger causality based dependence measure. These dependence measures will result in a non-symmetric distance function, especially during epileptic seizures. In this paper we propose to solve this limitation by decomposing the weighted connectivity network into its symmetric and anti-symmetric components using matrix decomposition and comparing the anti-symmetric component prior to and post seizure. Our analysis of epileptic seizure EEG data shows promising results.