Persistent Homology-Based Classification of Chaotic Multi-variate Time Series: Application to Electroencephalograms

In this work, we present a combination of dimension reduction techniques and persistent homology for detection of epileptic events in electroencephalograms for a special kind of epilepsy called petit-mal epilepsy. Persistent homology, one of the main methods in topological data analysis, extracts information about the structures appearing in a given data set. Since during an epileptic seizure of the above type the electrical brain activity is more synchronized, we take the resulting structure in the EEG signal as classification feature that is analyzed topologically by means of persistent homology. As preprocessing step, the dimension of the data is reduced by two alternative techniques, principal component analysis and dynamical component analysis, and their performance is compared. Our results show that in comparison to principal component analysis, dynamical component analysis captures the dynamics of the system when projected onto a low-dimensional subspace. Furthermore, the results prove that persistent homology is well-suited for the detection of petit-mal epileptic seizures by means of their inherent structure.