A Java High Performance Tool For Topological Data Analysis
The two signals have been previously classified by the PhysioNet users. Respectively, SIGI corresponds to an individual affected by epilepsy, while SIGII belongs to an healthy patient. This classification also is identified by our methodology. The analysis of the persistent entropy reveals that in WH0 of SIGI a phase transition occurs. The topological interpretation is that among the windows with id = 20, 21 and 22, the number of connected components tends to be one and the topological noise is minimized (all the features are persistent). Before and after this period, the number of connected component is higher and the barcodes are noisy. These three windows correspond exactly to the transition from the pre-ictal state to the ictal state. In both signals, Betti numbers for higher dimensions are present (β1, β2, β3) but in these signals the corresponding barcodes do not change significantly .
 YUE YANG AND HUIJIE YANG. Complex network-based time series analysis. Physica A: Statistical Mechanics and its Applications, 387(5-6):1381–1386, February 2008.
 MERELLI, EMANUELA AND RUCCO, MATTEO AND PIANGERELLI, MARCO AND TOLLER, DANIELE. A topological approach for multivariate time series characterization: the epilepsy case study. Proc. 9th EAI Conference on Bio-inspired Information and Communications Technologies (BICT 2015), 2015.
In this work we propose a methodology based on Topogical Data Analysis (TDA) for capturing when a complex system, represented by a multivariate time series, changes its internal organization. The modification of the inner organization among the entities belonging to a complex system can induce a phase transition of the entire system. In order to identify these reorganizations, we designed a new methodology that is based on the representation of time series by simplicial complexes. The topologization of multivariate time series successfully pinpoints out when a complex system evolves. Simplicial complexes are characterized by persistent homology techniques, such as the clique weight rank persistent homology and the topological invariants are used for computing the so-called persistent entropy. With respect to the global invariants, e.g. the Betti numbers, the entropy takes into account also the topological noise and then it captures when a phase transition happens in a system. In order to verify the reliability of the methodology, we have analyzed the EEG signals of PhysioNet database and we have found numerical evidences that the methodology is able to detect the transition between the pre-ictal and ictal states. Complex dynamical systems arise as mathematical descriptions of natural phenomena. Studying the time evolution of such systems provides broad insight into problems. A wide spectrum of behaviors is seen, from straightforward limit cycles to chaotic behavior stemming from sensitivity to initial conditions. However, nature is affected by noise that often obfuscates the true behaviors. In order to capture the behavior, we intend to represent the system by simplicial complexes. The construction of simplicial complexes from real data is only partially affected by the noise, while the topological invariants (i.e., the Betti numbers) are preserved. The protocol we propose consists in the following steps:
The EEG signals used in this study were collected at the Children’s Hospital Boston, and they consists of EEG recordings from pediatric subjects with intractable seizures. Subjects were monitored for up to several days following withdrawal of anti-seizure medication in order to characterize their seizures and assess their candidacy for surgical intervention. Because of the limitation on the number of pages, we report only on two signals, respectively SIGI and SIGII . We applied the procedure described in  to both signals and we found the optimal size of the segmentation is equal to 120secs, then we segmented the whole EEG track in 30 windows. For each window we computed the partial correlation coefficients and we used as threshold q = 0. The upper triangular part of each matrices was parsed and saved as edge list, hence the edge list was used as input for jHoles. jHoles provides the Betti barcodes both in graphical and textual formats, and we used the latter for computing the weighted persistent entropy over each homological dimension (H0, H1, H2, and H3). We plotted the WHj values for each matrix.
Weighted Persistent Entropy for the homological group H0. Left: Weighted persistent entropy for the SIGI, the marked peak corresponds to an ictal state. Right: Weighted persistent entropy for the SIGII. In SIGI a phase transition is well evident.