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Topological Data Analysis of Biomedical Big Data
Published in Ervin Sejdić, Tiago H. Falk, Signal Processing and Machine Learning for Biomedical Big Data, 2018
Angkoon Phinyomark, Esther Ibáñez-Marcelo, Giovanni Petri
EEG signals are valuable indicators of neural activity widely used in medical settings (e.g., to diagnose epilepsy) and BCIs [81]. EEG signals are most often used to diagnose epilepsy. Researchers in the topology community then have selected EEG signals collected from epileptic brains as a first case study [24,37,82,83]. The purpose of this study case is to determine whether topological features of EEG can identify the preictal state, i.e., the state before the onset of an epileptic seizure, or the ictal state, as well as detect the phase transition between the two states. Merelli et al. [24] analyzed the EEG signals of the PhysioNet database3 by representing brain complex systems with multivariate time series and transforming signals into complex networks that are analyzed by TDA and information theory. Specifically, the initial input time-series data were divided into a series of temporal segments, and the Pearson (partial) correlation coefficient matrices were computed together with applying a threshold. The threshold matrices were used as weighted edgelists, and then simplicial complexes were characterized by persistent homology, i.e., their clique weight rank persistent homology [56], providing a new entropy measure called the weighted persistent entropy [51]. The transient preictal to ictal can be detected by observing the number of connected components, i.e., the number of connected ones tends to be 1 (all the features are persistent) during a phase transition, while this number is higher before and after the period. In support of the finding of Merelli et al. [24], Piangerelli et al. [37] show that the persistent entropy can be used to discriminate the epileptic state from nonepileptic states (p-value = 1.8346e−36; area under the curve = 0.972). The topological entropy features also achieved higher accuracy than the sample entropy features. Instead of computing persistent homology from complex networks, a piecewise linear function combined with a lower-star filtration was used. They also found that Vietoris–Rips filtration can help to improve the understanding of which region plays the role of trigger for an epileptic seizure (as an EEG channel selection method). Furthermore, Wang et al. [82] studied the persistent homology of EEG signals, smoothed by a weighted Fourier series estimator, using the persistence landscape. The 2-persistence landscape distance, which is the measure of difference between two persistence landscapes, was used as a feature to identify differences between preseizure and seizure patterns. The results show that the proposed topological features were able to identify the significant site T3 (p-value = 0.0005), where the patient’s epileptic seizure originates, among eight channels without any prior information (in an unsupervised learning way). We can use these topological features to understand the underlying transition mechanisms and the ictogenesis, which are necessary in developing an automatic system to predict the onset of epileptic seizures. It is of great practical interest to further investigate the potential of topological features for other medical and research uses of EEG such as to monitor the depth of anesthesia or Alzheimer’s disease developments.
Feature Detection and Hypothesis Testing for Extremely Noisy Nanoparticle Images using Topological Data Analysis
Published in Technometrics, 2023
Andrew M. Thomas, Peter A. Crozier, Yuchen Xu, David S. Matteson
As topological data analysis is such a young field, there is no shortage of directions along which the methods in this article could be expanded. For example, one could employ a functional version of ; alternatively, some another functional summary of persistence diagrams may hold promise, especially used in conjunction with a global rank envelope test as in Biscio and Møller (2019) and Myllymäki et al. (2017). Furthermore, using persistent homology for point clouds such as the Vietoris-Rips filtration (Boissonnat, Chazal, and Yvinec 2018), would use the location information derived from the above algorithm in an essential way. Using a weighted Vietoris-Rips filtration may even furnish more precise results (Anai et al. 2020), as the marked point process output of the algorithm naturally yields weights for each point. After deriving an additional persistence diagram from the output , one may choose an appropriate functional summary (see Berry et al. 2020) and proceed from there.