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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
In previous sections, we saw that persistent homology describes the multiscale properties of a data set by capturing the birth and death times of topological features. These properties are summarized by the persistence diagrams, multisets of points in the ℝ2. For such diagrams to be relevant and interpretable for data analysis and applications, they must be robust: small perturbations of the data should correspond to small alterations in the corresponding persistence diagrams. Fortunately, the stability of persistence diagrams is one of the key features of the study of persistent homology and has been shown in a number of different works.
Persistent homology of images
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Persistence diagramThe k-intervals introduced in the previous section are defined in the literature as the persistence barcode. An alternative method for representing persistent vector spaces is given by the persistence diagram which is a two-dimensional scatter plot where each point corresponds to a persistent homology class3 (Fig. 2.6ii). The coordinates are filtration indexes of birth and death of the related homology class. To formalize the concept, a definition is given for the persistence module that was mentioned briefly in section 2.26.
PERCEPT: A New Online Change-Point Detection Method using Topological Data Analysis
Published in Technometrics, 2023
Xiaojun Zheng, Simon Mak, Liyan Xie, Yao Xie
A primary tool in TDA is persistent homology, which extracts topological features (e.g., connected components, holes, and their higher-dimensional analogues) from point cloud data. In what follows, we provide a brief overview of persistent homology, following Ghrist (2008) and Edelsbrunner and Harer (2008). For a given point cloud dataset, persistent homology provides a representation of this as a simplicial complex, defined as a set of vertices, edges, triangles, and their higher-dimensional counterparts. A common simplicial complex is the Rips complex, which depends on a single scale parameter ϵ. At a given radius , the Rips complex contains all edges between any two points whose distance is at most ϵ, and contains triangular faces for any three points whose pairwise distances are at most ϵ. Figure 1 (adapted from Han et al. 2018) illustrates this for a toy dataset. Clearly, a single ϵ cannot capture all geometric structures embedded in the data. Thus, a sequence of scale parameters is used to build a sequence (or filtration) of simplicial complexes. This filtration provides a means for extracting topological structure from the data: zero-dimensional holes (or connected components), one-dimensional holes, and their higher-dimensional analogues.
Topological machine learning for multivariate time series
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Chengyuan Wu, Carol Anne Hargreaves
Though the term ‘topology’ can be used to refer to a wide array of subjects, the topological tools used in TDA generally refer to algebraic topology (Letscher, 2012; Wu et al., 2020b), or to be specific persistent homology (Edelsbrunner & Morozov, 2012; Ghrist, 2008). Broadly speaking, persistent homology analyzes the ‘shape’ of the data to deduce the intrinsic properties of the data. Other prominent tools in TDA include Mapper (Ray & Trovati, 2017; Singh et al., 2007) and discrete Morse theory (Forman, 1998, 2002; Wu et al., 2020a). Due to the fact that TDA works quite differently from most other data analysis techniques, it can sometimes detect features that are missed by traditional methods of analysis (Nicolau et al., 2011).