Topological Data Analysis: Developments and Applications

Motta, Francis C.

doi:10.1007/978-3-319-58895-7_19

Cited by 5 publications

(2 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Topological data analysis (TDA) has witnessed many important advances over the last twenty years that aim to unravel and provide insight to the "shape" of the data (Edelsbrunner et al, 2002;Edelsbrunner and Harer, 2008;Wasserman, 2018;Chazal and Michel, 2021). The development of TDA tools such as barcodes and persistence diagrams (Ghrist, 2008;Bubenik, 2015;Adams et al, 2017) have opened many new perspectives for analyzing various types of data (Umeda, 2017;Gholizadeh and Zadrozny, 2018;Motta, 2018;Xu et al, 2021;Leykam and Angelakis, 2023). These tools enable practitioners to grasp the topological characteristics inherent in high-dimensional data, which often remain beyond the reach of classical data analysis methods.…”

Section: Introductionmentioning

confidence: 99%

Statistical inference for dependence networks in topological data analysis

El-Yaagoubi,

Chung,

Ombao

2023

Front. Artif. Intell.

View full text Add to dashboard Cite

Topological data analysis (TDA) provide tools that are becoming increasingly popular for analyzing multivariate time series data. One key aspect in analyzing multivariate time series is dependence between components. One application is on brain signal analysis. In particular, various dependence patterns in brain networks may be linked to specific tasks and cognitive processes. These dependence patterns may be altered by various neurological and cognitive impairments such as Alzheimer's and Parkinson's diseases, as well as attention deficit hyperactivity disorder (ADHD). Because there is no ground-truth with known dependence patterns in real brain signals, testing new TDA methods on multivariate time series is still a challenge. Our goal here is to develop novel statistical inference procedures via simulations. Simulations are useful for generating some null distributions of a test statistic (for hypothesis testing), forming confidence regions, and for evaluating the performance of proposed TDA methods. To the best of our knowledge, there are no methods that simulate multivariate time series data with potentially complex user-specified connectivity patterns. In this paper we present a novel approach to simulate multivariate time series with specific number of cycles/holes in its dependence network. Furthermore, we also provide a procedure for generating higher dimensional topological features.

show abstract

Section: Introductionmentioning

confidence: 99%

Statistical inference for dependence networks in topological data analysis

El-Yaagoubi,

Chung,

Ombao

2023

Front. Artif. Intell.

View full text Add to dashboard Cite

show abstract

“…To identify translational and orientational orderings during homogeneous nucleation in MD simulations, an unsupervised learning approach based on topological data analysis (TDA) signatures, and more precisely persistent homology (PH) [5,6] was developed. PH is an intrinsically flexible, yet highly informative, tool which detects meaningful topological features deduced from atomic configurations.…”

mentioning

confidence: 99%

Unsupervised topological learning approach of crystal nucleation in pure Tantalum

Becker¹,

Devijver²,

Molinier³

et al. 2022

Preprint

View full text Add to dashboard Cite

Nucleation phenomena commonly observed in our every day life are of fundamental, technological and societal importance in many areas, but some of their most intimate mechanisms remain however to be unraveled. Crystal nucleation, the early stages where the liquid-to-solid transition occurs upon undercooling, initiates at the atomic level on nanometer length and sub-picoseconds time scales and involves complex multidimensional mechanisms with local symmetry breaking that can hardly be observed experimentally in the very details. To reveal their structural features in simulations without a priori, an unsupervised learning approach founded on topological descriptors loaned from persistent homology concepts is proposed. Applied here to a monatomic metal, namely Tantalum (Ta), it shows that both translational and orientational ordering always come into play simultaneously when homogeneous nucleation starts in regions with low five-fold symmetry.Understanding homogeneous crystal nucleation under deep undercooling conditions remains a formidable issue, as crystallization is essentially heterogeneous in nature and initiated from impurities, surfaces, or near grain boundaries that often hinder its occurrence [1]. Unreachable until very recently, experimental observations of early stages of nuclei was achieved by a tour de force using time tracking of three-dimensional (3D) Atomic Electron Tomography [2] of metallic nanoparticles. Those complex phenomena remain to date out-of-reach experimentally for bulk systems, thus hindering our theoretical understanding. This line of research still belongs mostly to the domain of atomic-level simulations and more particularly to molecular dynamics (MD) with generic interaction models [3,4]. To reach statistically meaningful events, large scale simulations are required 1 .

show abstract

Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence

et al. 2018

View full text Add to dashboard Cite

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical MHD, it is important to verify that such simulations are in agreement with observations. One of the main challenges in this area is to identify robust quantitative measures to compare structures found in simulations with those inferred from astrophysical observations. A similar challenge is to compare quantitatively results from different simulations. Topological data analysis offers a range of techniques, including the Betti numbers and persistence diagrams, that can be used to facilitate such a comparison. After describing these tools, we first apply them to synthetic random fields and demonstrate that, when the data are standardized in a straightforward manner, some topological measures are insensitive to either large-scale trends or the resolution of the data. Focusing upon one particular astrophysical example, we apply topological data analysis to H i observations of the turbulent interstellar medium (ISM) in the Milky Way and to recent MHD simulations of the random, strongly compressible ISM. We stress that these topological techniques are generic and could be applied to any complex, multi-dimensional random field.

show abstract

Topological Data Analysis: Developments and Applications

Cited by 5 publications

References 37 publications

Statistical inference for dependence networks in topological data analysis

Statistical inference for dependence networks in topological data analysis

Unsupervised topological learning approach of crystal nucleation in pure Tantalum

Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence

Contact Info

Product

Resources

About