An introduction to persistent homology for time series

Равишанкер, Налини; Chen, Renjie

doi:10.1002/wics.1548

Cited by 8 publications

(3 citation statements)

References 59 publications

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“…Although the proposed state space is approximate, it captures network features across multiple scales and may provide insight into the full, latent mitochondrial state space. Persistent homology can also be extended to study the evolution of networks over time, enabling the detection of node movement along a tubule without changes to the underlying network structure [129,132]. Although persistent homology is not commonly used in the analysis of biological datasets, there are examples of its success in the neuroscience and protein structure literatures, as well as introductions to its practical usage and implementations in multiple programming languages.…”

Section: Integration Of Physical Properties Into Mitochondrial Networ...mentioning

confidence: 99%

Mitochondrial networks through the lens of mathematics

Lewis

Marshall

2023

Phys. Biol.

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Mitochondria serve a wide range of functions within cells, most notably via their production of ATP. Although their morphology is commonly described as bean-like, mitochondria often form interconnected networks within cells that exhibit dynamic restructuring through a variety of physical changes. Further, though relationships between form and function in biology are well established, the extant toolkit for understanding mitochondrial morphology is limited. Here, we emphasize new and established methods for quantitatively describing mitochondrial networks, ranging from unweighted graph-theoretic representations to multi-scale approaches from applied topology, in particular persistent homology. We also show fundamental relationships between mitochondrial networks, mathematics, and physics, using ideas of graph planarity and statistical mechanics to better understand the full possible morphologicalspace of mitochondrial network structures. Lastly, we provide suggestions for how examination of mitochondrial network form through the language of mathematics can inform biological understanding, and vice versa.

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Section: Integration Of Physical Properties Into Mitochondrial Networ...mentioning

confidence: 99%

Mitochondrial networks through the lens of mathematics

Lewis

Marshall

2023

Phys. Biol.

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“…regression analysis), we refer the reader to the paper [25], which is devoted to that topic. There are a variety of papers on TDA applied to time-series data, including [26], [18], and [22]; see [21] and [23] for nice surveys of some current ideas. The questions about hypothesis testing which motivate the present paper are not taken up in those works, however.…”

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confidence: 99%

Statistical inference for persistent homology applied to simulated fMRI time series data

Abdallah¹,

Regalski²,

Kang³

et al. 2023

FoDS

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<p style='text-indent:20px;'>Time-series data are amongst the most widely-used in biomedical sciences, including domains such as functional Magnetic Resonance Imaging (fMRI). Structure within time series data can be captured by the tools of topological data analysis (TDA). Persistent homology is the mostly commonly used data-analytic tool in TDA, and can effectively summarize complex high-dimensional data into an interpretable 2-dimensional representation called a <i>persistence diagram</i>. Existing methods for statistical inference for persistent homology of data depend on an independence assumption being satisfied. While persistent homology can be computed for each time index in a time-series, time-series data often fail to satisfy the independence assumption. This paper develops a statistical test that obviates the independence assumption by implementing a multi-level block sampled Monte Carlo test with sets of persistence diagrams. Its efficacy for detecting task-dependent topological organization is then demonstrated on simulated fMRI data. This new statistical test is therefore suitable for analyzing persistent homology of fMRI data, and of non-independent data in general.</p>

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“…Also, it would be interesting to extend the method to learn the time evolution, e.g. through recent generalization of the persistent homology to time series [25].…”

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confidence: 99%

Unsupervised topological learning approach of crystal nucleation in pure Tantalum

Becker¹,

Devijver²,

Molinier³

et al. 2022

Preprint

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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 .

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An introduction to persistent homology for time series

Cited by 8 publications

References 59 publications

Mitochondrial networks through the lens of mathematics

Mitochondrial networks through the lens of mathematics

Statistical inference for persistent homology applied to simulated fMRI time series data

Unsupervised topological learning approach of crystal nucleation in pure Tantalum

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