Topological data analysis and diagnostics of compressible magnetohydrodynamic turbulence

The predictions of mean-field electrodynamics can now be probed using direct numerical simulations of random flows and magnetic fields. When modelling astrophysical magnetohydrodynamics, 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 iobservations of the turbulent interstellar medium (ISM) in the Milky Way and to recent magnetohydrodynamic 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.

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Using Topological Data Analysis (TDA) and Persistent Homology to Analyze the Stock Markets in Singapore and Taiwan

Frontiers in Physics ◽

10.3389/fphy.2021.572216 ◽

2021 ◽

Vol 9 ◽

Author(s):

Peter Tsung-Wen Yen ◽

Siew Ann Cheong

Keyword(s):

Data Analysis ◽

Stock Markets ◽

Persistent Homology ◽

Betti Numbers ◽

Stock Index ◽

Topological Data Analysis ◽

Series Data ◽

Topological Features ◽

Market Crashes ◽

Topological Data

In recent years, persistent homology (PH) and topological data analysis (TDA) have gained increasing attention in the fields of shape recognition, image analysis, data analysis, machine learning, computer vision, computational biology, brain functional networks, financial networks, haze detection, etc. In this article, we will focus on stock markets and demonstrate how TDA can be useful in this regard. We first explain signatures that can be detected using TDA, for three toy models of topological changes. We then showed how to go beyond network concepts like nodes (0-simplex) and links (1-simplex), and the standard minimal spanning tree or planar maximally filtered graph picture of the cross correlations in stock markets, to work with faces (2-simplex) or any k-dim simplex in TDA. By scanning through a full range of correlation thresholds in a procedure called filtration, we were able to examine robust topological features (i.e. less susceptible to random noise) in higher dimensions. To demonstrate the advantages of TDA, we collected time-series data from the Straits Times Index and Taiwan Capitalization Weighted Stock Index (TAIEX), and then computed barcodes, persistence diagrams, persistent entropy, the bottleneck distance, Betti numbers, and Euler characteristic. We found that during the periods of market crashes, the homology groups become less persistent as we vary the characteristic correlation. For both markets, we found consistent signatures associated with market crashes in the Betti numbers, Euler characteristics, and persistent entropy, in agreement with our theoretical expectations.

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Topological data analysis of zebrafish patterns

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.1917763117 ◽

2020 ◽

Vol 117 (10) ◽

pp. 5113-5124 ◽

Cited By ~ 4

Author(s):

Melissa R. McGuirl ◽

Alexandria Volkening ◽

Björn Sandstede

Keyword(s):

Data Analysis ◽

Pattern Formation ◽

Large Scale ◽

Model Organism ◽

Topological Data Analysis ◽

Wild Type ◽

Cell Dynamics ◽

Global Pattern ◽

Agent Based ◽

Topological Data

Self-organized pattern behavior is ubiquitous throughout nature, from fish schooling to collective cell dynamics during organism development. Qualitatively these patterns display impressive consistency, yet variability inevitably exists within pattern-forming systems on both microscopic and macroscopic scales. Quantifying variability and measuring pattern features can inform the underlying agent interactions and allow for predictive analyses. Nevertheless, current methods for analyzing patterns that arise from collective behavior capture only macroscopic features or rely on either manual inspection or smoothing algorithms that lose the underlying agent-based nature of the data. Here we introduce methods based on topological data analysis and interpretable machine learning for quantifying both agent-level features and global pattern attributes on a large scale. Because the zebrafish is a model organism for skin pattern formation, we focus specifically on analyzing its skin patterns as a means of illustrating our approach. Using a recent agent-based model, we simulate thousands of wild-type and mutant zebrafish patterns and apply our methodology to better understand pattern variability in zebrafish. Our methodology is able to quantify the differential impact of stochasticity in cell interactions on wild-type and mutant patterns, and we use our methods to predict stripe and spot statistics as a function of varying cellular communication. Our work provides an approach to automatically quantifying biological patterns and analyzing agent-based dynamics so that we can now answer critical questions in pattern formation at a much larger scale.

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Spectral and topological analysis of the cortical representation of the head position: does hypnotizability matter?

10.1101/442053 ◽

2018 ◽

Author(s):

Esther Ibanez-Marcelo ◽

Lisa Campioni ◽

Diego Manzoni ◽

Enrica L Santarcangelo ◽

Giovanni Petri

Keyword(s):

Spectral Analysis ◽

Data Analysis ◽

Large Scale ◽

Topological Analysis ◽

Head Position ◽

Topological Data Analysis ◽

Proprioceptive Information ◽

Head Positions ◽

First Time ◽

Topological Data

The aim of the study was to assess the EEG correlates of head positions, which have never been studied in humans, in participants with different psychophysiological characteristics, as encoded by their hypnotizability scores. This choice is motivated by earlier studies suggesting different processing of the vestibular/neck proprioceptive information in subjects with high (highs) and low (lows) hypnotizability scores maintaining their head rotated toward one side (RH). We analysed EEG signals recorded in 20 highs and 19 lows in basal conditions (head forward) and during RH, using spectral analysis, which captures changes localized to specific recording sites, and Topological Data Analysis (TDA), which instead describes large-scale differences in processing and representing sensorimotor information. Spectral analysis revealed significant differences related to the head position for alpha1, beta2, beta3, gamma bands, but not to hypnotizability. TDA instead revealed global hypnotizability-related differences in the strengths of the correlations among recording sites during RH. Significant changes were observed in lows on the left parieto-occipital side and in highs in right fronto-parietal region. Significant differences between the two groups were found in the occipital region, where changes were larger in lows than in highs. The study reports findings of the EEG correlates of the head posture for the first time, indicates that hypnotizability modulates its representation/processing on large-scale and that spectral and topological data analysis provide complementary results.

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Persistent Homology Analysis of RNA

Computational and Mathematical Biophysics ◽

10.1515/mlbmb-2016-0002 ◽

2016 ◽

Vol 4 (1) ◽

Cited By ~ 1

Author(s):

Adane L. Mamuye ◽

Matteo Rucco ◽

Luca Tesei ◽

Emanuela Merelli

Keyword(s):

Data Analysis ◽

Rna Folding ◽

Persistent Homology ◽

Betti Numbers ◽

Structural Features ◽

Topological Data Analysis ◽

Analysis Tool ◽

Global Features ◽

Folding Space ◽

Topological Data

AbstractTopological data analysis has been recently used to extract meaningful information frombiomolecules. Here we introduce the application of persistent homology, a topological data analysis tool, for computing persistent features (loops) of the RNA folding space. The scaffold of the RNA folding space is a complex graph from which the global features are extracted by completing the graph to a simplicial complex via the notion of clique and Vietoris-Rips complexes. The resulting simplicial complexes are characterised in terms of topological invariants, such as the number of holes in any dimension, i.e. Betti numbers. Our approach discovers persistent structural features, which are the set of smallest components to which the RNA folding space can be reduced. Thanks to this discovery, which in terms of data mining can be considered as a space dimension reduction, it is possible to extract a new insight that is crucial for understanding the mechanism of the RNA folding towards the optimal secondary structure. This structure is composed by the components discovered during the reduction step of the RNA folding space and is characterized by minimum free energy.

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tmap: an integrative framework based on topological data analysis for population-scale microbiome stratification and association studies

Genome Biology ◽

10.1186/s13059-019-1871-4 ◽

2019 ◽

Vol 20 (1) ◽

Cited By ~ 4

Author(s):

Tianhua Liao ◽

Yuchen Wei ◽

Mingjing Luo ◽

Guo-Ping Zhao ◽

Haokui Zhou

Keyword(s):

Data Analysis ◽

Large Scale ◽

Association Studies ◽

Topological Data Analysis ◽

Association Patterns ◽

Integrative Framework ◽

Environmental Features ◽

Analytic Methods ◽

Population Scale ◽

Topological Data

AbstractUntangling the complex variations of microbiome associated with large-scale host phenotypes or environment types challenges the currently available analytic methods. Here, we present tmap, an integrative framework based on topological data analysis for population-scale microbiome stratification and association studies. The performance of tmap in detecting nonlinear patterns is validated by different scenarios of simulation, which clearly demonstrate its superiority over the most commonly used methods. Application of tmap to several population-scale microbiomes extensively demonstrates its strength in revealing microbiome-associated host or environmental features and in understanding the systematic interrelations among their association patterns. tmap is available at https://github.com/GPZ-Bioinfo/tmap.

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Euler characteristic surfaces

Foundations of Data Science ◽

10.3934/fods.2021027 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Gabriele Beltramo ◽

Primoz Skraba ◽

Rayna Andreeva ◽

Rik Sarkar ◽

Ylenia Giarratano ◽

...

Keyword(s):

Data Analysis ◽

Euler Characteristic ◽

Random Fields ◽

Topological Data Analysis ◽

Retinal Images ◽

Dimensional Parameter ◽

Parameter Spaces ◽

Higher Dimensional ◽

The Subject ◽

Topological Data

<p style='text-indent:20px;'>In this paper, we investigate the use of the Euler characteristic for the topological data analysis, particularly over higher dimensional parameter spaces. The Euler characteristic is a classical, well-understood topological invariant that has appeared in numerous applications, primarily in the context of random fields. The goal of this paper, is to present the extension of using the Euler characteristic in higher dimensional parameter spaces. The topological data analysis of higher dimensional parameter spaces using stronger invariants such as homology, has and continues to be the subject of intense research. However, as important theoretical and computational obstacles remain, the use of the Euler characteristic represents an important intermediary step toward multi-parameter topological data analysis. We show the usefulness of the techniques using generated examples as well as a real world dataset of detecting diabetic retinopathy in retinal images.</p>

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Identification of Relevant Genetic Alterations in Cancer using Topological Data Analysis

10.1101/2020.01.30.922310 ◽

2020 ◽

Author(s):

Raúl Rabadán ◽

Yamina Mohamedi ◽

Udi Rubin ◽

Tim Chu ◽

Oliver Elliott ◽

...

Keyword(s):

Data Analysis ◽

Large Scale ◽

Molecular Mechanisms ◽

Fold Increase ◽

Suppressor Gene ◽

Genetic Alterations ◽

Topological Data Analysis ◽

Integrated Analysis ◽

Cancer Genes ◽

Topological Data

AbstractLarge-scale cancer genomic studies enable the systematic identification of mutations that lead to the genesis and progression of tumors, uncovering the underlying molecular mechanisms and potential therapies. While some such mutations are recurrently found in many tumors, many others exist solely within a few samples, precluding detection by conventional recurrence-based statistical approaches. Integrated analysis of somatic mutations and RNA expression data across 12 tumor types reveals that mutations of cancer genes are usually accompanied by substantial changes in expression. We use topological data analysis to leverage this observation and uncover 38 elusive candidate cancer-associated genes, including inactivating mutations of the metalloproteinase ADAMTS12 in lung adenocarcinoma. We show that ADAMTS12−/− mice have a five-fold increase in the susceptibility to develop lung tumors, confirming the role of ADAMTS12 as a tumor suppressor gene. Our results demonstrate that data integration through topological techniques can increase our ability to identify previously unreported cancer-related alterations.

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