scholarly journals From mathematics to medicine: A practical primer on topological data analysis (TDA) and the development of related analytic tools for the functional discovery of latent structure in fMRI data

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255859
Author(s):  
Andrew Salch ◽  
Adam Regalski ◽  
Hassan Abdallah ◽  
Raviteja Suryadevara ◽  
Michael J. Catanzaro ◽  
...  
Keyword(s):  
Collaborative Research ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Fmri Data ◽  
Data Sets ◽  
Software Pipeline ◽  
Spatially Distributed ◽  
Mathematical Techniques ◽  
Topological Data

fMRI is the preeminent method for collecting signals from the human brain in vivo, for using these signals in the service of functional discovery, and relating these discoveries to anatomical structure. Numerous computational and mathematical techniques have been deployed to extract information from the fMRI signal. Yet, the application of Topological Data Analyses (TDA) remain limited to certain sub-areas such as connectomics (that is, with summarized versions of fMRI data). While connectomics is a natural and important area of application of TDA, applications of TDA in the service of extracting structure from the (non-summarized) fMRI data itself are heretofore nonexistent. “Structure” within fMRI data is determined by dynamic fluctuations in spatially distributed signals over time, and TDA is well positioned to help researchers better characterize mass dynamics of the signal by rigorously capturing shape within it. To accurately motivate this idea, we a) survey an established method in TDA (“persistent homology”) to reveal and describe how complex structures can be extracted from data sets generally, and b) describe how persistent homology can be applied specifically to fMRI data. We provide explanations for some of the mathematical underpinnings of TDA (with expository figures), building ideas in the following sequence: a) fMRI researchers can and should use TDA to extract structure from their data; b) this extraction serves an important role in the endeavor of functional discovery, and c) TDA approaches can complement other established approaches toward fMRI analyses (for which we provide examples). We also provide detailed applications of TDA to fMRI data collected using established paradigms, and offer our software pipeline for readers interested in emulating our methods. This working overview is both an inter-disciplinary synthesis of ideas (to draw researchers in TDA and fMRI toward each other) and a detailed description of methods that can motivate collaborative research.

Topological Data Analysis with Applications

2021 ◽  
Author(s):  
Gunnar Carlsson ◽  
Mikael Vejdemo-Johansson
Keyword(s):  
Data Analysis ◽  
Data Science ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Data Sets ◽  
Advanced Learners ◽  
Computer Scientists ◽  
Dramatic Rise ◽  
Modeling Data ◽  
Topological Data

The continued and dramatic rise in the size of data sets has meant that new methods are required to model and analyze them. This timely account introduces topological data analysis (TDA), a method for modeling data by geometric objects, namely graphs and their higher-dimensional versions: simplicial complexes. The authors outline the necessary background material on topology and data philosophy for newcomers, while more complex concepts are highlighted for advanced learners. The book covers all the main TDA techniques, including persistent homology, cohomology, and Mapper. The final section focuses on the diverse applications of TDA, examining a number of case studies drawn from monitoring the progression of infectious diseases to the study of motion capture data. Mathematicians moving into data science, as well as data scientists or computer scientists seeking to understand this new area, will appreciate this self-contained resource which explains the underlying technology and how it can be used.

scholarly journals Feasibility of topological data analysis for event-related fMRI

2019 ◽  
Vol 3 (3) ◽  
pp. 695-706 ◽  
Author(s):  
Cameron T. Ellis ◽  
Michael Lesnick ◽  
Gregory Henselman-Petrusek ◽  
Bryn Keller ◽  
Jonathan D. Cohen
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Time Frame ◽  
Topological Data Analysis ◽  
Fmri Data ◽  
Cognitive Representations ◽  
New Approach ◽  
Neural Data ◽  
Topological Features ◽  
Topological Data

Recent fMRI research shows that perceptual and cognitive representations are instantiated in high-dimensional multivoxel patterns in the brain. However, the methods for detecting these representations are limited. Topological data analysis (TDA) is a new approach, based on the mathematical field of topology, that can detect unique types of geometric features in patterns of data. Several recent studies have successfully applied TDA to study various forms of neural data; however, to our knowledge, TDA has not been successfully applied to data from event-related fMRI designs. Event-related fMRI is very common but limited in terms of the number of events that can be run within a practical time frame and the effect size that can be expected. Here, we investigate whether persistent homology—a popular TDA tool that identifies topological features in data and quantifies their robustness—can identify known signals given these constraints. We use fmrisim, a Python-based simulator of realistic fMRI data, to assess the plausibility of recovering a simple topological representation under a variety of conditions. Our results suggest that persistent homology can be used under certain circumstances to recover topological structure embedded in realistic fMRI data simulations.

scholarly journals Feasibility of Topological Data Analysis for event-related fMRI

2018 ◽  
Author(s):  
Cameron T. Ellis ◽  
Michael Lesnick ◽  
Gregory Henselman-Petrusek ◽  
Bryn Keller ◽  
Jonathan D. Cohen
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Time Frame ◽  
Topological Data Analysis ◽  
Fmri Data ◽  
Cognitive Representations ◽  
New Approach ◽  
Neural Data ◽  
Topological Features ◽  
Topological Data

AbstractRecent fMRI research shows that perceptual and cognitive representations are instantiated in high-dimensional multi-voxel patterns in the brain. However, the methods for detecting these representations are limited. Topological Data Analysis (TDA) is a new approach, based on the mathematical field of topology, that can detect unique types of geometric features in patterns of data. Several recent studies have successfully applied TDA to study various forms of neural data; however, to our knowledge, TDA has not been successfully applied to data from event-related fMRI designs. Event-related fMRI is very common but limited in terms of the number of events that can be run within a practical time frame and the effect size that can be expected. Here, we investigate whether persistent homology — a popular TDA tool that identifies topological features in data and quantifies their robustness — can identify known signals given these constraints. We use fmrisim, a Python-based simulator of realistic fMRI data, to assess the plausibility of recovering a simple topological representation under a variety of conditions. Our results suggest that persistent homology can be used under certain circumstances to recover topological structure embedded in realistic fMRI data simulations.

scholarly journals Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks

2021 ◽  
Vol 83 (3) ◽  
Author(s):  
Maria-Veronica Ciocanel ◽  
Riley Juenemann ◽  
Adriana T. Dawes ◽  
Scott A. McKinley
Keyword(s):  
Time Series ◽  
Data Analysis ◽  
Actin Filaments ◽  
Time Series Data ◽  
Polymer Networks ◽  
Topological Data Analysis ◽  
Series Data ◽  
Topological Features ◽  
Topological Data

AbstractIn developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

scholarly journals Classification of apatite structures via topological data analysis: a framework for a ‘Materials Barcode’ representation of structure maps

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Scott Broderick ◽  
Ruhil Dongol ◽  
Tianmu Zhang ◽  
Krishna Rajan
Keyword(s):  
Machine Learning ◽  
Data Analysis ◽  
Crystal Chemistry ◽  
Persistent Homology ◽  
Hierarchical Classification ◽  
Topological Data Analysis ◽  
Learning Tool ◽  
Coordination Polyhedra ◽  
Machine Learning Tool ◽  
Topological Data

AbstractThis paper introduces the use of topological data analysis (TDA) as an unsupervised machine learning tool to uncover classification criteria in complex inorganic crystal chemistries. Using the apatite chemistry as a template, we track through the use of persistent homology the topological connectivity of input crystal chemistry descriptors on defining similarity between different stoichiometries of apatites. It is shown that TDA automatically identifies a hierarchical classification scheme within apatites based on the commonality of the number of discrete coordination polyhedra that constitute the structural building units common among the compounds. This information is presented in the form of a visualization scheme of a barcode of homology classifications, where the persistence of similarity between compounds is tracked. Unlike traditional perspectives of structure maps, this new “Materials Barcode” schema serves as an automated exploratory machine learning tool that can uncover structural associations from crystal chemistry databases, as well as to achieve a more nuanced insight into what defines similarity among homologous compounds.

scholarly journals Topological data analysis for true step detection in periodic piecewise constant signals

2018 ◽  
Vol 474 (2218) ◽  
pp. 20180027 ◽  
Author(s):  
Firas A. Khasawneh ◽  
Elizabeth Munch
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Future Research ◽  
Accurate Identification ◽  
Piecewise Constant ◽  
Higher Dimensional ◽  
Powerful Approach ◽  
Hadamard Transforms ◽  
Topological Data

This paper introduces a simple yet powerful approach based on topological data analysis for detecting true steps in a periodic, piecewise constant (PWC) signal. The signal is a two-state square wave with randomly varying in-between-pulse spacing, subject to spurious steps at the rising or falling edges which we call digital ringing. We use persistent homology to derive mathematical guarantees for the resulting change detection which enables accurate identification and counting of the true pulses. The approach is tested using both synthetic and experimental data obtained using an engine lathe instrumented with a laser tachometer. The described algorithm enables accurate and automatic calculations of the spindle speed without any choice of parameters. The results are compared with the frequency and sequency methods of the Fourier and Walsh–Hadamard transforms, respectively. Both our approach and the Fourier analysis yield comparable results for pulses with regular spacing and digital ringing while the latter causes large errors using the Walsh–Hadamard method. Further, the described approach significantly outperforms the frequency/sequency analyses when the spacing between the peaks is varied. We discuss generalizing the approach to higher dimensional PWC signals, although using this extension remains an interesting question for future research.

scholarly journals Topological Data Analysis as a Morphometric Method: Using Persistent Homology to Demarcate a Leaf Morphospace

2018 ◽  
Vol 9 ◽  
Author(s):  
Mao Li ◽  
Hong An ◽  
Ruthie Angelovici ◽  
Clement Bagaza ◽  
Albert Batushansky ◽  
...  
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Morphometric Method ◽  
Topological Data

scholarly journals Flow Estimation Only from Image Data, based on Persistent Homology

2021 ◽  
Author(s):  
Anna Suzuki ◽  
Miyuki Miyazawa ◽  
James Minto ◽  
Takeshi Tsuji ◽  
Ippei Obayashi ◽  
...  
Keyword(s):  
Data Analysis ◽  
Persistent Homology ◽  
Image Data ◽  
Fracture Network ◽  
Topological Data Analysis ◽  
Geometric Information ◽  
Flow Estimation ◽  
Flow Phenomena ◽  
Network Patterns ◽  
Topological Data

Abstract Topological data analysis is an emerging concept of data analysis for characterizing shapes. A state-of-the-art tool in topological data analysis is persistent homology, which is expected to summarize quantified topological and geometric features. Although persistent homology is useful for revealing the topological and geometric information, it is difficult to interpret the parameters of persistent homology themselves and difficult to directly relate the parameters to physical properties. In this study, we focus on connectivity and apertures of flow channels detected from persistent homology analysis. We propose a method to estimate permeability in fracture networks from parameters of persistent homology. Synthetic 3D fracture network patterns and their direct flow simulations are used for the validation. The results suggest that the persistent homology can estimate fluid flow in fracture network based on the image data. This method can easily derive the flow phenomena based on the information of the structure.

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

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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