scholarly journals Topological Data Analysis of Collective and Individual Epithelial Cells using Persistent Homology of Loops

Soft Matter ◽  
2021 ◽  
Author(s):  
Dhananjay Bhaskar ◽  
William Y Zhang ◽  
Ian Y Wong
Keyword(s):  
Phase Transitions ◽  
Data Analysis ◽  
Epithelial Cells ◽  
Persistent Homology ◽  
A Priori ◽  
Topological Data Analysis ◽  
A Priori Information ◽  
Priori Information ◽  
Topological Data

Interacting, self-propelled particles such as epithelial cells can dynamically self-organize into complex multicellular patterns, which are challenging to classify without \emph{a priori} information. Classically, different phases and phase transitions have...

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.

scholarly journals Persistent homology approach distinguishes potential pattern between “Early” and “Not Early” hepatic decompensation groups using MRI modalities

2021 ◽  
Vol 7 (2) ◽  
pp. 488-491
Author(s):  
Yashbir Singh ◽  
William Jons ◽  
Gian Marco Conte ◽  
Jaidip Jagtap ◽  
Kuan Zhang ◽  
...  
Keyword(s):  
Data Analysis ◽  
Liver Failure ◽  
Algebraic Topology ◽  
Persistent Homology ◽  
Topological Data Analysis ◽  
Topological Representation ◽  
Hepatic Decompensation ◽  
Topological Data

Abstract Primary sclerosis cholangitis (PSC) predisposes individuals to liver failure, but it is challenging for radiologists examining radiologic images to predict which patients with PSC will ultimately develop liver failure. Motivated by algebraic topology, a topological data analysis - inspired framework was adopted in the study of the imaging pattern between the “Early Decompensation” and “Not Early” groups. The results demonstrate that the proposed methodology discriminates “Early Decompensation” and “Not Early” groups. Our study is the first attempt to provide a topological representation-based method into early hepatic decompensation and not early groups.

scholarly journals Revealing hidden medium-range order in amorphous materials using topological data analysis

2020 ◽  
Vol 6 (37) ◽  
pp. eabc2320
Author(s):  
Søren S. Sørensen ◽  
Christophe A. N. Biscio ◽  
Mathieu Bauchy ◽  
Lisbeth Fajstrup ◽  
Morten M. Smedskjaer
Keyword(s):  
Data Analysis ◽  
Amorphous Materials ◽  
Persistent Homology ◽  
Amorphous Solids ◽  
Range Order ◽  
Silicate Glasses ◽  
Topological Data Analysis ◽  
Medium Range Order ◽  
Medium Range ◽  
Topological Data

Despite the numerous technological applications of amorphous materials, such as glasses, the understanding of their medium-range order (MRO) structure—and particularly the origin of the first sharp diffraction peak (FSDP) in the structure factor—remains elusive. Here, we use persistent homology, an emergent type of topological data analysis, to understand MRO structure in sodium silicate glasses. To enable this analysis, we introduce a self-consistent categorization of rings with rigorous geometrical definitions of the structural entities. Furthermore, we enable quantitative comparison of the persistence diagrams by computing the cumulative sum of all points weighted by their lifetime. On the basis of these analysis methods, we show that the approach can be used to deconvolute the contributions of various MRO features to the FSDP. More generally, the developed methodology can be applied to analyze and categorize molecular dynamics data and understand MRO structure in any class of amorphous solids.

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.

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