🍩 Database of Original & Non-Theoretical Uses of Topology

(found 13 matches in 0.002508s)

  1. A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science (2023)

    Lander Ver Hoef, Henry Adams, Emily J. King, Imme Ebert-Uphoff
    Abstract Abstract Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code. Significance Statement Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.

  2. Persistent Homology for Breast Tumor Classification Using Mammogram Scans (2022)

    Aras Asaad, Dashti Ali, Taban Majeed, Rasber Rashid
    Abstract An Important tool in the field topological data analysis is known as persistent Homology (PH) which is used to encode abstract representation of the homology of data at different resolutions in the form of persistence diagram (PD). In this work we build more than one PD representation of a single image based on a landmark selection method, known as local binary patterns, that encode different types of local textures from images. We employed different PD vectorizations using persistence landscapes, persistence images, persistence binning (Betti Curve) and statistics. We tested the effectiveness of proposed landmark based PH on two publicly available breast abnormality detection datasets using mammogram scans. Sensitivity of landmark based PH obtained is over 90% in both datasets for the detection of abnormal breast scans. Finally, experimental results give new insights on using different types of PD vectorizations which help in utilising PH in conjunction with machine learning classifiers.

  3. Severe Slugging Flow Identification From Topological Indicators (2022)

    Simone Casolo
    Abstract In this work, topological data analysis is used to identify the onset of severe slug flow in offshore petroleum production systems. Severe slugging is a multiphase flow regime known to be very inefficient and potentially harmful to process equipment and it is characterized by large oscillations in the production fluid pressure. Time series from pressure sensors in subsea oil wells are processed by means of Takens embedding to produce point clouds of data. Embedded sensor data is then analyzed using persistent homology to obtain topological indicators capable of revealing the occurrence of severe slugging in a condition-based monitoring approach. A large dataset of well events consisting of both real and simulated data is used to demonstrate the possibilty of authomatizing severe slugging detection from live data via topological data analysis. Methods based on persistence diagrams are shown to accurately identify severe slugging and to classify different flow regimes from pressure signals of producing wells with supervised machine learning.

  5. Determining Structural Properties of Artificial Neural Networks Using Algebraic Topology (2021)

    David Pérez Fernández, Asier Gutiérrez-Fandiño, Jordi Armengol-Estapé, Marta Villegas
    Abstract Artificial Neural Networks (ANNs) are widely used for approximating complex functions. The process that is usually followed to define the most appropriate architecture for an ANN given a specific function is mostly empirical. Once this architecture has been defined, weights are usually optimized according to the error function. On the other hand, we observe that ANNs can be represented as graphs and their topological 'fingerprints' can be obtained using Persistent Homology (PH). In this paper, we describe a proposal focused on designing more principled architecture search procedures. To do this, different architectures for solving problems related to a heterogeneous set of datasets have been analyzed. The results of the evaluation corroborate that PH effectively characterizes the ANN invariants: when ANN density (layers and neurons) or sample feeding order is the only difference, PH topological invariants appear; in the opposite direction in different sub-problems (i.e. different labels), PH varies. This approach based on topological analysis helps towards the goal of designing more principled architecture search procedures and having a better understanding of ANNs.

  6. Topological Data Analysis of C. Elegans Locomotion and Behavior (2021)

    Ashleigh Thomas, Kathleen Bates, Alex Elchesen, Iryna Hartsock, Hang Lu, Peter Bubenik
    Abstract Video of nematodes/roundworms was analyzed using persistent homology to study locomotion and behavior. In each frame, an organism's body posture was represented by a high-dimensional vector. By concatenating points in fixed-duration segments of this time series, we created a sliding window embedding (sometimes called a time delay embedding) where each point corresponds to a sequence of postures of an organism. Persistent homology on the points in this time series detected behaviors and comparisons of these persistent homology computations detected variation in their corresponding behaviors. We used average persistence landscapes and machine learning techniques to study changes in locomotion and behavior in varying environments.

  7. TDAExplore: Quantitative Analysis of Fluorescence Microscopy Images Through Topology-Based Machine Learning (2021)

    Parker Edwards, Kristen Skruber, Nikola Milićević, James B. Heidings, Tracy-Ann Read, Peter Bubenik, Eric A. Vitriol
    Abstract Recent advances in machine learning have greatly enhanced automatic methods to extract information from fluorescence microscopy data. However, current machine-learning-based models can require hundreds to thousands of images to train, and the most readily accessible models classify images without describing which parts of an image contributed to classification. Here, we introduce TDAExplore, a machine learning image analysis pipeline based on topological data analysis. It can classify different types of cellular perturbations after training with only 20–30 high-resolution images and performs robustly on images from multiple subjects and microscopy modes. Using only images and whole-image labels for training, TDAExplore provides quantitative, spatial information, characterizing which image regions contribute to classification. Computational requirements to train TDAExplore models are modest and a standard PC can perform training with minimal user input. TDAExplore is therefore an accessible, powerful option for obtaining quantitative information about imaging data in a wide variety of applications.

  8. Stable Topological Summaries for Analyzing the Organization of Cells in a Packed Tissue (2021)

    Nieves Atienza, Maria-Jose Jimenez, Manuel Soriano-Trigueros
    Abstract We use topological data analysis tools for studying the inner organization of cells in segmented images of epithelial tissues. More specifically, for each segmented image, we compute different persistence barcodes, which codify the lifetime of homology classes (persistent homology) along different filtrations (increasing nested sequences of simplicial complexes) that are built from the regions representing the cells in the tissue. We use a complete and well-grounded set of numerical variables over those persistence barcodes, also known as topological summaries. A novel combination of normalization methods for both the set of input segmented images and the produced barcodes allows for the proven stability results for those variables with respect to small changes in the input, as well as invariance to image scale. Our study provides new insights to this problem, such as a possible novel indicator for the development of the drosophila wing disc tissue or the importance of centroids’ distribution to differentiate some tissues from their CVT-path counterpart (a mathematical model of epithelia based on Voronoi diagrams). We also show how the use of topological summaries may improve the classification accuracy of epithelial images using a Random Forest algorithm.

  9. Investigation of Flash Crash via Topological Data Analysis (2020)

    Wonse Kim, Younng-Jin Kim, Gihyun Lee, Woong Kook
    Abstract Topological data analysis has been acknowledged as one of the most successful mathematical data analytic methodologies in various fields including medicine, genetics, and image analysis. In this paper, we explore the potential of this methodology in finance by applying persistence landscape and dynamic time series analysis to analyze an extreme event in the stock market, known as Flash Crash. We will provide results of our empirical investigation to confirm the effectiveness of our new method not only for the characterization of this extreme event but also for its prediction purposes.

  12. Modelling Topological Features of Swarm Behaviour in Space and Time With Persistence Landscapes (2017)

    P. Corcoran, C. B. Jones
    Abstract This paper presents a model of swarm behavior that encodes the spatial-temporal characteristics of topological features, such as holes and connected components. Specifically, the persistence of topological features with respect to time is computed using zig-zag persistent homology. This information is in turn modelled as a persistence landscape, which forms a normed vector space and facilitates the application of statistical and data mining techniques. Validation of the proposed model is performed using a real data set corresponding to a swarm of fish. It is demonstrated that the proposed model may be used to perform retrieval and clustering of swarm behavior in terms of topological features. In fact, it is discovered that clustering returns clusters corresponding to the swarm behaviors of flock, torus, and disordered. These are the most frequently occurring types of behavior exhibited by swarms in general.

  13. Persistent Homology of Time-Dependent Functional Networks Constructed From Coupled Time Series (2017)

    Bernadette J. Stolz, Heather A. Harrington, Mason A. Porter
    Abstract We use topological data analysis to study “functional networks” that we construct from time-series data from both experimental and synthetic sources. We use persistent homology with a weight rank clique filtration to gain insights into these functional networks, and we use persistence landscapes to interpret our results. Our first example uses time-series output from networks of coupled Kuramoto oscillators. Our second example consists of biological data in the form of functional magnetic resonance imaging data that were acquired from human subjects during a simple motor-learning task in which subjects were monitored for three days during a five-day period. With these examples, we demonstrate that (1) using persistent homology to study functional networks provides fascinating insights into their properties and (2) the position of the features in a filtration can sometimes play a more vital role than persistence in the interpretation of topological features, even though conventionally the latter is used to distinguish between signal and noise. We find that persistent homology can detect differences in synchronization patterns in our data sets over time, giving insight both on changes in community structure in the networks and on increased synchronization between brain regions that form loops in a functional network during motor learning. For the motor-learning data, persistence landscapes also reveal that on average the majority of changes in the network loops take place on the second of the three days of the learning process.