🍩 Database of Original & Non-Theoretical Uses of Topology

(found 5 matches in 0.001406s)
  1. Extremal Event Graphs: A (Stable) Tool for Analyzing Noisy Time Series Data (2022)

    Robin Belton, Bree Cummins, Brittany Terese Fasy, Tomáš Gedeon
    Abstract Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise \$L_\\infty\\$ distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.
  2. Persistent Homology Analysis of Brain Transcriptome Data in Autism (2019)

    Daniel Shnier, Mircea A. Voineagu, Irina Voineagu
    Abstract Persistent homology methods have found applications in the analysis of multiple types of biological data, particularly imaging data or data with a spatial and/or temporal component. However, few studies have assessed the use of persistent homology for the analysis of gene expression data. Here we apply persistent homology methods to investigate the global properties of gene expression in post-mortem brain tissue (cerebral cortex) of individuals with autism spectrum disorders (ASD) and matched controls. We observe a significant difference in the geometry of inter-sample relationships between autism and healthy controls as measured by the sum of the death times of zero-dimensional components and the Euler characteristic. This observation is replicated across two distinct datasets, and we interpret it as evidence for an increased heterogeneity of gene expression in autism. We also assessed the topology of gene-level point clouds and did not observe significant differences between ASD and control transcriptomes, suggesting that the overall transcriptome organization is similar in ASD and healthy cerebral cortex. Overall, our study provides a novel framework for persistent homology analyses of gene expression data for genetically complex disorders.
  3. Gene Coexpression Network Comparison via Persistent Homology (2018)

    Ali Nabi Duman, Harun Pirim
    Abstract Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies.
  4. Topological Descriptors of Histology Images (2014)

    Nikhil Singh, Heather D. Couture, J. S. Marron, Charles Perou, Marc Niethammer
    Abstract The purpose of this study is to investigate architectural characteristics of cell arrangements in breast cancer histology images. We propose the use of topological data analysis to summarize the geometric information inherent in tumor cell arrangements. Our goal is to use this information as signatures that encode robust summaries of cell arrangements in tumor tissue as captured through histology images. In particular, using ideas from algebraic topology we construct topological descriptors based on cell nucleus segmentations such as persistency charts and Betti sequences. We assess their performance on the task of discriminating the breast cancer subtypes Basal, Luminal A, Luminal B and HER2. We demonstrate that the topological features contain useful complementary information to image-appearance based features that can improve discriminatory performance of classifiers.