Jessi Cisewski

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  Analyzing Data Full of Holes with Topological Data Analysis
  Jessi Cisewski
  Data exhibiting complicated spatial structures are common in many areas of science (e.g., cosmology, biology), but can be difficult to analyze. Persistent homology is an approach within the area of Topological Data Analysis (TDA) that offers a framework to represent, visualize, and interpret complex data by extracting topological features which may be used to infer properties of the underlying structures. For example, TDA is a beneficial technique for analyzing intricate and spatially complex web-like data such as fibrin or the large-scale structure (LSS) of the Universe. The output from persistent homology, called persistence diagrams, summarizes the different order holes in the data (e.g., connected components, loops, voids). I will present a framework for inference or prediction using functional transformations of persistence diagrams and discuss how persistent homology can be used to locate cosmological voids and filament loops in the LSS of the Universe.
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  Hypothesis tests for complicated spatial structures using persistent homology
  Jessi Cisewski
  Complicated spatial structures (CSS) are common in biological data (e.g. fibrin clots, fibroblasts), but are difficult to quantitatively analyze without losing important information. Topological data analysis (TDA) provides a way for biologists to better understand, visualize, and interpret such data. TDA is a statistical framework for extracting topological information from data and using it to estimate properties of the underlying structures. It has potential to dramatically improve the analysis of biological data by retrieving and quantifying crucial information that is missed in ad-hoc methods by specifically targeting shape-related features.
  We present a framework for hypothesis testing of CSS using persistent homology. The randomness in the data (due to measurement error or topological noise) is transferred to randomness in the topological summaries, which provides an infrastructure for inference. These tests allow for statistical comparisons between CSS. We present several possible test statistics using persistence diagrams and carryout a simulation study to investigate the suitableness of the proposed test statistics.