FRG: Collaborative Research: Statistical Approaches to Topological Data Analysis that Address Questions in Complex Data

Montana State University, Bozeman MT

Investigators

Abstract

As both real and simulated data become increasingly complex due to improved instrumentation and deeper understanding of the underlying data-generating mechanisms, improved statistical methodology is required for proper analysis. Fields such as astronomy and biology that have spatial intricate, web-like data (e.g., the large-scale structure of the Universe, fibrin networks) can benefit from methodology that exploits the web-like information. The field of Topological Data Analysis (TDA) has great potential for the innovations needed to address these important and challenging scientific questions. This project will extend existing TDA algorithms, statistical theory and applications, and make the advancement easily accessible by incorporating the work into the freely available R package TDA. Moreover, the research will train undergraduate and graduate students in an interdisciplinary and collaborative environment. The goals of this project are (1) to extend existing algorithms in TDA to allow statistically rigorous inferences and improved visualization, (2) to develop the statistical theory necessary to apply hypothesis testing to sets of topological descriptors, (3) to develop justifiable algorithms for parameter selection, and (4) to apply these methods to complex data, especially to critical areas in astrophysics. These developments will make TDA more accessible to scientists and data analysts across disciplines and will give TDA a rigorous statistical foundation. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.