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

由于仪器的改进和对基本数据生成机制的深入理解，真实的和模拟数据都变得越来越复杂，因此需要改进统计方法以进行适当的分析。天文学和生物学等领域具有空间复杂的网络状数据（例如，宇宙的大规模结构，纤维蛋白网络）可以从利用网络状信息的方法中受益。 拓扑数据分析（TDA）领域在解决这些重要且具有挑战性的科学问题所需的创新方面具有巨大的潜力。 该项目将扩展现有的TDA算法，统计理论和应用，并通过将工作纳入免费提供的R软件包TDA，使进步更容易获得。 此外，本研究将在跨学科和协作的环境中培养本科生和研究生。本项目的目标是（1）扩展TDA中现有的算法，以允许统计上严格的推理和改进的可视化，（2）发展必要的统计理论，将假设检验应用于拓扑描述符集，（3）发展合理的参数选择算法，（4）将这些方法应用于复杂数据，特别是天体物理学中的关键领域。 这些发展将使跨学科的科学家和数据分析师更容易获得TDA，并将为TDA提供严格的统计基础。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.