A statistical framework for the analysis of the evolution in shape and topological structure of random objects

用于分析随机物体形状和拓扑结构演化的统计框架

• 批准号: 2311338
• 负责人: Anne van Delft
• 金额: $ 32.96万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2311338&HistoricalAwards=false
• 关键词: statistical; framework; analysis; evolution; shape

Modern data sets often consist of sequential collections of point clouds that are samples from underlying objects with intrinsic geometry, such as curves, surfaces, or manifolds. Analyzing the dynamics of these time series of random objects requires qualitative inference methods that capture information on the geometric properties, i.e., the evolution of descriptors of the 'shape.' Analyzing shape is of paramount interest in many research areas such as genomics, climatology, neuroscience, and finance. In this project, we develop novel methodology and provide probabilistic and statistical foundations to model, analyze, and predict the evolution over time of geometric and topological features of data sets. The research will broaden the scope of the methodological interface between mathematics, computer science, statistics, and probability theory and will have direct applications to genomics and cell biology. We focus our theoretical work to support applications coming from two areas in genomics; cell differentiation in development and tumor evolution. This will be done in collaboration with the Herbert and Florence Irving Institute for cancer dynamics (IICD) at Columbia University. The research findings are also expected to influence model-building and data analysis techniques in geospatial data. Besides the theoretical contribution, we will provide software packages to make the inference methods available to a broad audience. The PIs further propose to design classes and produce expository notes from a cross-disciplinary perspective, and provide projects at the interface of mathematical statistics and topological data analysis for summer undergraduate mentoring.Over the past few decades, there has been substantial interest in the area of geometric data analysis known as topological data analysis (TDA); this provides qualitative multiscale shape descriptors for point clouds. However, in order to draw reliable qualitative inferences on shape and topological features, it is crucial to account for the (evolving) spatial and temporal dependence present in the data. To address these questions, we take the point of view that the fundamental datum is a function, i.e., the observations are points in a function space. This perspective integrates statistical methodology and TDA in the context of functional time series (FTS). We provide novel methodology to model, analyze and predict data generated from nonstationary metric space-valued stochastic processes. Our framework establishes the statistical and probabilistic foundations for applying multiscale geometric descriptors to meaningfully capture their evolving geometric features as well as the investigation of topological invariants. This new methodology will allow practitioners to perform statistical inference to address important scientific questions.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.

现代数据集通常由点云的顺序收集组成，这些集合是来自具有内在几何形状的基础对象的样本，例如曲线，表面或歧管。分析这些时间序列的随机对象的动力学需要定性推理方法，以捕获有关几何特性的信息，即“形状”的描述符的演变。在许多研究领域（例如基因组学，气候学，神经科学和金融）中，分析形状是至关重要的。在这个项目中，我们开发了新颖的方法，并提供概率和统计基础，以建模，分析和预测数据集的几何和拓扑特征的演变。该研究将扩大数学，计算机科学，统计和概率理论之间方法学接口的范围，并将直接应用于基因组学和细胞生物学。我们将理论工作集中在基因组学领域的两个领域的应用程序上；细胞分化发育和肿瘤进化。这将与哥伦比亚大学的赫伯特和佛罗伦萨欧文癌症动力学研究所（IICD）合作完成。研究结果还有望影响地理空间数据中的模型构建和数据分析技术。除了理论上的贡献外，我们还将提供软件包，以使广泛受众的推理方法可用。 PI进一步建议从跨学科的角度设计课程并制作说明性注释，并在夏季本科指导的数学统计数据和拓扑数据分析的界面上提供项目。在过去的几十年中，在过去的几十年中，对几何数据分析的领域一直具有重大兴趣，称为拓扑数据分析（TDA）；这为点云提供了定性的多尺寸形状描述符。但是，为了对形状和拓扑特征进行可靠的定性推断，重要的是说明数据中存在的（不断发展的）空间和时间依赖性。为了解决这些问题，我们认为基本数据是一个函数，即观察值是功能空间中的点。该观点在功能时间序列（FTS）的背景下集成了统计方法和TDA。我们提供了新的方法来建模，分析和预测从非组织度量空间价值的随机过程产生的数据。我们的框架建立了用于应用多尺寸几何描述符的统计和概率基础，以有意义地捕获其不断发展的几何特征以及对拓扑不变的研究。这种新方法将使从业人员能够进行统计推断以解决重要的科学问题。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被视为值得通过评估来获得支持。