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

现代数据集通常由点云的连续集合组成，这些点云是具有固有几何图形的底层对象的样本，例如曲线、曲面或流形。分析这些随机物体的时间序列的动力学需要定性推理方法，这些方法捕捉有关几何特性的信息，也就是说，形状描述符的演变。形状分析在基因组学、气候学、神经科学和金融学等许多研究领域都是非常重要的。在这个项目中，我们开发了新的方法，并提供了概率和统计基础来建模、分析和预测数据集的几何和拓扑特征随时间的演变。这项研究将拓宽数学、计算机科学、统计学和概率论之间的方法论接口的范围，并将直接应用于基因组学和细胞生物学。我们的理论工作重点是支持来自基因组学两个领域的应用：发育中的细胞分化和肿瘤进化。这项研究将与哥伦比亚大学赫伯特和弗洛伦斯·欧文癌症动力学研究所(IICD)合作完成。预计研究结果还将影响地理空间数据的建模和数据分析技术。除了理论上的贡献，我们还将提供软件包，让广大受众能够使用推理方法。PIS还建议从跨学科的角度设计课程和制作说明性笔记，并为夏季本科生辅导提供数理统计和拓扑数据分析的界面项目。在过去的几十年里，人们对称为拓扑数据分析(TDA)的几何数据分析领域非常感兴趣，这为点云提供了定性的多尺度形状描述符。然而，为了对形状和拓扑特征做出可靠的定性推断，关键是要考虑到数据中存在的(不断演变的)空间和时间相关性。为了解决这些问题，我们认为基本基准是一个函数，即观测值是函数空间中的点。这一视角在功能时间序列(FTS)的背景下整合了统计方法论和TDA。我们提供了新的方法来对非平稳度量空间值随机过程产生的数据进行建模、分析和预测。我们的框架为应用多尺度几何描述子有意义地捕捉其演化的几何特征以及拓扑不变量的研究奠定了统计和概率基础。这一新的方法将允许从业者进行统计推断，以解决重要的科学问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.