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，这些点云是具有内在几何形状的底层对象（如曲线，曲面或流形）的样本。分析随机对象的这些时间序列的动态需要捕获关于几何特性的信息的定性推断方法，即，形的描述词的演变。“分析形状在许多研究领域都是最重要的，如基因组学，气候学，神经科学和金融。在这个项目中，我们开发了新的方法，并提供概率和统计的基础来建模，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。我们专注于我们的理论工作，以支持来自基因组学的两个领域的应用程序;细胞分化的发展和肿瘤的演变。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。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，我们将提供软件包，使推理方法提供给广大观众。PI进一步建议从跨学科的角度设计课程并制作简要说明，并为暑期本科生提供数理统计和拓扑数据分析接口的项目指导。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产生了浓厚的兴趣;这为点云提供了定性的多尺度形状描述符。然而，为了得出可靠的定性推断的形状和拓扑特征，它是至关重要的占（不断演变的）空间和时间的依赖性存在于数据中。为了解决这些问题，我们采取的观点是，基本数据是一个功能，即，观测值是函数空间中的点。这种观点集成了统计方法和TDA的功能时间序列（FTS）的背景下。我们提供了新的方法来建模，分析和预测数据产生的非平稳度量空间值随机过程。我们的框架建立了应用多尺度几何描述符，有意义地捕捉他们不断变化的几何特征，以及拓扑不变量的调查统计和概率的基础。这个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。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.