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

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.