Modeling Complex Functional Data and Random Objects in Metric Spaces

技术的快速发展导致了各个学科和社会的复杂数据激增。这对统计分析提出了挑战，也是本研究项目背后的推动力。此类复杂数据的例子包括大脑连接相关矩阵、出租车行程网络、微生物组成和死亡年龄分布，这些数据的代数运算（如求和和标量乘法）没有明确定义，因此无法通过依赖代数运算提取相关信息的传统统计工具直接分析。该项目旨在发展统计方法，以理论和有效的计算实现为支持，解决这些数据分析需求。该研究项目有望带来实质性的见解，包括表征早期神经发育中感兴趣的大脑区域的共同进化，或比较不同国家的死亡率或收入分布。该方法还将能够检测复杂数据组之间的差异，例如正常衰老和病理性衰老期间大脑连接网络之间的差异，并确定不同数据对象之间的关联，例如身体质量组成和身体活动强度分布。该项目还将为本科生和研究生提供统计学培训和研究的机会。该研究项目将开发功能数据和随机对象的统计建模和推理方法，这些对象在默认情况下不具有向量空间结构的度量空间中取值。线性的缺乏消除了对欧几里得数据开发的现有方法的适用性，需要开发新的工具来分析这些数据。稀疏多元函数数据的建模是目前研究的热点之一。将开发一种因子分析方法，通过非参数回归来处理极端的时间稀疏性，这将用于通过选择合适的度量来估计横截面协方差矩阵。研究的另一个重点是度量空间中随机对象的统计建模和推理。主成分分析（PCA）方法将被开发来使用度量几何模型一般对象数据。将研究所提出的对象PCA在Wasserstein空间随机分布样本中的应用，这将受益于与功能数据分析技术相结合的Wasserstein几何。将设计推理方法来测试基于深度剖面的对象数据样本的同质性和独立性，深度剖面独特地表征了广泛度量空间中随机对象的规律。这些发展将伴随着理论分析和论证，以及可扩展和稳定的算法，这些算法将被制成公开可用的软件。该奖项反映了美国国家科学基金会的法定使命，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。

The rapid advancement of technology has led to a surge in complex data across various disciplines and society. This poses challenges for statistical analysis and is the driving force behind this research project. Examples of such complex data include brain connectivity correlation matrices, taxi trip networks, microbial compositions, and age-at-death distributions, for which algebraic operations such as sums and scalar multiplications are not well-defined and hence cannot be directly analyzed by conventional statistical tools that rely on algebraic operations to extract relevant information. This project aims to develop statistical methodology to address these data analytic needs, supported by theory and efficient computational implementations. This research project is anticipated to lead to substantial insights, including characterization of the co-evolution of brain regions of interest in early neurodevelopment, or comparison of mortality or income distributions across different countries. The methodology will also enable the detection of differences between groups of complex data, such as between brain connectivity networks during normal aging and pathological aging, and determine associations between different data objects, such as body mass compositions and physical activity intensity distributions. The project will also provide opportunities of statistical training and research for undergraduate and graduate students. This research project will develop statistical modeling and inference methods for functional data and random objects that take values in a metric space which by default does not possess vector space structures. The lack of linearity eliminates the applicability of existing methods that have been developed for Euclidean data and necessitates the development of novel tools for the analysis of such data. One focus of the research is modeling of sparse multivariate functional data. A factor analysis approach will be developed that lends itself to handling extreme temporal sparsity through nonparametric regression, which will be used to estimate the cross-sectional covariance matrices with the choice of a suitable metric. Another focus of the research is statistical modeling and inference for random objects in metric spaces. Principal component analysis (PCA) methods will be developed to model general object data using metric geometry. An application of the proposed object PCA to samples of random distributions in the Wasserstein space will be investigated, which benefits from the Wasserstein geometry potentially in conjunction with functional data analysis techniques. Inference methods will be devised for testing homogeneity and independence for samples of object data based on depth profiles, which uniquely characterize the law of random objects for a wide range of metric spaces. These developments will be accompanied by theoretical analysis and justification as well as scalable and stable algorithms that will be made into publicly available software.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.