Statistical Models and Methods for Complex Data in Metric Spaces

随机对象形式的大数据越来越多地在科学和社会中遇到。充分和有原则的分析方法对于提取相关信息，做出预测以及评估样本之间是否存在差异非常重要。然而，这些数据的复杂性对统计分析提出了重大挑战，传统方法无法应用。在这个项目中，这些挑战将通过开发先进的非参数统计方法来解决和克服，这些方法可以处理对象数据的复杂性。将研究的具体对象包括网络和分布，以及脑成像和基因组学数据、气候变化数据和其他当前感兴趣领域的数据的应用。对于随时间反复观察随机物体的情况，例如反复观察同一受试者的MRI脑部扫描，也将开发潜在时间动力学的量化。该研究将包括测试和估计方法、理论、有效的计算实现和数据应用，预计将导致实质性的新见解。该项目将为下一代数据分析人员和研究统计人员提供培训，并将提供执行这些方法的用户友好代码。该项目将为新兴的度量统计领域奠定基础，作为随机对象样本的统计方法和理论的综合框架，随机对象是在度量空间中取值的随机变量/数据。随机对象包括分布、网络、树、协方差矩阵和曲面形式的数据，以及黎曼流形（如球体）上的数据。这类数据的统计分析具有挑战性，因为由于缺乏向量空间结构，无法应用传统的统计方法。具体而言，对于位于测地线空间中的随机物体，将开发从重心到特定物体的传输的一般回归模型。在这些模型中，预测因子和响应都是输运，具体的例子是球形数据和分布的回归模型。该项目还将包括研究多元分布的回归模型作为响应，与欧几里得预测器配对，使用瓦瑟斯坦空间的切片优化。这种方法将用气候学和生命科学数据中的应用来说明。另一个目标是在测地线空间中发展输运过程，构成一种新型的随机过程。对于这些过程，将得到一般类型随机对象的锚点表示和分布对象在Wasserstein空间中的潜在高斯过程表示。此外，将开发Frechet回归的随机效应模型，为纵向对象数据提供第一个这样的模型，并在脑成像和分布数据分析中应用。在整个过程中，由于欧几里得和代数结构的缺失所带来的挑战将用经验过程理论和其他工具来解决。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。

Big data in the form of random objects are increasingly encountered across sciences and society. Adequate and principled approaches for their analysis are important to extract relevant information, to make predictions, and to assess whether there are differences between samples. However, the complexity of such data poses major challenges for statistical analysis and traditional methods cannot be applied. In this project, these challenges will be addressed and overcome through the development of advanced nonparametric statistical methodology that can handle the complexity of object data. Specific objects that will be studied include networks and distributions, with applications for brain imaging and genomics data, climate change data and data from other areas of current interest. For situations where random objects are repeatedly observed over time, such as repeatedly observed MRI brain scans for the same subject, quantifications of the underlying time dynamics will also be developed. The research will include methods for testing and estimation, theory, efficient computational implementation, and data applications, which are expected to lead to substantial new insights. The project will provide training for the next generation of data analysts and research statisticians and user-friendly code implementing the methods will be made available. The project will contribute to the foundations of the emerging field of metric statistics as a comprehensive framework for statistical methodology and theory for samples of random objects, which are random variables/data that take values in a metric space. Random objects encompass data in the form of distributions, networks, trees, covariance matrices and surfaces, and data on Riemannian manifolds such as spheres. The statistical analysis of such data is challenging as one cannot apply traditional statistical methods due to the absence of a vector space structure. Specifically, for random objects that are situated in a geodesic space, a general class of regression models for transports from the barycenter to specific objects will be developed. In these models both predictors and responses are transports and specific examples are regression models for spherical data and distributions. The project will also include the study of regression models for multivariate distributions as responses, paired with Euclidean predictors, using slice-optimization in Wasserstein space. This approach will be illustrated with applications in climatology and life sciences data. Another goal is the development of transport processes in a geodesic space, constituting a new type of stochastic process. For such processes, anchor point representations for general types of random objects and latent Gaussian process representations for distributional objects in Wasserstein space will be obtained. Additionally, a random effects model for Frechet regression will be developed, providing the first such model for longitudinal object data, with applications in brain imaging and distributional data analysis. Throughout, the challenges resulting from the absence of Euclidean and algebraic structures will be addressed with empirical process theory and other tools.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.