Statistical Modelling of Multivariate Functional and Distributional Data

现代记录设备正在收集更复杂的数据，当这些数据在空间或时间上进行测量时，分辨率也在不断提高。在为我们周围发生的相关物理现象提供更详细信息的同时，它们也提出了与这些数据的可解释建模和可行计算相关的统计挑战。在空间、时间或其他连续体上测量的数据被恰当地称为功能数据，并构成现代统计学的一个重要分支。这个项目将开发重要的方法来分析两种类型的功能数据。第一组项目旨在估计所谓的多元功能数据组成部分之间的依赖模式，其中一组共同的特征是随着时间的推移对研究中的每个主题进行测量，例如神经成像扫描，其中信号随时间在各种空间位置被记录下来。另一类重要的功能数据是分布或直方图的样本，它们经常出现在人口统计数据分析中，例如死亡率分布，但也出现在其他重要领域，如神经科学和金融。本项目概述了降维和回归的方法，这些方法尊重众所周知的正性和分布的曲线下面积约束，产生可解释的数据摘要和模型拟合，为从业者提供对数据中包含的信息的更清晰的理解。功能磁共振成像和脑电图都能在多个大脑位置产生与时间相关的信号，从而产生多元功能数据。量化连接模式来定义大脑网络，例如，为了识别正常和病理特征，是一个重要的神经科学问题，可以使用多元功能数据技术来解决。该项目旨在推进功能图形模型的使用，以估计潜在的大脑依赖网络，包括与现有方法相比提高计算效率。这些方法同样适用于产生类似结构数据的其他领域，例如纵向医学研究，其中一组共同的测量值在一段时间内重复记录。本建议还考虑了分布数据的方法，分布数据可以被认为是曲线或曲面的集合，每个曲线或曲面对应于一个概率分布。例如，神经成像数据自然地提供了这样的分布样本，因为大脑区域内的髓鞘形成水平或信号相关性是高维数据，可以通过直方图或分布在受试者水平上有效地总结。本项目以此类分布数据为样本，通过分布回归研究可解释降维的统计方法以及分布响应函数对相关协变量的依赖关系。一个关键的工具是分布的Wasserstein度量，它在应用环境中取得了广泛的成功，但在统计中尚未得到充分利用。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。

Modern recording devices are collecting data of greater complexity and, when these data are measured over space or time, also at ever-increasing resolution. While providing more detailed information about the associated physical phenomena occurring around us, they also pose statistical challenges related to interpretable modeling and feasible computation for such data. Data measured over space, time, or some other continuum, are fittingly termed functional data, and constitute an important subfield of modern statistics. This project will develop important methodology for the analysis of two types of functional data. The first set of projects aim at the estimation of dependency patterns between components of so-called multivariate functional data, where a common set of features is measured over time for each subject in a study, such as neuroimaging scans where signals are recorded over time at a variety of spatial locations. Another important class of functional data are samples of distributions or histograms, which regularly arise in the analysis of demographic data as mortality distributions, for example, but also in other important fields such as neuroscience and finance. This project outlines methods for dimension reduction and regression that respect the well-known positivity and area-under-the-curve constraints for distributions, yielding interpretable data summaries and model fits that provide the practitioner with a clearer understanding of the information contained in their data.Both fMRI and EEG yield time-dependent signals at multiple brain locations, resulting in multivariate functional data. Quantifying connectivity patterns to define brain networks, for example in order to identify normal and pathological characteristics, is an important neuroscientific problem that can be addressed using multivariate functional data techniques. This project seeks to advance the use of functional graphical models to estimate underlying brain dependency networks, including improved computational efficiency compared to existing methods. These methods are equally applicable in other domains that produce data of similar structure, such as longitudinal medical studies, where a common set of measurements is recorded repeatedly over time. Also considered in this proposal are methods for distributional data, which can be thought of as collections of curves or surfaces, each corresponding to a probability distribution. For example, neuroimaging data naturally provide such distributional samples, as levels of myelination or signal correlations within brain regions are high-dimensional data that can be effectively summarized at the subject level by a histogram or distribution. Given a sample of such distributional data, this project investigates statistical methods of interpretable dimension reduction and dependency of distributional response functions on relevant covariates through distributional regression. A key tool is the Wasserstein metric for distributions, which has been widely successful in applied settings, but has not been utilized to its full extent in statistics.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.