Models for Complex Functional and Object Data

由于大数据的复杂性和规模，整个社会和所有科学越来越多地遇到大数据，并对统计分析提出了新的挑战。推动这项研究的具有挑战性的数据分析任务源于大脑成像、基因组学、社会科学和许多其他当前感兴趣的领域。要理解这些数据并提取相关信息，需要适用于对复杂数据的大样本进行分析的原则性统计方法。例如，网络或死亡年龄分布没有定义常见的代数运算，如求和或求差。在许多情况下，这样的数据对象也可能随着时间的推移而被重复观察，因此对它们的时间动态的量化是非常有意义的。例如，人们可能有兴趣确定是否发生突然变化，以及这些变化在时间上的位置。将开发满足这些数据分析需求的统计方法，以及理论和有效的计算实施。这一新的方法论有望带来实质性的新见解。例如，将有可能量化温度、死亡率或历年收入分布的变化，或大脑连接网络随年龄的变化等现象，这将有助于区分正常和病理性的大脑老化。新的方法还将使人们有可能发现复杂数据组之间的差异，例如各国死亡率分布之间的差异，包括确定集群。该项目还为本科生和研究生提供了研究培训机会。本研究的重点是随机对象的统计方法和理论的发展，即度量空间值随机变量，包括对象值泛函数据和纵向数据。由于缺乏欧几里得结构，现有的高维和泛函数据分析方法一般不适用于度量空间值随机对象。这推动了新方法的发展，以应对缺乏欧几里得结构的挑战。主要的研究方向一方面是随机对象的回归和变点模型，另一方面是包括复杂功能数据的随机对象轨迹的方法。将要研究的新的回归和变点模型包括作为预测变量的分布；点过程的回归模型；Frechet回归的推断和单指数建模；以及各种情景下对象数据序列的变点分析。对于对象值函数数据，重点将是开发用于随机对象的时间扭曲模型和用于各种空间中的纵向随机对象的模型，包括数据在时间上仅被稀疏和不规则观测的情况。要开发的随机对象的原则性统计分析的工具和理论将依赖于度量空间中M估计量的经验过程理论、U统计和相关方法。这些发展将导致创建一个适用于对象数据的数据分析的工具箱和相关的免费软件。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Big data are increasingly encountered across society and all of the sciences and pose novel challenges for statistical analysis, due to their complexity and size. Challenging data analysis tasks that motivate this research originate in brain imaging, genomics, the social sciences and many other areas of current interest. To make sense of such data and extract relevant information requires principled statistical methodology that is suitable for the analysis of large samples of complex data. Examples include networks or age-at-death distributions for which common algebraic operations such as sums or differences are not defined. In many instances such data objects may also be repeatedly observed over time, and the quantification of their time dynamics is then of great interest. For example, one might be interested to determine whether sudden changes occur and where these are located in time. Statistical methodology will be developed that addresses these data analytic needs, along with theory and efficient computational implementations. This new methodology is expected to lead to substantial new insights. For example, it will be possible to quantify phenomena such as changes in temperature, mortality or income distributions over calendar years, or changes in brain connectivity networks as a function of age, which will aid in distinguishing normal and pathological brain aging. The new methodology will also make it possible to detect differences between groups of complex data, for example between the mortality distribution of countries, including the identification of clusters. The project also provides research training opportunities for undergraduate and graduate students. The focus of this research is the development of statistical methods and theory for random objects, i.e., metric space valued random variables, including object-valued functional and longitudinal data. Due to the lack of Euclidean structure, existing methods from high-dimensional and functional data analysis are generally not applicable for metric-space valued random objects. This motivates the development of novel approaches that address the challenge of a lack of Euclidean structure. Major lines of inquiry will be regression and change-point models for random objects on one hand and methods for trajectories of random objects including complex functional data on the other. New regression and change-point models to be studied include distributions as predictors; regression models for point processes; inference and single index modeling for Frechet regression; and change-point analysis for sequences of object data under various scenarios. For object-valued functional data, an emphasis will be the development of time warping models for random objects and of models for longitudinal random objects in various spaces, including the case where the data are only sparsely and irregularly observed in time. Tools and theory for principled statistical analysis of random objects to be developed will rely on empirical process theory for M estimators in metric spaces, U statistics and related approaches. These developments will lead to the creation of a toolbox suitable for data analysis of object data and associated freely 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.