Models for Complex Functional and Object Data

大数据越来越多地出现在社会和所有科学领域，由于其复杂性和规模，对统计分析提出了新的挑战。激发这项研究的数据分析任务起源于脑成像，基因组学，社会科学和许多其他当前感兴趣的领域。为了理解这些数据并提取相关信息，需要适用于分析复杂数据的大样本的原则性统计方法。例子包括网络或死亡年龄分布，其中常见的代数运算，如和或差没有定义。在许多情况下，这样的数据对象也可以随着时间的推移被重复观察，并且它们的时间动态的量化于是引起了极大的兴趣。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位于何处。 统计方法将开发，解决这些数据分析的需要，沿着理论和有效的计算实现。这一新方法有望带来大量新的见解。例如，将有可能量化一些现象，如历年来温度、死亡率或收入分布的变化，或作为年龄函数的大脑连接网络的变化，这将有助于区分正常和病理性大脑老化。新的方法还将使人们能够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，包括确定群组。该项目还为本科生和研究生提供研究培训机会。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，包括对象值函数和纵向数据。由于缺乏欧氏结构，现有的方法从高维和功能的数据分析一般不适用于度量空间值的随机对象。这激发了新方法的发展，以解决缺乏欧几里得结构的挑战。 调查的主要路线将是回归和变点模型的随机对象，一方面和方法的轨迹随机对象，包括复杂的功能数据。新的回归和变点模型进行研究，包括分布作为预测;回归模型的点过程;推理和单指数建模的弗雷歇回归;和变点分析的对象数据序列在各种情况下。对于对象值函数数据，重点将是随机对象的时间弯曲模型和各种空间中纵向随机对象的模型的开发，包括数据在时间上仅稀疏和不规则观察的情况。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工具箱，适合对象数据和相关的免费软件的数据分析。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的知识价值和更广泛的影响审查标准。

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.