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From Functional Data to Random Objects

从功能数据到随机对象

基本信息

批准号：
1712864
负责人：
Hans-Georg Mueller
金额：
$ 15万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712864&HistoricalAwards=false
关键词：
Functional Data Random Objects

项目摘要

Large and complex data that are increasingly collected across the sciences and by companies pose novel challenges for statistical analysis, due to their complexity and size. Specific applications that motivate this research come from brain imaging, genomics, and the social sciences. To make sense of such data and extract relevant features, statistical methodology that is suitable for the analysis of large samples of complex random objects is needed. Examples of these objects include networks, distribution functions, and covariance matrices. A challenge is that common algebraic operations such as sums or differences are not defined for such objects. In many instances, objects may also be repeatedly observed over time, and the quantification of their time dynamics is then of interest. In this project, statistical methodology that addresses these basic data analytic needs is developed under minimal assumptions. These developments also include the theoretical foundations of this methodology and computational implementations. This methodology is expected to lead to new insights by quantifying phenomena such as changes in mortality or income distributions over calendar years, or changes in brain connectivity networks with aging to allow researchers to distinguish normal and pathological aging processes. Procedures are also developed to test for significant differences between groups of random objects, for example, comparisons between mortality distributions of countries, including the identification of clusters. The methodology to be developed is based on delicate extensions of basic statistical notions such as population and sample mean, variance, regression and analysis of variance to the case of more complex spaces of random objects.Over the past decade, there have been rapid advances and substantial developments for functional data, including advanced methods for functional regression. The developments and methodology for Functional Data Analysis are limited to Hilbert space valued random variables, such as square integrable random functions, which limits their applicability. This research is motivated by the increasing prevalence of examples where random objects are not in a Hilbert space. Key objects of interest are distributions, networks and covariance matrices, in addition to general metric space valued random objects. Core concepts that will be applied and appropriately extended to these random objects include Frechet mean, Frechet variance, and Frechet regression. For longitudinally observed random objects, the notion of a general Frechet integral will serve to quantify projections in general spaces. Such projections will be studied for their use in representing time-varying random objects. The tools that will be developed are based only on distances, and are therefore suitable for general metric space valued objects. For special classes of objects such as distributions, additional characterizations such as manifold representations and Wasserstein covariance will also be developed and illustrated in applications.

由于数据的复杂性和规模，越来越多的科学领域和公司收集了大量复杂的数据，这给统计分析带来了新的挑战。激发这项研究的具体应用来自脑成像、基因组学和社会科学。为了理解这些数据并提取相关特征，需要适用于复杂随机对象的大样本分析的统计方法。这些对象的例子包括网络、分布函数和协方差矩阵。一个挑战是，诸如和或差之类的常见代数操作没有为这些对象定义。在许多情况下，随着时间的推移，物体也可能被反复观察，因此它们的时间动态的量化是有意义的。在这个项目中，在最小的假设下制定了处理这些基本数据分析需求的统计方法。这些发展还包括该方法的理论基础和计算实现。这种方法有望通过量化诸如历年死亡率或收入分布的变化，或大脑连接网络随年龄增长的变化等现象，从而使研究人员能够区分正常和病理衰老过程，从而产生新的见解。还制定了检验随机对象组之间的重大差异的程序，例如，比较各国的死亡率分布，包括确定聚类。待开发的方法是基于对基本统计概念的精细扩展，如总体和样本均值、方差、回归和方差分析，以适用于更复杂的随机对象空间。在过去的十年中，功能数据有了快速的发展和实质性的发展，包括功能回归的先进方法。功能数据分析的发展和方法仅限于希尔伯特空间值随机变量，如平方可积随机函数，这限制了它们的适用性。这项研究的动机是越来越普遍的例子，其中随机对象不在希尔伯特空间。除了一般的度量空间值随机对象外，感兴趣的关键对象是分布、网络和协方差矩阵。将应用并适当扩展到这些随机对象的核心概念包括Frechet mean， Frechet variance和Frechet regression。对于纵向观察的随机物体，一般Frechet积分的概念将用于量化一般空间中的投影。我们将研究这种投影在表示时变随机物体时的用途。将开发的工具仅基于距离，因此适用于一般度量空间值对象。对于特殊类型的对象，如分布，附加的表征，如流形表示和瓦瑟斯坦协方差也将在应用中开发和说明。