Collaborative Research: Statistical Inference for Functional and High Dimensional Data with New Dependence Metrics

协作研究：使用新的依赖性度量对功能和高维数据进行统计推断

• 批准号: 1607489
• 负责人: Xiaofeng Shao
• 金额: $ 18.5万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607489&HistoricalAwards=false
• 关键词: Collaborative; Research; Statistical; Inference; Functional

Due to the rapid development of information technologies and their applications in many scientific fields such as climate science, medical imaging, and finance, statistical analysis of high-dimensional data and infinite-dimensional functional data has become increasingly important. A key challenge associated with the analysis of such big data is how to measure and infer complex dependence structure, which is a fundamental step in statistics and becomes more difficult owing to the data's high dimensionality and huge size. The main goal of this research project is to develop new dependence measures for quantifying dependence of large scale data sets such as temporally dependent functional data and high dimensional data, and utilize these new measures to develop novel statistical tools for conducting sparse principal component analysis, dimensional reduction, and simultaneous hypothesis testing. Building on the new dependence metrics that can capture nonlinear and non-monotonic dependence, the methodologies under development are expected to lead to more accurate prediction and inference, as well as more effective dimension reduction in the analysis of functional and high dimensional data. The research consists of three projects addressing different challenges in the analysis of functional and high dimensional data. In Project 1, the investigators introduce a new operator-valued quantity to characterize the conditional mean (in)dependence of one function-valued random element given another, and apply the newly developed dependent metrics to do dimension reduction for functional time series under a new framework of finite dimensional functional data. In Project 2, the investigators explore a new dimension reduction framework for regression models with high dimensional response, which requires less stringent linear model assumptions and is more flexible in terms of capturing possible nonlinear dependence between the response and the covariates. In Project 3, the investigators develop new tests for the mutual independence of high dimensional data via distance covariance and rank distance covariance using both sum of squares and maximum type test statistics. Overall, the three lines of research are all related to big data, and they touch upon various aspects of modern statistics; the project aims to push the current frontiers in areas including sparse principal component analysis, inference for dependent functional data, and high dimensional multivariate analysis to another level.

由于信息技术的快速发展及其在许多科学领域（例如气候科学，医学成像和金融）中的应用，因此对高维数据和无限维功能数据的统计分析变得越来越重要。与此类大数据分析相关的一个关键挑战是如何衡量和推断复杂的依赖结构，这是统计中的基本步骤，并且由于数据的高维度和巨大的规模而变得更加困难。该研究项目的主要目标是开发新的依赖度量，以量化大规模数据集的依赖性，例如时间依赖的功能数据和高维数据，并利用这些新措施来开发新的统计工具来进行稀疏主组件分析，尺寸减少和同时假设测试。基于可以捕获非线性和非单调依赖性的新依赖指标，预计正在开发的方法将导致更准确的预测和推理，并在功能和高维数据的分析中更有效的降低降低。该研究由三个针对功能和高维数据分析的不同挑战的项目组成。在项目1中，研究人员介绍了一个新的操作员值数量，以表征一个给定另一个功能值的随机元素的条件平均值（在）依赖性的依赖性，并应用新开发的依赖指标来减少功能时间序列的尺寸，以在有限尺寸函数数据的新框架下进行功能时间序列。在项目2中，研究人员探索了具有高维响应的回归模型的新维度缩小框架，这需要较少严格的线性模型假设，并且在捕获响应与协变量之间可能的非线性依赖性方面更加灵活。在项目3中，研究人员使用距离协方差和秩距离协方差使用正方和最大类型测试统计数据开发了高维数据相互独立的新测试。总体而言，这三条研究都与大数据有关，它们涉及现代统计的各个方面。该项目旨在将当前边界推向包括稀疏主成分分析，依赖功能数据的推断以及高维多元分析的领域。