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

由于信息技术的快速发展及其在气候科学、医学成像、金融等诸多科学领域的应用，高维数据和无限维功能数据的统计分析变得越来越重要。分析此类大数据的一个关键挑战是如何测量和推断复杂的依赖结构，这是统计学的一个基本步骤，但由于数据的高维和庞大规模，这一步骤变得更加困难。本研究项目的主要目标是开发新的依赖度量来量化大规模数据集（如时间依赖的功能数据和高维数据）的依赖，并利用这些新度量来开发新的统计工具来进行稀疏主成分分析、降维和同步假设检验。基于能够捕获非线性和非单调相关性的新依赖度量，正在开发的方法有望导致更准确的预测和推理，以及在分析功能和高维数据时更有效的降维。该研究由三个项目组成，解决了功能和高维数据分析中的不同挑战。在项目1中，研究者引入了一个新的算子值量来表征一个函数值随机元素给定另一个函数值随机元素的条件平均（In）依赖性，并在有限维函数数据的新框架下应用新开发的相关度量对功能时间序列进行降维。在项目2中，研究人员为具有高维响应的回归模型探索了一个新的降维框架，该框架不需要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方面更加灵活。在项目3中，研究者使用平方和和最大类型检验统计量，通过距离协方差和秩距离协方差开发了高维数据相互独立性的新测试。总的来说，这三条研究线都与大数据有关，它们涉及现代统计学的各个方面；该项目旨在将稀疏主成分分析、相关功能数据推理和高维多元分析等领域的当前前沿推向另一个层次。

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.