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、医学影像、金融等诸多科学领域的应用，高维数据和无穷维函数数据的统计分析变得越来越重要。与这种大数据分析相关的一个关键挑战是如何测量和推断复杂的依赖结构，这是统计学中的基本步骤，并且由于数据的高维性和巨大的尺寸而变得更加困难。该研究项目的主要目标是开发新的相关性度量，用于量化大规模数据集的相关性，如时间相关的功能数据和高维数据，并利用这些新的度量开发新的统计工具，用于进行稀疏主成分分析，降维和同时假设检验。建立在新的依赖性度量，可以捕捉非线性和非单调的依赖性，正在开发的方法，预计将导致更准确的预测和推理，以及更有效的降维功能和高维数据的分析。该研究由三个项目组成，旨在解决功能和高维数据分析中的不同挑战。在项目1中，研究人员引入了一个新的算子值量来表征一个函数值随机元素对另一个函数值随机元素的条件均值（不）依赖，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函数时间序列进行降维。在项目2中，研究人员探索了一种新的降维框架，用于高维响应的回归模型，该框架需要不太严格的线性模型假设，并且在捕获响应和协变量之间可能的非线性相关性方面更加灵活。在项目3中，研究人员通过距离协方差和秩距离协方差，使用平方和和最大类型检验统计量，开发了高维数据相互独立性的新检验。总的来说，这三条研究路线都与大数据有关，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，相关函数数据推理和高维多元分析等领域的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.