Variable Selection in High Dimensional Feature Space with Applications to Covariance Matrix Estimation and Functional Data Analysis

在高维统计建模中，变量选择起着重要的作用。研究者研究了基于高维统计建模的正则化机制的变量选择技术，开发了高维特征空间的变量筛选新方法，并探索了这些技术在高维稀疏推理中的应用。提出了三个相互关联的研究课题。首先，研究了一种统一的惩罚似然变量选择方法，同时选择显著变量并估计其回归系数，并开发了适用于超高维特征空间的可靠筛选技术。其次，提出了因子模型来估计大规模的协方差矩阵，并利用变量选择技术来构建因子，并对协方差选择进行了研究。第三，研究者研究了变量选择技术在函数数据分析中的应用，其中回归系数函数表现出各种稀疏性。对大量数据集的分析现在普遍出现在许多科学学科和工程问题中，对统计理论和方法提出了许多挑战，而这些挑战在较小规模的研究中是不存在的。本提案的主要目标是对高维变量选择这一重要且具有挑战性的主题做出方法和理论贡献。这些新的发展提供了对各种高维正则化方法的进一步理解，并允许科学家通过有效的降维和增加的可解释性来分析高维数据。高维度的挑战来自科学和人文学科的不同领域，从基因组学和健康科学到经济学和金融学。关于高维变量选择的拟议工作不仅将有助于更好地确定对公共卫生或市场风险等重要因素，而且还将使各个领域的广泛科学家和研究人员受益。

Variable selection plays an important role in high dimensional statistical modeling which nowadays arises in many scientific investigations. The investigator studies variable selection techniques built upon the machinery of regularization for high dimensional statistical modeling, develops new approaches to variable screening for high dimensional feature space, and explores the applications of these techniques to high dimensional sparse inference. Three interrelated research topics are proposed for investigation. First, the investigator studies a unified approach to variable selection with penalized likelihood which simultaneously selects significant variables and estimates their regression coefficients, and develops sure screening techniques applicable to ultra-high dimensional feature space. Second, factor models are proposed to estimate large scale covariance matrices while variable selection techniques are invoked to construct the factors, and covariance selection is also investigated. Third, the investigator studies the applications of variable selection techniques to functional data analysis where the regression coefficient functions exhibit various kinds of sparsity.The analysis of vast data sets now commonly arising in many scientific disciplines and engineering problems poses numerous challenges to statistical theory and methodology that are not present in smaller scale studies. A major goal of this proposal is to make methodological and theoretical contributions to the important and challenging topic of high dimensional variable selection. These new developments provide further understanding of various regularization methods in high dimensions, and allow scientists to analyze high dimensional data with efficient dimension reduction and increased interpretability. The challenges of high dimensionality arise from diverse fields of sciences and humanities ranging from genomics and health sciences to economics and finance. The proposed work on variable selection in high dimensions will not only help better identify factors that are important to, for instance, public health or market risk, but also benefit a broad range of scientists and researchers in various fields.