Efficient Bi-Level Variable Selection in High-Dimensional Models

本文针对高维回归模型中协变量存在组结构的情况，提出了一类新的双水平变量选择方法。现有的变量选择方法要么是针对单个变量选择，要么是针对群体选择，而不是针对两者。此外，用于评估统计过程的标准方法假设模型中的变量数量是固定的，并且比样本量小得多，这通常不适用于高维设置。高维数据的分析提出了新的和具有挑战性的统计理论和计算问题。所提出的方法是能够在选定的组中同时进行组和个体变量选择。对于建议的双层选择方法，计算算法将被开发，并在一类重要的回归模型的理论性质将被调查。在稀疏模型中，即使协变量的数目远大于样本容量，所提出的方法也能够以高概率同时正确地选择重要的组和变量。高维数据出现在许多科学领域，包括生物学、经济学、金融学、信息技术和健康科学。在所有这些领域，从数据中识别重要特征是科学发现过程中的关键步骤。该研究的预期应用是对高维基因组数据的分析。特别是，拟议的研究将获得全基因组关联研究和遗传途径回归分析的新方法。这是了解基因和遗传途径如何导致常见和复杂疾病（如各种类型的癌症）的两种最重要的方法。 这项研究旨在将新的统计方法转化为分析高维基因组数据的新方法。

This project proposes a class of novel bi-level variable selection methodin high-dimensional regression models when there is group structure in covariates. The existing variable selection methods are designed for either individual variable selection or group selection, but not for both. Furthermore, standard methods for evaluating a statistical procedure assume that the number of variables in a model is fixed and much smaller than the sample size which, in general, are not applicable for high-dimensional settings. Analysis of high-dimensional data presents novel and challenging theoretical and computational questions in statistics. The proposed methods are capable of simultaneous group and individual variable selection within selected groups. For the proposed bi-level selection methods, computational algorithms will be developed and the theoretical properties in a class of important regression models will be investigated. The proposed methods are expected to be able to correctly select the important groups and variables simultaneously with high probability in sparse models even when the number of covariates is much larger than the sample size.High-dimensional data arise in many scientific fields, including biology, economics, finance, information technology, and health sciences. In all these fields, the identification of important features from data is a crucial step in the process of scientific discovery. The intended applications of the proposed study are to the analysis of high-dimensional genomic data. In particular, the proposed research will obtain novel methods for genome wide association studies and genetic pathway regression analysis. These are two of the most important approaches for understanding how genes and genetic pathways cause common and complex diseases such as various types of cancers. The proposed research aims to translate novel statistical approaches into new methodologies for analyzing high-dimensional genomic data.