Nonparametric classification, tuning parameter selection, and asymptotic stability for high-dimensional data

技术创新是推动科学研究和社会进步的主要力量。从计算生物学和健康研究到金融工程和风险管理，科学和人文的各个领域经常可以看到前所未有的大小和复杂性的高通量数据。这种高维数据在当代统计学中引发了许多重要问题，其中特征选择起着关键作用。在高维数据及其在分类和变量选择方面的应用这一主题方面，拟议项目有以下三个相互关联的目标。(1)引入高维数据的非参数分类框架。本研究的目的是在高维环境下将非参数成分与经典的参数方法(如惩罚Logistic回归、线性判别分析)相结合，在不增加太多计算负担的情况下进行分类。研究了超额风险的渐近性质。(2)研究高维变量选择中调整参数选择的交叉验证的渐近性质。本文的目的是系统地研究各种惩罚函数(套索、SCAD、MCP等)中选择调谐参数的主要交叉验证方法的渐近行为。都被使用过。通过描述经典交叉验证法的性质，提出了一种新的改进交叉验证法，用于在求解路径中选择最优调整参数，以实现模型选择的一致性。(3)引入最大惩罚似然估计的渐近稳定性概念。尽管关于高维环境下的最大惩罚似然估计量的文献很多，但对估计量稳定性的研究一直非常有限。研究人员旨在引入一类一般最大惩罚似然估计量的渐近稳定性概念，研究这些估计量在应用不同惩罚函数时的行为，并评估这些估计量的性能。目前，大数据分析普遍存在于许多科学领域，这给统计学领域带来了挑战和机遇。这一建议的一个主要目标是为高维分类和变量选择这一重要而具有挑战性的主题做出方法论和理论贡献。拟议的研究将对许多科学学科产生广泛影响，包括健康/生命科学、经济学、金融学、天文学和社会学等。在这些领域中，变量选择、特征提取、稀疏挖掘是知识发现的关键。调查人员一直在与哥伦比亚大学医学中心纽约州精神病研究所、纪念斯隆-凯特琳癌症中心计算生物学中心和哥伦比亚大学计算学习系统中心的研究人员互动。拟议的调查结果将用于了解精神健康问题，识别癌症疾病的风险因素，并预测复杂工程系统的故障。在教育方面，拟议的工作将纳入关于最先进的高维统计学习的新课程。它还将纳入本科生和研究生的培训，特别是在博士论文和本科生研究项目方面代表性不足的群体的培训。

Technological innovations have provided a primary force in advancement of scientific research and in social progress. High-throughput data of unprecedented size and complexity are frequently seen in diverse fields of science and humanity, ranging from computational biology and health studies to financial engineering and risk management. Such high-dimensional data have initiated many important problems in contemporary statistics where feature selection plays pivotal roles. The proposed project has the following three interrelated objectives in the theme of high-dimensional data with applications in classification and variable selection. (1) To introduce a nonparametric classification framework for high-dimensional data. The target of this research is to integrate the nonparametric component to the classical parametric methods for classification (e.g., penalized logistic regression, linear discriminant analysis) under high-dimensional settings without incurring much computational burden. Asymptotic properties are investigated regarding the excess risk. (2) To investigate the asymptotic properties of cross-validation for tuning parameter selection in high-dimensional variable selection. The goal here is to perform a systematic study on the asymptotic behavior of major cross-validation methods for choosing the tuning parameter when various penalty functions (LASSO, SCAD, MCP, etc.) are used. By delineating the properties of the classical cross-validation, a new modified cross-validation method for the purpose of choosing the optimal tuning parameter in the solution path is developed that achieves model selection consistency. (3) To introduce the notion of asymptotic stability for maximum penalized likelihood estimators. Despite the extensive literature on the maximum penalized likelihood estimators in high-dimensional settings, the research on the stability of the estimators has been very limited. The investigators aim to introduce the notion of asymptotic stability for a general class of maximum penalized likelihood estimators, study the behavior and evaluate the performance of these estimators when different penalty functions are applied.The analysis of "big data" now pervasive across many scientific disciplines poses challenges as well as opportunities to the field of statistics. A major goal of this proposal is to make methodological and theoretical contributions to the important and challenging topic of high-dimensional classification and variable selection. The proposed research will have broad impacts on many disciplines of science, including health/life sciences, economics, finance, astronomy and sociology, among others. In these fields, variable selection, feature extraction, sparsity explorations are crucial for knowledge discovery. The investigators have been interacting with researchers at New York State Psychiatric Institute at the Columbia University Medical Center, Computational Biology Center of the Memorial Sloan-Kettering Cancer Center and Center for Computational Learning Systems at Columbia University. The results of the proposed investigations will be used for understanding mental health issues, for identifying risk factors in diseases of cancer and for predicting failures in complex engineering systems. On the educational side, the proposed work will be incorporated into new courses on the state-of-the-art high-dimensional statistical learning. It will also be integrated into the training of undergraduate and graduate students, especially of under-represented groups, in terms of Ph.D. dissertations and undergraduate research projects.