CAREER: High-Dimensional Variable Selection in Nonlinear Models and Classification with Correlated Data

大规模数据集的估计和预测在统计学及相关领域中普遍存在，并提出了很大的挑战。为了应对这些挑战，提出了四个相互关联的研究课题进行调查。首先，针对维度随样本量呈指数增长的超高维环境下的重尾数据，提出了稳健的变量选择方法。为了解决重尾现象，研究了在不同模型设置下具有稳健损失和一般惩罚函数的正则化方法。研究了这些方法的风险性质，刻画了惩罚函数和损失函数的最优性。提出并研究了稳健的独立性筛选方法。其次，研究了具有功能预测因子和/或功能反应的高维函数回归模型中的变量选择。提出了模型拟合步骤，并对所提方法的采样特性进行了深入研究。第三，研究了超高维模型中惩罚经验风险最小化中正则化参数的选择问题。描述了模型拟合和模型复杂性之间的适当折衷。本研究还回答了AIC和BIC等传统模型选择标准是否继续适用于超高维的问题。第四，在阈值分类规则的统一框架下，深入研究了具有相关特征的高维分类问题，找出了使分类误差最小的最佳阈值选择。由于手写数字识别和单核苷酸多态性(SNP)基因分型实验等现代技术的出现，具有大量变量的海量数据集在计算生物学、经济学、金融学、机器学习和气候学等科学领域中变得越来越常见。如何有效地分析这些数据集在方法学和计算方面都提出了巨大的挑战，这在较小规模的研究中是不存在的。这项建议的一个主要目标是提出新的或扩展的方法，并研究其深度和宽度的抽样性质，用于在各种回归和分类问题的高维模型建立和模型评估。PI在统计学之外的许多领域都有广泛的研究兴趣，如计算生物学、金融、计量经济学和机器学习。所提出的方法将在真实数据集上进行测试，并扩展到这些不同的领域。此外，PI计划开发软件包来实施拟议的方法，并将其公之于众。拟议的工作将使各个领域的广泛科学家和研究人员受益。PI还计划将教育活动与拟议的研究相结合，例如让少数民族学生、本科生和研究生参与拟议的项目，并将尖端的高维统计方法纳入新课程。

Estimation and prediction with large-scale data sets commonly arise in statistics and related fields and pose great challenges. To address these challenges, four interrelated research topics are proposed for investigation. First, the investigator proposes robust variable selection methods for heavy-tailed data in the ultra-high dimensional setting of dimensionality increasing exponentially with the sample size. To address the heavy-tailedness, regularization methods with robust losses and general penalty functions in various model settings are investigated. The risk properties of these methods are studied and the optimality of penalty function and loss function is characterized. Robust independence screening methods are also proposed and studied. Second, variable selection in high-dimensional functional regression models with functional predictors and/or functional response is investigated. Model fitting procedures are proposed and sampling properties of the proposed methods are thoroughly investigated. Third, the investigator studies the regularization parameter selection in penalized empirical risk minimization in both settings of correctly specified and misspecified models in ultra-high dimensions. The appropriate tradeoff between the model fitting and model complexity is characterized. This study also answers the question on whether conventional model selection criteria such as AIC and BIC continue to work in ultra-high dimensions. Fourth, high-dimensional classification with correlated features is extensively studied under the unified framework of thresholding classification rules, and the optimal choice of threshold that minimizes the classification error is identified. The investigator studies Gaussian classification and generalizes the methods and results to the case of correlated discrete features.Thanks to the advent of modern technologies such as the handwritten digital recognition and single-nucleotide polymorphism (SNP) genotyping experiments, massive data sets with a large number of variables are becoming more and more common in various scientific fields such as computational biology, economics, finance, machine learning, and climatology. How to effectively analyze these data sets poses great challenges in both methodology and computation that are not present in smaller scale studies. A major goal of this proposal is to propose new or extended methodologies and investigate their sampling properties in depth and width for high-dimensional model building and model evaluation in various settings of regression and classification problems. The PI has broad research interests in many fields outside statistics such as computational biology, finance, econometrics, and machine learning. The proposed methods will be tested on real data sets and extended to these different areas. In addition, the PI plans to develop software packages to implement the proposed methods, and make them publicly available. The proposed work will benefit a broad range of scientists and researchers in various fields. The PI also plans to integrate education activities with the proposed research, such as involving minority students, undergraduate students, and graduate students in the proposed projects and incorporating cutting-edge high-dimensional statistical methods into new courses.