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

利用大规模数据集进行估计和预测是统计及相关领域中常见的问题，并提出了巨大的挑战。为了解决这些挑战，提出了四个相互关联的研究课题进行调查。首先，研究者提出了在维数随样本量呈指数增长的超高维设置下的重尾数据的稳健变量选择方法。为了解决重尾性问题，研究了各种模型设置下具有鲁棒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，并刻画了惩罚函数和损失函数的最优性。还提出并研究了鲁棒独立性筛选方法。其次，研究了具有功能预测因子和/或功能响应的高维函数回归模型中的变量选择。提出了模型拟合程序，并对所提方法的抽样特性进行了深入研究。第三，研究了在超高维正确指定模型和错误指定模型两种情况下惩罚经验风险最小化的正则化参数选择。在模型拟合和模型复杂性之间进行了适当的权衡。该研究还回答了传统的模型选择标准（如AIC和BIC）是否继续在超高维度中工作的问题。第四，在统一的阈值分类规则框架下，对具有相关特征的高维分类进行了广泛的研究，确定了使分类误差最小的阈值的最优选择。研究者研究高斯分类，并将方法和结果推广到相关离散特征的情况。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、气候学等各个科学领域越来越普遍。如何有效地分析这些数据集在方法论和计算上都提出了巨大的挑战，而这些挑战在较小规模的研究中并不存在。本提案的主要目标是提出新的或扩展的方法，并在深度和宽度上研究它们的采样特性，用于各种回归和分类问题的高维模型构建和模型评估。PI在统计以外的许多领域都有广泛的研究兴趣，如计算生物学、金融学、计量经济学和机器学习。所提出的方法将在实际数据集上进行测试，并扩展到这些不同的领域。此外，PI计划开发软件包来实现所提出的方法，并使它们公开可用。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，将少数民族学生、本科生、研究生等纳入研究课题，并在新课程中引入尖端的高维统计方法等教育活动与研究课题相结合。

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.