CAREER: Nonparametric Models Building, Estimation, and Selection with Applications to High Dimensional Data Mining

非参数方法越来越多地应用于回归、分类和密度估计，无论是在统计学还是其他相关领域，如数据挖掘和机器学习。然而，非参数模型的一个关键困难是由于维度的诅咒而对高维数据进行模型拟合。另一个困难是模型推断和解释，即如何评估或测试复杂曲面拟合中的单个变量影响。对于协方差结构复杂的异构数据，非参数模型估计更具挑战性。本提案的目标是开发新的和广泛适用的程序，以同时对数据挖掘中的非参数模型及其相关范例进行模型选择和估计。在再现核希尔伯特空间（RKHS）的框架中，PI为几个模型族提出了一系列新的正则化技术：用于相关数据的平滑样条方差分析模型、半参数回归模型、用于监督和半监督学习的支持向量机。所提出的方法通过其实现模型稀疏性和函数平滑的统一框架，其易于处理的理论性质以及易于适应高维问题，构成了标准方法的关键进步。PI将研究所提出的估计量的渐近行为，探索调整正则化参数的数据驱动程序，并开发计算算法和软件来实现所提出的程序。PI还将通过广泛的模拟研究和实际数据分析来检查新方法的有限样本性能。在当前的信息时代，科学和工业数据库的数量和复杂性呈指数级增长。因此，数据形式的维度会越来越高。这些数据的分析对统计学家提出了新的挑战，并成为现代统计学中最重要的研究课题之一。这个项目的目的是显著增加分析复杂高维数据的可用工具。在本项目中，PI旨在实现以下三个目标：(1)在统一的数学框架内应对非参数模型估计和选择的挑战；(2)开发具有理想统计特性的灵活方法和高性能统计软件，用于挖掘海量数据；(3)将以上两项活动的研究机会和成果整合到研究生、本科和高中阶段的学科和跨学科统计教育中。这项研究将拓宽对非参数推断和模型选择的传统理解，为社会学、经济学、环境、生物学和医学等各个领域的研究人员和实践者提供最先进的数据分析工具，并帮助下一代学生准备必要的现代统计视角。

Nonparametric methods are increasingly applied to regression, classification and density estimation, both in statistics and other related areas such as data mining and machine learning. However, a key difficulty with nonparametric models is model fitting for high dimensional data due to the curse of dimensionality. Another difficulty is model inference and interpretation, i.e., how to evaluate or test individual variable effects on the complex surface fit. For heterogeneous data with complicated covariance structure, nonparametric model estimation is even more challenging. The objectives of this proposal are to develop novel and widely applicable procedures to simultaneous model selection and estimation for nonparametric models and their related paradigms in data mining. In the framework of reproducing kernel Hilbert space (RKHS), the PI proposes a host of new regularization techniques for several families of models: smoothing spline ANOVA models for correlated data, semiparametric regression models, support vector machines for supervised and semi-supervised learning. The proposed methodologies constitute key advances over standard methods through their unified framework for achieving model sparsity and function smoothing altogether, their tractable theoretical properties, and their easy adaptation to high dimensional problems. The PI will study asymptotic behaviors of the proposed estimators, explore data-driven procedures for tuning regularization parameters, and develop computation algorithms and softwares to implement the proposed procedures. The PI will also examine finite sample performance of new methods via extensive simulation studies and real data analysis.In the current information era, the volume and complexity of scientific and industrial databases have been exponentially expanding. As a consequence, the data form keeps gaining higher and higher dimensionality. Analysis of such data poses new challenges to statisticians and is becoming one of the most important research topics in modern statistics. The purpose of this project is to significantly increase the available tools for analyzing complex high dimensional data. In this project, the PI aims to accomplish the following three goals: (1) meet the challenges of nonparametric model estimation and selection within a unified mathematical framework; (2) develop flexible methods with desired statistical properties and high-performance statistical softwares for mining massive data; (3) integrate research opportunities and findings from the above two activities into disciplinary and interdisciplinary statistical education at graduate, undergraduate and high school levels. This research will broaden traditional understanding of nonparametric inferences and model selection, provide a broad range of researchers and practitioners in various fields including sociology, economics, environmental, biological and medical sciences with state-of-the-art data analysis tools, and help to prepare the next-generation students with the necessary modern statistical perspectives.