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数灾难而导致的高维数据的模型拟合。另一个困难是模型推理和解释，即，如何评估或测试单个变量对复杂曲面拟合的影响。对于具有复杂协方差结构的异质数据，非参数模型估计更具挑战性。本建议的目标是开发新的和广泛适用的程序，同时模型选择和估计的非参数模型及其相关的范式在数据挖掘。在再生核希尔伯特空间（RKHS）的框架下，PI为几个模型家族提出了一系列新的正则化技术：相关数据的平滑样条ANOVA模型，半参数回归模型，监督和半监督学习的支持向量机。所提出的方法构成了标准方法的关键进步，通过其统一的框架，实现模型稀疏性和函数平滑，其易于处理的理论特性，以及其易于适应高维问题。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真实的数据分析来检验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容量和复杂性已经呈指数级增长。因此，数据形式不断获得越来越高的维度。对这些数据的分析对统计人员提出了新的挑战，并正在成为现代统计学中最重要的研究课题之一。该项目的目的是显著增加用于分析复杂高维数据的可用工具。在这个项目中，PI的目标是实现以下三个目标：（1）在一个统一的数学框架内迎接非参数模型估计和选择的挑战;（2）开发具有所需统计特性的灵活方法和高性能的统计软件来挖掘海量数据;（3）将上述两项活动的研究机会和成果纳入研究生的学科和跨学科统计教育，本科和高中水平。这项研究将拓宽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从业人员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观点。

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.