Collaborative Research: Penalization Methods for Screening, Variable Selection and Dimension Reduction in High-Dimensional Regression via Multiple Index Models

该项目旨在为高维回归中的筛选、降维和变量选择开发有效的惩罚方法。研究人员主要关注多指标模型，因为这种类型的模型结合了线性和非参数回归的优点，同时避免了它们的缺点。采用一种新的惩罚方法进行模型拟合，对多指标模型的参数和非参数成分进行正则化。一项初步研究表明，这种方法比其他现有方法更有利。当面对超高维度时，研究者在应用建议的惩罚之前使用前向变量筛选程序将维度降低到可管理的大小。研究人员计划研究这种方法的理论性质，并为其实现开发快速有效的计算算法。所提出的方法进一步扩展到涉及分类反应或随机效应的应用。科学技术的进步导致了生物信息学、气候研究、互联网等各个领域的海量数据的爆炸式增长。传统的聚类、回归和分类统计方法在处理大量变量时变得无效。近年来，大量的研究工作致力于开发诸如降维和变量选择等统计方法来分析这类海量数据。研究人员提出了一种新的惩罚方法，并开发了高效的计算算法。该项目的结果不仅推动了统计研究，而且还帮助其他科学家和研究人员更好地理解和分析他们的大量数据，从而加强他们的科学发现。

The project aims to develop effective penalization methods for screening, dimension reduction, and variable selection in high dimensional regression. The investigators focus mainly on multiple index models, because this type of models combines the strengths of linear and nonparametric regression while avoiding their drawbacks. A novel penalization approach is employed for model fitting, which regularizes both the parametric and nonparametric components of a multiple index model. A pilot study shows that this approach is more advantageous than other existing ones. When facing ultra-high dimensionality, the investigators use a forward variable screening procedure to reduce the dimension to a manageable size before applying the proposed penalization. The investigators plan to study the theoretical properties of this approach and develop fast and efficient computing algorithms for its implementation. The proposed approach is further extended to applications involving categorical responses or random effects.Advances in science and technology have led to an explosive growth of massive data across a variety of areas such as bioinformatics, climate research, internet, etc. Traditional statistical methods for clustering, regression and classification become ineffective when dealing with a large number of variables. Lately, a tremendous amount of research effort has been dedicated to the development of statistical methods such as dimension reduction and variable selection for analyzing this type of massive data. The investigators join the effort by proposing a novel penalization approach and developing efficient computing algorithms. The results from this project not only advance statistical research but also help other scientists and researchers better understand and analyze their massive data and hence enhance their scientific discovery.