Statistical Modeling with High-dimensional Data: Variable Selection and Regularization

对于高维数据，首选简约模型，因为它们更具可解释性，同时减少预测误差。在大多数数据分析的现代发展中，正则化也是一个重要的组成部分，特别是当预测器的数量很大时。非正则拟合肯定会得到严重过拟合和无用的模型。研究人员对高维统计建模中的变量选择问题采用正则化方法，使所得到的模型具有良好的预测精度，同时具有稀疏表示。具体而言，研究人员开发了：(1)蛋白质组学数据分析中新的融合变量选择方法，已成为革命性的癌症诊断工具；(2)自动采用支持向量表示的核逻辑回归模型；(3)在估计多分位数回归函数中进行同时变量选择的几种新技术。研究者还研究了这些新的变量选择技术的理论。高效的算法和软件被开发出来供公众使用。现代科学创新使科学家能够收集大量高维数据。从大量的数据中提取有用的信息对科学调查是至关重要的。因此，变量选择和降维在高维统计建模中起着至关重要的作用。变量选择问题出现在广泛的领域，机器学习，药物发现，生物标志物发现，遗传学，蛋白质组学，脑成像分析，金融建模，环境科学，仅举几例。该研究项目旨在开发最先进的统计工具，帮助各个领域的研究人员分析他们的数据。

With high-dimensional data parsimonious models are preferred because they are much more interpretable and at the same time reduce prediction errors. Regularization is also an essential component in most modern developments for data analysis, in particular when the number of predictors is large. Non-regularized fitting is guaranteed to give badly over-fitted and useless models. The investigators take a regularization approach to the variable selection problem in high-dimensional statistical modeling such that the resulting model enjoys excellent prediction accuracy and at the same time has a sparse representation. In particular, the investigators develop: (1) new fused variable selection methods in proteomics data analysis which has been arevolutionary cancer diagnostic tool; (2) a novel kernel logistic regression model which automatically adopts a support-vector representation; (3) several new techniques for performing simultaneous variable selection in estimating multiple quantile regression functions. The investigators also study the theory of these new variable selection techniques. Efficient algorithms and software are developed for public use.Modern scientific innovations allow scientists to collect massive and high-dimensional data. It is critical in scientific investigations to extract useful information from the huge amount of data. For this reason, variable selection and dimension reduction play a fundamental role in high-dimensional statistical modeling. Variable selection problems arise from a wide range of fields, machine learning, drug discovery, biomarker finding, genetics, proteomics, brain imaging analysis, financial modeling, environmental sciences, to name a few. The research project aims to develop state-of-the-art statistical tools that help researchers in various fields to analyze their data.