Collaborative Research: Penalized Methods for Variable Selection and Estimation in High-Dimensional Models

本计画研究高维模型中变量选择与估计的惩罚方法。拟合高维模型的一般方法是使用正则化惩罚。一些重要的惩罚变量选择和估计方法已经被提出，但这些方法的性质还没有得到系统的研究。为了在科学研究中应用这些方法，重要的是要了解它们的属性。特别是，重要的是要知道在什么条件下，这些方法正确地选择了重要的变量，并以有效的方式估计其影响。评估统计程序的标准方法假设模型中的变量数量是固定的，并且比样本量小得多。该公式不适用于高维模型。分析高维模型的问题提出了新的和具有挑战性的理论问题，在数理统计。目前使用惩罚的变量选择方法假设了一种已知形式的统计模型，这可能是对现实的歪曲。如果参数模型被错误指定或没有对模型进行参数假设，那么调查会发生什么是很重要的。特别是，重要的是要知道是否有条件下，惩罚的方法正确地选择变量，尽管错误的规格和在什么条件下错误的规格导致他们产生误导性的结果。将惩罚方法推广到非参数和半参数模型也是很重要的。高维数据在许多重要的应用中出现，特别是生物和生物医学研究。随着生物技术的迅速发展，越来越多的大型数据集正在生成。从高维和噪声数据集中识别统计和生物学上有意义的模式是一个重大挑战。研究人员将拟议的研究应用于全基因组关联（GWA）分析，拷贝数变异（CNV）检测以及基因表达谱的删失生存数据分析。CNV的GWA分析和检测使得能够鉴定负责疾病（例如许多形式的癌症）的发展和进展的基因和途径。将基因表达谱与存活率相关联是有用的，因为存活率可能是许多癌症研究中最重要的临床终点。在估计临床结果与遗传和基因组数据之间的关系时，可以处理高维问题的统计方法的发展有助于更好地理解疾病的遗传基础，更好的诊断和更好的生存预测。

This project studies penalized methods for variable selection and estimation in high-dimensional models. A general approach for fitting high-dimensional models is to use regularization penalties. Several important penalized methods for variable selection and estimation have been proposed, but the properties of these methods have not been systematically studied. To apply the methods in scientific investigations, it is important to understand their properties. In particular, it is important to know under what conditions, the methods correctly select the important variables and estimate their effects in an efficient way. Standard methods for evaluating a statistical procedure assume that the number of variables in a model is fixed and much smaller than the sample size. This formulation is not applicable to high-dimensional models. The problem of analyzing high-dimensional models presents novel and challenging theoretical questions in mathematical statistics. Current variable selection methods using penalties assume a known form of the statistical model, which can be a misrepresentation of the reality. It is important to investigate what happens if a parametric model is misspecified or if no parametric assumptions are made about the model. In particular, it is important to know whether there are conditions under which penalized methods select variables correctly despite misspecification and under what conditions misspecification causes them to yield misleading results. It is also important to extend the penalized methods to nonparametric and semiparametric models.High-dimensional data arise in many important applications, notably biological and biomedical investigations. With rapid advances in biotechnology, more and more large data sets are being generated. The identification of statistically and biologically significant patterns from high-dimensional and noisy data sets is a major challenge. The investigators apply the proposed research to genome-wide association (GWA) analysis, detection of copy number variation (CNV), and analysis of censored survival data with gene expression profiles. GWA analysis and detection of CNV enable the identification of genes and pathways responsible for the development and progression of a disease, such as many forms of cancer. Correlating a gene expression profiles with survival is useful, because survival is perhaps the most important clinical endpoint in many cancer studies. The development of statistical methods that can deal with high-dimensional problems in estimating the relationship between clinical outcomes and genetic and genomic data contribute to better understanding of the genetic basis of diseases, better diagnoses, and better survival prediction.