Regularization Methods in High Dimensions with Applications to Functional Data Analysis, Mixed Effects Models and Classification

从历史上看，统计学处理的问题是从一个小的数据集中提取尽可能多的信息。然而，在过去的十年里，由于图像处理、计算生物学、气候学、经济学和金融学等各个领域的技术进步，统计学中最重要的活跃研究课题之一现在涉及到处理具有大量预测因子的数据集。这样的大规模问题可以抽象为统计回归和分类问题，其解释变量的数量远远大于观测的数量。在这些情况下，某种形式的正规化是必不可少的。研究人员研究了一类一般的惩罚函数，以及在回归和分类环境下所产生的正则化方法的理论性质。此外，还开发了两个具体的惩罚函数，每个函数都采用了不同的方法。研究了这些方法在最常见的线性回归环境下的理论和经验性质。最后，研究人员将研究方法扩展到高维环境中较少探索的领域，即混合效应模型、函数线性回归和分类问题。预计拟议的研究将对统计学实践和教育以及统计学以外的领域产生广泛影响。整个提案的共同主题是为高层面问题制定一般性的正规化处罚和相关方法。这些研究人员在统计学之外的许多领域都有直接的联系，如计算生物学、金融学、营销学、机器学习和计量经济学。调查人员将系统地开发软件，通过R等自由软件包实施所提出的方法，然后使它们随时可用，并在所有这些领域宣传它们。高维数据变得越来越普遍，因此开发的方法和软件将得到广泛应用。这项研究还将有助于培养和发展未来的数据分析师(包括统计学家和分析数据的统计之外的研究人员)。

Historically statistics has dealt with the problem of extracting as much information as possible from a small data set. However, over the last decade, because of technological advances in various fields such as image processing, computational biology, climatology, economics and finance, one of the most important active research topics in statistics now involves dealing with data sets with enormous numbers of predictors. Such large scale problems may be abstracted as statistical regression and classification problems with the number of explanatory variables much larger than the number of observations. In these situations some form of regularization is essential. The investigators study a general class of penalty functions and the theoretical properties of the resulting regularization methods in regression and classification settings. In addition, two specific penalty functions that each motivate a different methodology are developed. The theoretical and empirical properties of these methods in the most common linear regression setting are investigated. Finally, the investigators study extending the methodologies to areas that are less well explored in the high dimensional setting, namely, mixed effects models, functional linear regression, and classification problems.The proposed research is expected to have a broad impact on the practice and education, both of statistics, as well as on fields outside statistics. The common theme underlying this entire proposal is that of developing general regularization penalties and related methodologies for high dimensional problems. The investigators together have direct connections in many fields outside statistics such as Computational Biology, Finance, Marketing, Machine Learning, and Econometrics. The investigators will systematically develop software to implement the proposed methods through free software packages, like R, and then make them readily available and publicize them in all these fields. High dimensional data are becoming increasingly common, so the developed methodologies and software will be widely utilized. The research will also contribute to the training and development of future data analysts (including both statisticians and researchers outside statistics who analyze data).