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数据集中提取尽可能多的信息。然而，在过去十年中，由于图像处理，计算生物学，气候学，经济学和金融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处理具有大量预测因子的数据集。这种大规模的问题可以抽象为统计回归和分类问题，解释变量的数量远远大于观察的数量。在这些情况下，某种形式的正则化是必不可少的。研究人员研究了一类通用的惩罚函数，以及回归和分类设置中正则化方法的理论特性。此外，开发了两个具体的惩罚函数，每个激励不同的方法。这些方法在最常见的线性回归设置的理论和经验性质进行了研究。最后，研究人员研究扩展的方法，在高维设置，即，混合效应模型，功能线性回归，和classification problems.拟议的研究预计将有广泛的影响，统计的实践和教育，以及统计以外的领域。整个提议的共同主题是为高维问题开发一般正则化惩罚和相关方法。研究人员在统计学之外的许多领域都有直接联系，如计算生物学，金融，市场营销，机器学习和计量经济学。研究人员将系统地开发软件，通过自由软件包（如R）实现所提出的方法，然后使它们随时可用，并在所有这些领域推广。随着高维数据的日益普遍，所开发的方法和软件将得到广泛的应用。该研究还将有助于培训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学之外的研究人员）。

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).