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

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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).