Regularized Statistical Methods for Data of Grouped or Dynamic Nature

High-dimensional;Dynamic System;Variable Selection;Convex Regularization;Ordinary Differential Equation;Time-varying;Statistics and Numeric Data;Science;Statistics

Li, YunShedden, Kerby A. ;

University of Michigan

关键词: High-dimensional;    Dynamic System;    Variable Selection;    Convex Regularization;    Ordinary Differential Equation;    Time-varying;    Statistics and Numeric Data;    Science;    Statistics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/93831/yrlee_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

This dissertation consists of two parts. In the first part,one new convex regularized variable selection method is proposedfor high-dimensional grouped data. Existing group variable selection methods via convex penalties, such as Yuan and Lin (2006) and Zhao et al. (2009), have the limitation of selecting variables in an ``all-in-all-out;;;; fashion and lack of selection flexibility within a group. In Chapter II, we propose a new group variable selection method via convex penalties that not only removes unimportant groups effectively, but also keeps the flexibility of selecting variables within an important group. Both the efficient numerical algorithm and high-dimensional theoretical estimation bounds are provided. Simulation resultsindicate that the proposed method works well in terms of bothvariable selection and prediction accuracy.In the second part of the dissertation,we develop the parameter estimation methodsfor the dynamic ordinary differential equations (ODEs).Ramsay et al. (2007) proposed a popular parameter cascading method that tries to strike a balance between the data and the ODE structure via a ``loss + penalty;; framework. In Chapter III, we investigate this method in detail and take an alternative view through variance evaluation on it.We found, through both theoretical evaluation and numerical experiments, that the penalty term in Ramsay et al. (2007) could unnecessarilyincrease estimation variation. Consequently, we propose a simpleralternative structure for parameter cascading that achieves theminimum variation.We also provide theoretical explanations behind the observed phenomenon and report numerical findings on both simulations and one real dynamic data set.In Chapter IV, we consider the estimation problem with time-varying ODE parameters. This is often necessary when there are unknown sources of disturbances that lead to deviations from the standard constant-parameter ODE system. To keep the structure of the parameters simple, we propose a novel regularization method for estimating time-varying ODE parameters. Our numerical studies suggest that the proposed approach works better than competing methods.We also provide finite-sample estimation error bounds under certain regularity conditions. Thereal-data applications of the proposed method lead to satisfactory and meaningful results.