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Title: Reweighting methods in high dimensional regression
Author: Fang, Zhou
ISNI:       0000 0004 2745 4491
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2012
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In this thesis, we focus on the application of covariate reweighting with Lasso-style methods for regression in high dimensions, particularly where p ≥ n. We apply a particular focus to the case of sparse regression under a-priori grouping structures. In such problems, even in the linear case, accurate estimation is difficult. Various authors have suggested ideas such as the Group Lasso and the Sparse Group Lasso, based on convex penalties, or alternatively methods like the Group Bridge, which rely on convergence under repetition to some local minimum of a concave penalised likelihood. We propose in this thesis a methodology that uses concave penalties to inspire a procedure whereupon we compute weights from an initial estimate, and then do a single second reweighted Lasso. This procedure -- the Co-adaptive Lasso -- obtains excellent results in empirical experiments, and we present some theoretical prediction and estimation error bounds. Further, several extensions and variants of the procedure are discussed and studied. In particular, we propose a Lasso style method of doing additive isotonic regression in high dimensions, the Liso algorithm, and enhance it using the Co-adaptive methodology. We also propose a method of producing rules based regression estimates for high dimensional non-parametric regression, that often outperforms the current leading method, the RuleFit algorithm. We also discuss extensions involving robust statistics applied to weight computation, repeating the algorithm, and online computation.
Supervisor: Meinshausen, Nicolai Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Computer science (mathematics) ; Statistics (see also social sciences) ; machine learning ; statistical methodology