Abstract
We provide a comprehensive review of traditional and modern
approaches to estimation and model selection in the linear
regression framework. We are particularly interested in methods
that estimate regression coefficients under various constraints,
and the impact these constraints have on the resulting coefficient
estimates and models selected. We propose a novel approach that
allows for a more flexible penalty structure, and provide an
estimation algorithm that utilizes linear programming. Finally, our
flexible estimator is illustrated in various applications that
exhibit spatial structure.
Table of Contents
- Traditional and Modern Approaches to Estimation and Model
Selection in Linear Regression
- Introduction
- Notation
- Linear Regression (ordinary least squares)
- Definition
- Estimation
- Variable Selection
- Shortcomings
- Penalized Linear Regression
- Definition
- Ridge Regression (L2)
- Lasso (L1)
- Motivation
- Definition
- Computation
- Theoretical Details
- Choosing Lambda
- Drawbacks
- L1 Extensions
- Adaptive Lasso
- Elastic Net
- Group Lasso
- Fused Lasso
- Lasso and Generalized Linear Models
- Results
- Illustrations
- Simulations
- Description of Scenarios
- Simulation 1
- Simulation 2
- Discussion
- A Flexible Dantzig Selector
- Introduction
- Background
- Generalized Lasso
- Definition
- Estimation
- Incorporating a Ridge Penalty (L2)
- Penalty Matrix, M
- Shortcomings
- Discussion
- Dantzig Selector
- Definition
- Properties
- Computation
- Comparisions with Lasso
- Extensions
- Methods
- Flexible Dantzig Selector
- Motivation
- Definition
- Flexible Penalty Matrix (M)
- Computation
- Speed Tests
- Variant 1: Proportional Weighting
- Variant 2: Adaptive Weighting
- Variant 3: Weighting Observations
- Bootstrap-Enhanced Estimation
- Randomized Estimation
- Results
- Illustrations
- Simulations
- Description of Scenarios
- Simulation - Goal 1
- Simulation - Goal 2
- Data Analysis
- Spatially-Informed Test Statistic Thresholding
- The Alzheimer's Disease Neuroimaging Initiative (ADNI)
- Discussion
- Conclusion
About this Master's Thesis
Rights statement
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