Exploring the Application of Penalized Regression Techniques in Sparse and High-Dimensional Statistical Problems

Posted: Jul 19, 2015

• Julian West
• Katherine Brooks
• Kayla Hayes

Abstract

The proliferation of high-dimensional data across scientific disciplines has created unprecedented challenges for traditional statistical methods. In fields ranging from genomics to finance, researchers routinely encounter datasets where the number of potential predictors (p) vastly exceeds the number of observations (n). This p >> n scenario renders conventional regression techniques inapplicable due to identifiability issues and overfitting concerns. Penalized regression methods have emerged as powerful tools for addressing these challenges by imposing constraints on model complexity while performing variable selection and parameter estimation simultaneously. Traditional approaches such as LASSO (Least Absolute Shrinkage and Selection Operator) and ridge regression have demonstrated considerable success in high-dimensional settings. However, these methods exhibit limitations when dealing with complex correlation structures among predictors or when domain knowledge suggests specific relationships between variables. The LASSO tends to select at most n variables when p > n and may arbitrarily select one variable from a group of highly correlated predictors. Ridge regression, while providing stable coefficient estimates, does not perform variable selection, resulting in models that lack interpretability in high-dimensional contexts. This research addresses these limitations by developing a novel adaptive regularization framework that integrates structural information into the penalty function. Our approach extends beyond conventional penalized regression by incorporating domain-specific constraints that reflect known relationships among predictors.