PHD Math Defence: Doubly Sparse Regularized Regression Incorporating Graphical Structure Among Predictors

CANDIDATE:   MATTHEW STEPHENSON

ABSTRACT:

Recent research has demonstrated that information learned from building a graphical model on the predictor set of a regularized linear regression model can be leveraged to improve prediction of a continuous outcome.  This thesis proposes the doubly sparse regression incorporating graphical structure among predictors (DSRIG) model, and its logistic regression counterpart, doubly sparse logistic regression incorporating graphical structure among predictors (DSLRIG).  In general, the regularization scheme of these models works by building an undirected graph over the predictor set and then using the resulting neighbourhoods of the graph to form a set of (overlapping) groups.  Sparsity is encouraged both within and among the groups that contribute to the overall estimation of the regression parameters.  Together, DSRIG and DSLRIG provide a unified framework for the fitting of many other commonly used regularization schemes. 

In this thesis, a combination of simulation and analysis of real world data are used to evaluate and compare model performance. Ultimately, the DSRIG and DSLRIG models improve outcome prediction and parameter estimation compared to previously proposed methods.  A finite sample error bound is derived for DSRIG in the case of a quantitative outcome and predictors distributed as multivariate normal.  Guidelines for the implementation of DSRIG in the analysis of real world data are also provided.