Graphical models have proven to be a useful tool in understanding the conditional depen-dency structure of multivariate distributions. In this thesis, we consider two types of undirectedgraphical models that are motivated by particular types of applications. The First model weconsider is a mixed graphical model, linking both continuous and discrete variables. The pro-posed model is simple enough to be suitable for high-dimensional data, yet exible enoughto represent all possible graph structures. We develop a computationally efficient regression-based algorithm for fitting the model by focusing on the conditional log-likelihood of eachvariable given the rest. The parameters have a natural group structure, and sparsity in thefitted graph is attained by applying group lasso penalty which can be approximated by aweighted L1 penalty. We demonstrate the effectiveness of our method through an extensivesimulation study and apply it to a music annotation data set (CAL500), obtaining a sparseand interpretable graphical model. The second model we consider is a covariate dependentIsing model which studies the conditional dependency within the binary data and how theyare dependent on the additional covariates. This results in subject-specific Ising models, wherethe subjects;; covariates inuence the strength of association between the binary variables. Weadd L1 penalty to induce sparsity in the fitted graphs and in the number of selected covariates.Two algorithms to fit the model are proposed and compared on a set of simulated data, andasymptotic results are established. The results on a study of genes involved in breast cancerand their biological significance are discussed.1
PDF
download
878987797
028-85220240
