부산대학교 도서관

Graphical models have established themselves as fundamental tools through which to understand complex relationships in high-dimensional datasets. Applications abound, a pertinent example being neuroimaging where Gaussian graphical models are employed to model statistical dependencies across spatially remote brain regions. Often such models are estimated under regularization penalties which help to enforce properties such as sparsity. Much of the current methodology is rooted on the assumption that the same covariance structure characterizes all observations and may be summarized using a single graphical model. However, such an assumption is untenable in the context of many applications. In order to address this issue, we propose a host of algorithms through which to accurately estimate Gaussian graphical models in the context of data with heterogeneous covariance structure. Formally, this thesis is focused in studying graphical models in two distinct manifestations of heterogeneity in covariance structure. The first relates to the task of estimating time-varying graphical models and two algorithms are proposed to this end. A related challenge is associated with the choice of regularization parameter: in the presence of variable covariance structure such a parameter is both difficult to estimate and potentially time-varying itself. In order to address these challenges, a novel framework is proposed through which to iteratively tune this parameter. The second manifestation is related to the presence of heterogeneity in covariance structure across multiple related graphical models. In such a setting scientific objectives consist in inferring the covariance structure shared across all graphs together with the idiosyncrasies of each specific graph. A related objective which is often overlooked consists of quantifying variability across all graphs. A novel algorithm is proposed through which to simultaneously address the aforementioned objectives.