Functional Graphical Models

Graphical models have attracted increasing attention in recent years, especially in settings involving high-dimensional data. In particular, Gaussian graphical models are used to model the conditional dependence structure among multiple Gaussian random variables. As a result of its computational efficiency, the graphical lasso (glasso) has become one of the most popular approaches for fitting high-dimensional graphical models. In this paper, we extend the graphical models concept to model the conditional dependence structure among p random functions. In this setting, not only is p large, but each function is itself a high-dimensional object, posing an additional level of statistical and computational complexity. We develop an extension of the glasso criterion (fglasso), which estimates the functional graphical model by imposing a block sparsity constraint on the precision matrix, via a group lasso penalty. The fglasso criterion can be optimized using an efficient block coordinate descent algorithm. We establish the concentration inequalities of the estimates, which guarantee the desirable graph support recovery property, that is, with probability tending to one, the fglasso will correctly identify the true conditional dependence structure. Finally, we show that the fglasso significantly outperforms possible competing methods through both simulations and an analysis of a real-world electroencephalography dataset comparing alcoholic and nonalcoholic patients.