Functional directed graphical models and applications in root-cause analysis and diagnosis

Directed graphical models aim to represent the probabilistic relationships between variables in a system. Learning a directed graphical model from data includes parameter learning and structure learning. Several methods have been developed for directed graphical models with scalar variables. However, the case in which the variables are infinite-dimensional has not been studied thoroughly. Nowadays, in many applications, the variables are infinite-dimensional signals that need to be treated as functional random variables. This article proposes a novel method to learn directed graphical models in the functional setting. When the structure of the graph is known, function-to-function linear regression is used to estimate the parameters of the graph. When the goal is to learn the structure, a penalized least square loss function with a group LASSO penalty, for variable selection, and an L2 penalty, to handle group selection of nodes, is defined. Cyclic coordinate accelerated proximal gradient descent algorithm is employed to minimize the loss function and learn the structure of the directed graph. Through simulations and a case study, the advantage of the proposed method is proven.