Sparse directed acyclic graphs incorporating the covariates

Abstract

Directed acyclic graphs (DAGs) have been widely used to model the causal relationships among variables using multivariate data. However, covariates are often available together with these data which may influence the underlying causal network. Motivated by such kind of data, in this paper, we incorporate the covariates directly into the DAGs to model the dependency relationships among nodal variables. Specifically, the causal strengths are assumed to be a linear function of the covariates, which enhances the interpretability and flexibility of the model. We fit the model in the $$l_1$$ penalized maximum likelihood framework and employ a coordinate descent based algorithm to solve the resulting optimization problem. The consistency of the estimator are also established under the regime where the order of nodal variables are known. Finally, we evaluate the performance of the proposed method through a series of simulations and a lung cancer data example.