Functional Structural Equation Model.

In this article, we introduce a functional structural equation model for estimating directional relations from multivariate functional data. We decouple the estimation into two major steps: directional order determination and selection through sparse functional regression. We first propose a score function at the linear operator level, and show that its minimization can recover the true directional order when the relation between each function and its parental functions is nonlinear. We then develop a sparse functional additive regression, where both the response and the multivariate predictors are functions and the regression relation is additive and nonlinear. We also propose strategies to speed up the computation and scale up our method. In theory, we establish the consistencies of order determination, sparse functional additive regression, and directed acyclic graph estimation, while allowing both the dimension of the Karhunen-Loéve expansion coefficients and the number of random functions to diverge with the sample size. We illustrate the efficacy of our method through simulations, and an application to brain effective connectivity analysis.