Nonparametric and high-dimensional functional graphical models

Abstract

We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear conditional means. In this article, we provide a more flexible model which relaxes the linearity assumption by replacing it by an arbitrary additive form. The use of functional principal components offers an estimation strategy that uses a group lasso penalty to estimate the relevant edges of the graph. We establish statistical guarantees for the resulting estimators, which can be used to prove consistency if the dimension and the number of functional principal components diverge to infinity with the sample size. We also investigate the empirical performance of our method through simulation studies and a real data application

The completion of covariance kernels

“…They exhibit a complex variety in the details of the structural assumptions they make. As an interesting development, we mention here Solea and Dette (2022), but a comprehensive review of these different approaches is beyond the scope of this article.…”