Forward Variable Selection for Sparse Ultra-High Dimensional Varying Coefficient Models

ABSTRACT

Varying coefficient models have numerous applications in a wide scope of scientific areas. While enjoying nice interpretability, they also allow for flexibility in modeling dynamic impacts of the covariates. But, in the new era of big data, it is challenging to select the relevant variables when the dimensionality is very large. Recently, several works are focused on this important problem based on sparsity assumptions; they are subject to some limitations, however. We introduce an appealing forward selection procedure. It selects important variables sequentially according to a reduction in sum of squares criterion and it employs a Bayesian information criterion (BIC)-based stopping rule. Clearly, it is simple to implement and fast to compute, and possesses many other desirable properties from theoretical and numerical viewpoints. The BIC is a special case of the extended BIC (EBIC) when an extra tuning parameter in the latter vanishes. We establish rigorous screening consistency results when either BIC or EBIC is used as the stopping criterion. The theoretical results depend on some conditions on the eigenvalues related to the design matrices, which can be relaxed in some situations. Results of an extensive simulation study and a real data example are also presented to show the efficacy and usefulness of our procedure. Supplementary materials for this article are available online.

Keywords:

• BIC
• B-spline
• Independence screening
• Marginal model
• Sub-Gaussion error

Supplementary materials

Section 1: Selection Consistency

Section 2: Proofs

Section 3: Additional simulation results

Acknowledgment

The authors thank the associate editor and three referees for their constructive comments that led to substantial improvement of this article, Dr. Jialiang Li for providing the genetic dataset, and Dr. Wei Dai for sharing the codes for implementation of the methods of Fan, Ma, and Dai (2014).

Funding

This research is partially supported by the Hitotsubashi International Fellow Program and the Mathematics Division, National Center of Theoretical Sciences (Taipei Office). Cheng is supported by the Ministry of Science and Technology grant 101-2118-M-002-001-MY3. Honda is supported by the JSPS Grant-in-Aids for Scientific Research (A) 24243031 and (C) 25400197. Zhang is supported by the National University of Singapore research grant R-155-000-128-112.