A varying coefficient model with matrix valued covariates

Abstract

Modern data are often collected in a matrix form. In this paper, we consider modelling the varying coefficient regression with matrix valued covariate X and scalar index variable U. The proposed model simultaneously makes principal component analysis for both the row and column dimensions of the matrix objects, maintaining the matrix structure while achieving substantial dimension reduction. We develop an iterative estimation method for the involved principal parameters and nonparametric functions. Under regularity conditions, the asymptotic distributions of the estimators are derived. In addition, by incorporating the estimation with the adaptive group Lasso and the group SCAD penalties, variables of X in entire rows or columns are selected. The proximal gradient algorithm is further utilised to solve the regularised optimisation problems. The asymptotic properties of the penalised estimators are also studied. Our model and estimation methods are demonstrated by simulated experiments. Real applications to the primary biliary cirrhosis (PBC) data reveal that the effects of the blood measurements to the survival time vary with levels of age.

KEYWORDS:

• Varying coefficient model
• matrix covariates
• principal component analysis
• variable selection
• proximal gradient algorithm

Acknowledgments

The author thanks the editor, associate editor and referees for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

H.-F. Zhang is partially supported by National Natural Science Foundation of China, [grant numbers 12201521,72033002] and Fundamental Research Funds for the Central Universities, [grant number 2682022CX049].