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.