Dimension Reduction for Longitudinal Data Based on Martingale Difference Divergence

Within-subject correlation and correlation among variables are two inherent characteristics of longitudinal datasets, which contain lots of important data information. In order to use these two kinds of correlation for dimension reduction, in this paper, we propose a sufficient dimension folding method based on martingale difference divergence in the spirit of dimension folding of matrix-valued data. It can be shown that the method can find the central mean dimension folding subspace in the population level, and can reduce the dimensions of both predictors and observation times simultaneously. Further, the estimated basis directions ensures the root-n consistency. To implement the proposed method, the Kronecker product assumption is introduced, so that the process can be transformed to a constrained low-dimensional optimization problem, which can be quickly solved by exisiting nonlinear optimization algorithms. Furthermore, a consistent BIC criterion is proposed to determine the structural dimension. Simulation studies show that the proposed method is efficient and can have higher accuracy on subspace estimation and structural dimension determination. Finally, an application on primary biliary cirrhosis data is used to illustrate the effectiveness of the proposed method.