Ensemble sufficient dimension folding methods for analyzing matrix-valued data

The construction of novel sufficient dimension folding methods for analyzing matrix-valued data is considered. For a matrix-valued predictor, traditional dimension reduction methods fail to preserve the matrix structure. However, dimension folding methods can preserve the data structure and improve estimation accuracy. Folded-outer product of gradient (folded-OPG) ensemble estimator and two refined estimators, folded-minimum average variance estimation (folded-MAVE) ensemble and folded-sliced regression (folded-SR) ensemble are proposed to recover central dimension folding subspace (CDFS). Due to ensemble idea, estimation accuracies are improved for finite samples by repeatedly using the data. A modified cross validation method is used to determine the structural dimensions of CDFS. Simulated examples demonstrate the performance of folded ensemble methods by comparing with existing inverse dimension folding methods. The efficacy of folded-MAVE ensemble method is also evaluated by comparing with inverse dimension folding methods for analyzing the Standard & Poor’s 500 stock data set.