Sparse dimension reduction with group identification

Estimating the central mean subspace without requiring to designate a model is achieved via MAVE method. The original p predictors are replaced with d-linear combinations (LC) of predictors in MAVE, where d<p without loss of any information about the regression. However, it is known that the interpretation of the estimated effective dimension reduction (EDR) direction is not easy due to each EDR direction is a LC of all the original predictors. The PACS method is an oracle procedure. In this article, a group variable selection method (SMAVE-PACS) is proposed. The sufficient dimension reduction (SDR) concepts and group variable selection are emerged through SMAVE-PACS. SMAVE-PACS produces sparse and accurate solutions with the ability of group identification. SMAVE-PACS extended PACS to multi-dimensional regression under SDR conditions. In addition, a method for estimating the structural dimension was proposed. The effectiveness of the SMAVE-PACS is checked through simulation and real data.