A Forward Approach to Estimating Effective Dimension Reduction Space in High-dimensional Regression

• Activity Number: 276
• Type: Contributed
• Date/Time: Tuesday, August 10, 2004 : 2:00 PM to 3:50 PM
• Sponsor: General Methodology
• Author(s): Peng Zeng*+ and Yu Zhu
• Companies: Purdue University and Purdue University
• Keywords: dimension reduction ; e.d.r subspace ; SIR ; SAVE ; regression

Abstract

Dimension reduction is crucial to the success of high dimensional regression. Suppose Y is the response and x is the d-dimensional input vector. It is reasonable to assume that Y depends on x through k linear combinations, i.e., Y=f(b_1 x,.,b_k x,e), where e is the random error and f the unknown link function. The space spanned by b_1, b_2, ., b_k is called the effective dimension reduction (e.d.r) space. Two approaches exist to recover the e.d.r space, namely the forward approach and the inverse approach. The former involves the fitting of f, thus can become infeasible even when d is moderate. The latter has led to several methods such as SIR and SAVE, which successfully avoid the fitting of f. Due to their inverse fashion, some disadvantages exist. We propose a novel method based on the forward approach. Our method can also avoid the fitting of f (thus it is link-free) and the curse of dimensionality, and is more transparent than SIR and SAVE. Theoretical results have been established under the model assumption. Simulation results using synthetic and real data will be reported to compare our method with SIR and SAVE.