Covariate Information Matrix for Sufficient Dimension Reduction

Building upon recent research on the applications of the density information matrix, we develop a tool for sufficient dimension reduction (SDR) in regression problems called covariate information matrix (CIM). CIM exhaustively identifies the central subspace (CS) and provides a rank ordering of the reduced covariates in terms of their regression information. Compared to other popular SDR methods, CIM does not require distributional assumptions on the covariates, or estimation of the mean regression function. CIM is implemented via eigen-decomposition of a matrix estimated with a previously developed efficient nonparametric density estimation technique. We also propose a bootstrap-based diagnostic plot for estimating the dimension of the CS. Results of simulations and real data applications demonstrate superior or competitive performance of CIM compared to that of some other SDR methods. Supplementary materials for this article are available online.

本研究立足于近期关于密度信息矩阵（density information matrix）应用的前沿研究，开发了一款面向回归问题的充分降维（Sufficient Dimension Reduction, SDR）工具，命名为协变量信息矩阵（covariate information matrix, CIM）。CIM可全面识别中心子空间（central subspace, CS），并基于回归信息对降维后的协变量进行秩排序。相较于其他主流SDR方法，CIM无需对协变量施加分布假设，亦无需估计均值回归函数。CIM通过对某一矩阵进行特征分解实现，该矩阵由此前开发的高效非参数密度估计技术所构建。此外，本文还提出了一种基于自助法（bootstrap）的诊断图，用于估计中心子空间的维度。模拟实验与真实数据应用结果表明，相较于部分其他SDR方法，CIM的性能更优或具备竞争力。本文的补充材料可在线获取。