The Local Influence Analysis Of Direction Regression

Dimension reduction of high-dimensional data is a very important issue in the nonparametric regression.Sufficient dimension reduction is a method of dimension reduction finding out a small number of linear combination of the original independent variables without loss of regression information.In the current dimension reduction theory,directional regression is a popular method.The principle of directional regression is natural and simple,which integrates the dimension reduction methods based on the first and second order conditional expectations.This method is accurate with slight computational burden.Hence,it is extensively applied to nonparametric regression.In this thesis,based on the joint perturbation to data points,a local influence analysis method for directional regression is proposed.Specifically,on the basis of the concepts such as space displacement function,influence graph,perturbation direction,lifted line and quasi-curvature,the expression of the local influence assessment statistic for the data points is constructed under the directional regression.Here,the space displacement function is used to measure the discrepancy between the perturbed and unperturbed estimates of the dimension reduction space.The quasi-curvature is used to assess the change of the dimension reduction space estimate caused by a minor perturbation along some direction near the position of non-perturbation,and the maximizer of the quasi-curvature is use to construct the influence assessment statistic.In this process,we need to deduce the expression of the quasi-curvature that is a quadratic form of the perturbation direction vector,based on the matrix differentiations and perturbation theory of the eigenvectors.The proposed method is illustrated by a simulation study.The practical application is illustrated by an analysis of a dataset about the identification of handwritten digit.