Robust Dimension Reduction Based On MCD Method In Sufficient Dimension Reduction

Dimension reduction is one of the important stages of high dimensional data analysis.Sufficient dimension reduction is to find small number of linear combinations of the original predictors such that using them to replace the original predictors in regression will not lead to any loss of the information.The sliced inverse regression,sliced average variance estimate and principal Hessian directions are all classical sufficient dimension reduction methods.However,they all depend on some strict assumptions about the distribution of the predictors.Hence,both the observations with extremely outlying values of predictors(extremely high-leverage points)and the ones with big errors of regression(outliers)may be harmful to the results of the sufficient dimension reduction when the above methods are used.Therefore,in order to avoid the threat from these data points,thesis improves the robustness of the above sufficient dimension reduction methods and proposes robust dimension reduction methods with the fast robust algorithms designed,based on the idea of minimum covariance determinant.For the above three classical sufficient dimension reduction methods,their robust versions are provided:minimum covariance inverse regression estimation(FMSIR),minimum covariance mean variance estimation(FMSAVE)and minimum covariance principal Hessian direction estimation(FMPHD).The above robust dimension reduction methods based on FAST-MCD algorithm detect not only the outliers with big errors but also the extremely high-leverage points which are harmful to the sufficient dimension reduction.After deleting the detected outliers and high-leverage points,the classical dimension reduction methods are used.To enhance the efficiency of detection of the outliers and high-leverage points,the detection and data trimming are made for not only the original data but also the data after dimension reduction,which are put in an iterative process.In the thesis,the three algorithms of robust dimension reduction are given.In the numerical simulation,the performance of robust dimension reduction estimation is better than that of classical dimension reduction methods when there exist outliers.In addition,we apply the proposed method to the Istanbul stock exchange data set and get better dimension reduction results.