Some Extensions For Multi-response Sufficient Dimension Reduction Methods

With the development of information age,data tends to be high-dimensional and complex,and traditional data analysis methods are facing huge difficulties.In non-parametric regression,high-dimensional data seriously affects the result of regression.Therefore,preserving its sample information while reducing the dimension of high-dimensional data as much as possible is an important research topic.Sufficient Dimension Reduction(SDR)was proposed under this background.The main idea of SDR is to find the combinations of the independent variables as few as possible which can represent the sample information.In recent years,SDR has also been extended to the case where the response variable is multiple.The latter one can use the information of the response variable more comprehensive to achieve a more accurate estimation of the dimension reduction space.However,few people have discussed how different response variables affect independent variables.Unreasonable setting of response variable may cause insufficient use of the key information and affect the dimension reduction result for independent variables,and even make it worse.In multiple multivariate regression,a new sufficient dimension reduction method based on the Minimum Average Variance Estimation(MAVE)is proposed in this paper,called Minimum Weight Average Variance Estimation(MWAVE).In the case that the independent variable and the response variable are both multivariate,MWAVE estimates the dimension reduction space by a iterative least squares algorithm.The key process is calculating the outer product of gradient of Y's objective function and estimating the weight vector and weighting parameter for the multivariate response Y.The estimation methods of dimension reduction space and structure dimension are also given in this paper.The new method can reduce the dimensions of both response variable and independent variable at the same time without any assumptions about the variable.Through the simulation,it is found that MWAVE can seek the response variable which is most relevant to the independent variable.We also find that MWAVE is better than the existing multiple multivariate regression dimension reduction methods in estimating the dimension reduction space,especially in small sample.