Improved Elastic Net Penalty And Adaptive Elastic Net Regression Based On Residual Bayesian Bootstrap

The problem of feature selection and modeling prediction of high-dimensional data is a great challenge in academic research and engineering practice.The penalized likelihood regression model represented by（adaptive）elastic net model is a powerful tool to solve this problem.Theory and practice show that the adaptive elastic net model has good performance in feature selection,grouping variable identification,coefficient estimation and so on.Under certain regularization conditions,the adaptive elastic net estimators of regression coefficients are proved to have consistency and asymptotic normality.However,how to approximate the sampling distribution of adaptive elastic net estimators based on limited samples is rarely studied.The purpose of this thesis is to extend the elastic net model to obtain a generalized model with lower coefficient estimation deviation.At the same time,the bootstrap distribution of the adaptive elastic net estimators is constructed by using the Bayesian bootstrap residual method to approximate its sampling distribution,and based on this,the confidence intervals of the regression coefficients are calculated.Firstly,the range of L₂ regularization parameters of elastic net model is appropriately extended to obtain the generalized model,and the conditions for the existence of solutions of the generalized model are established.The generalized model can improve the bias of lasso estimators and ordinary elastic net estimators.Theoretically,the generalized model is proved to have grouping effect.Through simulation experiments,we can verify that the generalized model has better performance in coefficient estimation,feature selection and prediction accuracy.Secondly,the Monte Carlo simulation of bootstrap distribution of adaptive elastic net estimators can be obtained by Bayesian bootstrap on the least squares residual,so as to approximate the sampling distribution of adaptive elastic net estimators.Theoretically,the adaptive elastic net estimators of regression coefficients obtained based on Bayesian bootstrap residual samples are proved to be consistent and asymptotically normal,which proves that the method of approximating its sampling distribution by its bootstrap distribution is asymptotically effective in the sense of Mallows Distance.The superiority and effectiveness of the residual Bayesian bootstrap method are verified by simulation experiments.