The Study And Application Of Robust Sparse Principal Component Regression

Principal component regression is a multivariate regression method based on principal component analysis.It can effectively avoid multiple colinearity in the model and establish the internal connection between dependent and independent variables.Therefore,it has a wide range of applications and extraordinary practical significance.However,the accuracy of the regression results is easily affected by outliers.And with the increase of data dimension and scale,the traditional principal component regression is difficult to quickly find the variables that really affect the principal components.Therefore,it is of great theoretical significance and application value to improve the robustness and sparsity of principal component regression.Based on robust sparse principal component analysis and robust sparse principal component regression,this paper proposes a robust sparse principal component regression method.The practicability of this method is demonstrated by Monte Carlo numerical simulation and case analysis.It can help us reduce outliers interference.On the premise of not destroying the excellent properties of the principal components,it can mine the most important factors affecting the variables and establishing a robust regression model.It strives to improve the interpretability of principal components,and then provides strong support for solving practical problems.The research ideas of this paper can be summarized into three parts:The first part introduces the basic principles and algorithms of sparse principal component analysis,robust principal component analysis,and robust regression respectively,and then compares them with traditional methods through Monte Carlo numerical simulation.The result reflects their superiority in the sparsity and robustness.The second part briefly introduces the basic ideas of robust sparse principal component analysis and robust principal component regression.Through numerical simulation,the validity of the method are demonstrated,and the two are organically combined to form a robust sparse principal component regression method.In the third part,the paper applies the robust sparse principal component regression method on an example,whose theme is"factor analysis of railway passenger traffic volume in China".First it uses the robust sparse principal components analysis to cluster the influencing factors,thereby enhancing the interpretability of the principal components.Then a robust regression model is established by using robust principal component regressionIn the section of the numerical simulation,comparing the traditional principal component analysis method with the sparse principal component analysis method,the result of comparison shows that the generating mechanism of raw data can be easily identified by the sparse principal component analysis method.Comparing the traditional principal component analysis method with the robust principal component analysis method,the result of comparison shows that the robust principal component analysis method can easily resist the adverse effects of outliers in the independent variables.Comparing OLS estimates,M estimates,LMS estimates,and LTS estimates,the result of comparison shows that OLS estimates are sensitive to outliers,M estimators perform well only when resisting outliers in the dependent variables,and LMS and LTS estimates are robust for all outliers.In the section of the case application,the clustering and regression results are in accordance with the practical significance,which shows that the robust sparse principal component regression method can improve the interpretability of principal components and obtain a robust regression model.It is extremely effective for solving practical problems.The innovation of this paper is that,starting from the two aspects of sparsity and robustness,we optimize the traditional principal component regression and form a robust and sparse principal component regression method.The origin of the method and the principle and algorithm of each basic method are described in detail.The numerical simulation of each method shows its effectiveness.The robust sparse principal component regression method is applied to real cases,which provides a new modeling method for solving the problem of factor analysis.