Research On Sparse Principal Component Analysis Under Adaptive Lasso Penalty

When applying regression analysis to deal with actual problems,in order to reduce model errors,more independent variables are generally introduced.But it is inevitable that some independent variables will have little or no effect on the dependent variable,which will not only increase the complexity of the calcula-tion Degree.It will also reduce the interpretability and prediction accuracy of the model,and will also increase the cost of obtaining data.Therefore,it becomes very important to select some important features from many feature variables.Tibshirani proposed the Lasso regression method,which can compress certain co-efficients to 0 by adjusting the regression coefficients to realize variable selection.The Lasso method has become one of the most widely used variable selection methods because of its excellent sparsity ability and efficient calculation speed.This has also led many statisticians to do more in-depth research on the basis of Lasso.Zouhui proposed the Adaptive Lasso method,which is an improvement of the Lasso method.By applying different degrees of adjustment to different regression coefficients,Zouhui demonstrated that Adaptive Lasso has a better performance in the accuracy of variable selection and model prediction.Principal components analysis(PCA)is a common data processing technique and dimensionality reduction method,which reduces the number of features by obtaining a few principal components.But because each of its principal com-ponents is a linear combination of the original features,the meaning of each principal component is usually difficult to explain,so the sparseness of the prin-cipal components is a very important issue.Zouhui proposed the sparse principal component analysis(SPCA)method,which transforms the solution of principal component sparseness into solving an elastic net regression problem,which not only greatly improves the efficiency of calculating principal component sparse-ness problem,but also communicates principal component analysis and regres-sion analysis.The channels between the problems give a new perspective to the principal component analysis problem and its sparseness problem.This paper is from the perspective of Zouhui's solution to sparse principal component analysis,combining the Adaptive Lasso method with the SPCA method,so that when dealing with the sparseness of the principal components.The result can also inherit the advantages of Adaptive Lasso in the variable selection problem,so as to achieve better sparsity results.The second chapter of this article mainly introduces several classic regression and variable selection methods,as well as their basic definitions and properties.At the same time,combined with the image of the theoretical solution,it is more clear to show the influence of the penalty function on various variable selection methods.Finally,the advantages of the Adaptive Lasso method over the Lasso method are explained,which is a theoretical preparation for the subsequent re-search.The third chapter of the article mainly introduces the theory and example veri-fication of principal component analysis and sparse principal component analysis.Mainly introducing the definition of principal component analysis and related the-orems as well as specific algorithm steps,and showing the poor interpret ability of principal components through specific examples.After that,the theoretical method of Zouhui's sparse principal component analysis(SPCA)is introduced,and specific definitions,related theorems and specific algorithms are given.Fi-nally,it is verified by an example and compared with the PCA example.It shows that SPCA obtains excellent sparse principal component results based on the variance contribution rate with a small loss,which greatly improves the inter-pretability of principal components.The main innovation of the article is to combine Adaptive Lasso with SPCA,and propose a sparse principal component analysis method under Adaptive Lasso penalty,and give the corresponding algorithm.At the same time,instance verifi-cation and simulation data show that under the same situation,sparse principal component analysis under Adaptive Lasso penalty has better experimental results than SPCA,which can select features better and lose less variance contribution rate.In short,the new method is a reasonable improvement of the SPCA method.