Research On Sparse Principal Component Analysis

Principal component analysis(PCA)is a statistical analysis method,which is mainly applied in data analysis and dimension reduction.But the principal component depends on all the original variables,thus it is often difficult to interpret the results.In order to enhance the interpretability of the principal component,the sparsity is introduced in the principal component analysis by Jolliffe in 2003.Meanwhile,the penalty regression method is also proposed to solve this new problem that named sparse principal component.The introduction of the sparsity makes some loadings of the principal components tend to zero,so the interpretability of the principal component is enhanced.On the other hand,the principal component analysis is sensitive to outliers.When the data contains large sparse noise,the result will be in inconsistent with the actual data,so the principal component obtained from the original principal component analysis is unreasonable.In 2009,Wright et al used the low-rank matrix recovery method to reduce the noise in the original data,and then used the principal component analysis to the restored low-rank matrix to enhance the robust of principal component analysis.In this paper,the sparse and robust of principal component analysis are mainly studied.The main contents are as follows:First,sparse principal component analysis bases on TL1 penalty.In the classical sparse principal component analysis,Zou et al.used the L1 penalty to solve the sparse principal component,but the principal component obtained by L1 penalty is not sparse enough.The penalty(a+1)|?|/a+|?|(Transformed L1 penalty,referred to as TL1 penalty)isa penalty function between the L0 penalty and L1 penalty,which can be approximated to L0,L1/2 and Li by changing the value of a.In this paper,TL1 penalty is used to replace L1 penalty in sparse PCA,and TL1 penalty principal component analysis is constructed.We use the threshold iterative algorithm to compare the sparse and variance cumulative contribution of principal component obtained from principal component analysis,spare principal component analysis and TL1 penalty spare principal component analysis,and then these methods are also applied to the selection of main vegetables.The experimental results show that the TL1 penalty spare principal component analysis is more effective than others.Second,robust principal component analysis bases on TL1 penalty.In principal component analysis and robust principal component analysis(RPCA),we use TL*norm((?))to replace the rank of matrix,propose TL*penalty principal component analysis and TL*penalty robust principal component analysis,and use the augmented Lagrange multiplier algorithm to solve these two problems in experiments.Firstly,when the noise is Gaussian noise,we compare the variance cumulative contribution rate with same number of eigenvalues and the number of eigenvalues which is extracted with similar variance cumulative contribution rate,the experiments show that the TL*penalty principal component analysis is better than principal component analysis.When the noise is a large sparse noise,the relative error of the low-rank matrix and the original low-rank matrix and the sparsity and the number of iterations of the sparse noise matrix are compared.The results show that the TL*penalty robust principal component analysis is more effective than the robust principal component analysis.