sparse PCA.ppt

Sparse Principal Component Analysis Zou, Hastie, Tibshirani Outline Microarray expression data PCA on expression data SPCA Definition Motivation Method Linear regression Lasso/elastic net 2 Examples DNA Microarrays Gene Protein Monitors gene expression levels on a genomic scale. (~abundance of each protein in the cell) DNA Microarrays One (micro-)array = one snapshot of the cell Each spot represents a gene Intensity of colors Indicate expression level of the corresponding gene in a certain experiment Noisy, each expression level is only an approximation. Microarray Expression Data PCA on Expression Data PCA on Expression Data Drawbacks of PCA PCs (eigen-genes) are linear combinations of all n variables (genes). Each PC corresponds to a loading vector (columns of V) Loadings = coefficients corresponding to variables in the linear combination Difficult to interpret SPCA: Motivation Idea: Reduce the number of explicitly used variables (genes). Approach: Modify PCA so that PCs have sparse loadings = sparse PCA (SPCA) SPCA Writes PCA as a regression-type optimization problem. Uses lasso a variable selection technique produces sparse models Result: modified PCA Modified PCs with sparse loadings. Linear Regression Problem Input variables Response variable Regression coefficients Multivariate linear model Linear Regression Problem Least Squares Solution Lasso Solution Lasso Solution Elastic Net Solution Elastic Net Solution SPCA Goal: Construct a regression framework in which PCA can be reconstructed exactly. Use lasso/elastic-net to construct modified PCs with sparse loadings. Reconstruction of PCA in a Regression Framework Idea: Each PC is a linear combination of the p variables. Its loadings can be recovered by regression PC on the p variables. Theorem 1 Reconstruction of PCA in a Regression Framework By theorem 1, we can reconstruct the loadings of PCs exactly by a linear regression problem. not an alternative to PCA as it