Research On The Properties And Applications Of The L₂-Fused Lasso

Variable selection is an important part in modern statistics, the predecessors have done a lot on variable selection in the linear models. Especially, the research on Lasso and its related corrective methods have become extremely popular in statistics. Taking into account the order of the variables, Tibshirani and Saunders (2005) proposed Fused Lasso, combining the Lasso penalty and a L1penalty on coefficients’ successive differ-ences, which was borrowed from Land and Friedman (1996). This method encourages sparsity of the coefficients and also sparsity of their differences which produces a piece-wise constant solution. We will do some research on L2-2-Fused Lasso, penalizing the L1-norm of coefficients and the L2-2-norm of their differences. This method brings a sparse coefficients, but does not shrink the differences of coefficients to0. Theoreti-cally, we study whether the L2-2-Fused Lasso has Oracle properties or not. Then we use real data examples to check its performances. Also we compare it with Lasso, Fused Lasso and other variable selection methods.In section1we review the background and the development of relevant variable selection methods. In section2we introduce some related preparation knowledge, in-cluding Linear models, Lasso, Elastic Net and Fused Lasso. In section3we give the definition of L2-2-Fused Lasso and rewrite it into the form of Lasso, which shows that it can solve the case of p>> n. Then, we prove the asymptotic property of L2-2-Fused Lasso in details and discuss whether it has the consistency property or not. We find that it is inconsistent when the optimal estimation rate is attainable. In section4diabetes data and leukemia data are used to illustrate its performances. At last, we summarize the full paper and propose some questions for further study.