Exact Post-Selection Inference for Sequential Regression Procedures

We propose new inference tools for forward stepwise regression, least angle regression, and the lasso. Assuming a Gaussian model for the observation vector y, we first describe a general scheme to perform valid inference after any selection event that can be characterized as y falling into a polyhedral set. This framework allows us to derive conditional (post-selection) hypothesis tests at any step of forward stepwise or least angle regression, or any step along the lasso regularization path, because, as it turns out, selection events for these procedures can be expressed as polyhedral constraints on y. The p-values associated with these tests are exactly uniform under the null distribution, in finite samples, yielding exact type I error control. The tests can also be inverted to produce confidence intervals for appropriate underlying regression parameters. The R package "selectiveInference", freely available on the CRAN repository, implements the new inference tools described in this paper.Comment: 26 pages, 5 figure

"Pre-conditioning" for feature selection and regression in high-dimensional problems

We consider regression problems where the number of predictors greatly exceeds the number of observations. We propose a method for variable selection that first estimates the regression function, yielding a "pre-conditioned" response variable. The primary method used for this initial regression is supervised principal components. Then we apply a standard procedure such as forward stepwise selection or the LASSO to the pre-conditioned response variable. In a number of simulated and real data examples, this two-step procedure outperforms forward stepwise selection or the usual LASSO (applied directly to the raw outcome). We also show that under a certain Gaussian latent variable model, application of the LASSO to the pre-conditioned response variable is consistent as the number of predictors and observations increases. Moreover, when the observational noise is rather large, the suggested procedure can give a more accurate estimate than LASSO. We illustrate our method on some real problems, including survival analysis with microarray data

DISCRIMINANT STEPWISE PROCEDURE

Stepwise procedure is now probably the most popular tool for automatic feature selection. In the most cases it represents model selection approach which evaluates various feature subsets (so called wrapper). In fact it is heuristic search technique which examines the space of all possible feature subsets. This method is known in the literature under different names and variants. We organize the concepts and terminology, and show several variants of stepwise feature selection from a search strategy point of view. Short review of implementations in R will be given