The LOGSELECT Procedure

The LOGSELECT procedure implements the group LASSO method, which is described in the section Group LASSO Selection in Chapter 3: Shared Concepts. The current section provides some background about the LASSO method that you need in order to understand the group LASSO method.

LASSO (least absolute shrinkage and selection operator) selection arises from a constrained form of ordinary least squares regression in which the sum of the absolute values of the regression coefficients is constrained to be smaller than a specified parameter. More precisely, let denote the matrix of covariates, and let denote the response. Then for a given parameter t, the LASSO regression coefficients are the solution to the following constrained least squares problem:

For generalized linear models, the LASSO regression coefficients are the solution to the following constrained optimization problem,

where L is the log-likelihood function defined in the section Log-Likelihood Functions.

Provided that the LASSO parameter t is small enough, some of the regression coefficients will be exactly zero. Hence, you can think of the LASSO method as selecting a subset of the regression coefficients for each LASSO parameter. By increasing the LASSO parameter in discrete steps, you obtain a sequence of regression coefficients for which the nonzero coefficients at each step correspond to selected parameters. For more information about the LASSO method, see, for example, Hastie, Tibshirani, and Friedman (2009).