An l2 penalty is applied to the regression coefficients during ridge regression. Ridge regression coefficient estimates are given by the following:
is the l2 penalty, λ is the tuning parameter, N is the number of rows, and p is the number of variables.
An l1 penalty is applied to the regression coefficients during Lasso. Coefficient estimates for the Lasso are given by the following:
is the l1 penalty, λ is the tuning parameter, N is the number of rows, and p is the number of variables
The Elastic Net combines both l1 and l2 penalties. Coefficient estimates for the Elastic Net are given by the following:
is the l1 penalty, is the l2 penalty, and λ is the tuning parameter. The α tuning parameter determines the mix of the l1 and l2 penalties, N is the number of rows, and p is the number of variables.
For the adaptive Lasso, this weighted form of the l1 penalty is used in determining the coefficients.
The adaptive Elastic Net uses this weighted form of the l1 penalty and also imposes a weighted form of the l2 penalty. The weighted form of the l2 penalty for the adaptive Elastic Net is
Options for Tuning Parameter Grid Scale