| scad {lqa} | R Documentation |
The SCAD Penalty
Description
Object of the penalty class to handle the SCAD penalty (Fan \& Li, 2001)
Usage
scad(lambda = NULL, ...)
Arguments
lambda |
two-dimensional tuning parameter. The first component corresponds to the regularization parameter |
... |
further arguments. |
Details
The SCAD penalty is formally defined as
P_{\tilde{\lambda}}^{sc} (\boldsymbol{\beta}) = \sum_{j=1}^p p_{\tilde{\lambda},j}^{sc} (|\beta_j|), \quad \tilde{\lambda} = (\lambda, s),
where p_{\tilde{\lambda},j}^{sc} (|\beta_j|) is complicated to be specified directly.
Fan \& Li (2001) just give the penalty by the first derivatives of its components as
\frac{d p_{\tilde{\lambda},j}^{sc} (|\beta_j|)}{d |\beta_j|} = \lambda\left\{1_{|\beta_j| \leq \lambda}(|\beta_j|) +
\frac{(a\lambda - |\beta_j|)_+}{(a-1)\lambda}1_{|\beta_j| > \lambda} (|\beta_j|) \right\},
where we use the notation b_+ := \max \{0, b\} and 1_A(x) denotes the indicator function.
The penalty depends on two tuning parameters, \lambda>0 and a>2.
It is continuously differentiable in \beta_j, but not in their tuning
parameters. If |\beta_j| \leq \lambda then the lasso penalty is applied to \beta_j.
Afterwards this penalization smoothly clipped apart until
the threshold a is reached. For |\beta_j| > a there is no penalization at all at this coefficient.
Fan \& Li (2001) suggest to use a = 3.7.
The SCAD penalty leaves large values of \beta_j not excessively penalized and makes the solution continuous.
Value
An object of the class penalty. This is a list with elements
penalty |
character: the penalty name. |
lambda |
double: the (nonnegative) tuning parameter. |
getpenmat |
function: computes the diagonal penalty matrix. |
Author(s)
Jan Ulbricht
References
Fan, J. \& R. Li (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360.