Penalty function
Smoothly Clipped Absolute Deviation function.
SCAD(x, lambda)
x |
a vector where the function is to be evaluated. |
lambda |
a number representing a tuning parameter. |
SCAD is an even continuous function equal to 0 at x=0, and defined piecewise with derivative lambda in [0, lambda], (a*lambda - x)/(a-1) in [lambda, a*lambda], and 0 for x larger than a*lambda. As suggested by (Li, Sudjianto, 2005), we set a=3.7.
A vector containing the SCAD values at x.
In MLE problems, the penalty value lambda should tend to 0 when the sample size tends to infinity to insure that the asymptotic properties of Penalized-MLE and MLE are the same (see Li, Sudjianto, 2005).
O. Roustant, D. Ginsbourger, Ecole des Mines de St-Etienne.
R. Li and A. Sudjianto (2005), Analysis of Computer Experiments Using Penalized Likelihood in Gaussian Kriging Models, Technometrics, 47 no. 2, 111-120.
SCAD.derivative and km for a famous example
x <- seq(-8,8, length=200)
a <- 3.7
lambda <- 1.5
y <- SCAD(x, lambda)
plot(x, y, type="l", ylim=c(0,6))
x.knots <- c(-a*lambda, -lambda, 0, lambda, a*lambda)
points(x.knots, SCAD(x.knots, lambda), pch=19, cex=0.5)
text(6, SCAD(6, lambda)+0.3, paste("lambda =", lambda))
for (i in 1:2) {
lambda <- lambda - 0.5
y <- SCAD(x, lambda)
lines(x, y, type="l")
x.knots <- c(-a*lambda, -lambda, 0, lambda, a*lambda)
points(x.knots, SCAD(x.knots, lambda), pch=19, cex=0.5)
text(6, SCAD(6, lambda)+0.3, paste("lambda =", lambda))
}
abline(v=0, h=0, lty="dotted")
title("SCAD function")Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.