Tukey's Bi-square Score (Psi) and "Chi" (Rho) Functions and Derivatives
tukeyPsi1() computes Tukey's bi-square score (psi) function, its first
derivative or it's integral/“principal function”. This is
scaled such that ψ'(0) = 1, i.e.,
psi(x) ~= x around 0.
tukeyChi() computes Tukey's bi-square loss function,
chi(x) and its first two derivatives. Note that in the general
context of M-estimators, these loss functions are called
rho-functions.
tukeyPsi1(x, cc, deriv = 0) tukeyChi (x, cc, deriv = 0)
x |
numeric vector. |
cc |
tuning constant |
deriv |
integer in \{-1,0,1,2\} specifying the order of the
derivative; the default, |
a numeric vector of the same length as x.
tukeyPsi1(x, d) and tukeyChi(x, d+1) are just
re-scaled versions of each other (for d in -1:1), i.e.,
χ^{(ν)}(x, c) = (6/c^2) ψ^{(ν-1)}(x,c),
for ν = 0,1,2.
We use the name ‘tukeyPsi1’, because tukeyPsi is
reserved for a future “Psi Function” class object, see
psiFunc.
Matias Salibian-Barrera, Martin Maechler and Andreas Ruckstuhl
lmrob and Mpsi; further
anova.lmrob which needs the deriv = -1.
op <- par(mfrow = c(3,1), oma = c(0,0, 2, 0),
mgp = c(1.5, 0.6, 0), mar= .1+c(3,4,3,2))
x <- seq(-2.5, 2.5, length = 201)
cc <- 1.55 # as set by default in lmrob.control()
plot. <- function(...) { plot(...); abline(h=0,v=0, col="gray", lty=3)}
plot.(x, tukeyChi(x, cc), type = "l", col = 2)
plot.(x, tukeyChi(x, cc, deriv = 1), type = "l", col = 2)
plot.(x, tukeyChi(x, cc, deriv = 2), type = "l", col = 2)
mtext(sprintf("tukeyChi(x, c = %g, deriv), deriv = 0,1,2", cc),
outer = TRUE, font = par("font.main"), cex = par("cex.main"))
par(op)
op <- par(mfrow = c(3,1), oma = c(0,0, 2, 0),
mgp = c(1.5, 0.6, 0), mar= .1+c(3,4,1,1))
x <- seq(-5, 5, length = 201)
cc <- 4.69 # as set by default in lmrob.control()
plot. <- function(...) { plot(..., asp = 1); abline(h=0,v=0, col="gray", lty=3)}
plot.(x, tukeyPsi1(x, cc), type = "l", col = 2)
abline(0:1, lty = 3, col = "light blue")
plot.(x, tukeyPsi1(x, cc, deriv = -1), type = "l", col = 2)
plot.(x, tukeyPsi1(x, cc, deriv = 1), type = "l", col = 2); abline(h=1,lty=3)
mtext(sprintf("tukeyPsi1(x, c = %g, deriv), deriv = 0, -1, 1", cc),
outer = TRUE, font = par("font.main"), cex = par("cex.main"))
par(op)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.