Smooth Weighting Function - Generalized Biweight
“The Biweight on a Stick” — Compute a smooth (when h > 0) weight function typically for computing weights from large (robust) “distances” using a piecewise polynomial function which in fact is a 2-parameter generalization of Tukey's 1-parameter “biweight”.
smoothWgt(x, c, h)
x |
numeric vector of abscissa values |
c |
“cutoff”, a typically positive number. |
h |
“bandwidth”, a positive number. |
Let w(x;c,h) := smoothWgt(x, c, h). Then,
w(x; c,h) := 0 if |x| >= c + h/2,
w(x; c,h) := 1 if |x| <= c - h/2,
w(x; c,h) := (1 - (|x| - (c-h/2))^2)^2 if c-h/2 < |x| < c+h/2.
smoothWgt() is scale invariant in the sense that
w(S x; S c, S h) = w(x; c, h),
when S > 0.
a numeric vector of the same length as x with weights between
zero and one. Currently all attributes including
dim and names are dropped.
Martin Maechler
## a somewhat typical picture: curve(smoothWgt(x, c=3, h=1), -5,7, n = 1000) csW <- curve(smoothWgt(x, c=1/2, h=1), -2,2) # cutoff 1/2, bandwidth 1 ## Show that the above is the same as ## Tukey's "biweight" or "bi-square" weight function: bw <- function(x) pmax(0, (1 - x^2))^2 cbw <- curve(bw, col=adjustcolor(2, 1/2), lwd=2, add=TRUE) cMw <- curve(Mwgt(x,c=1,"biweight"), col=adjustcolor(3, 1/2), lwd=2, add=TRUE) stopifnot(## proving they are all the same: all.equal(csW, cbw, tol=1e-15), all.equal(csW, cMw, tol=1e-15))
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