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\name{smoothWgt}
\alias{smoothWgt}
\title{Smooth Weighting Function - Generalized Biweight}% The Biweight on a Stick
\description{
\dQuote{The Biweight on a Stick} ---
Compute a smooth (when \eqn{h > 0}) weight function typically for
computing weights from large (robust) \dQuote{distances} using a
piecewise polynomial function which in fact is a
2-parameter generalization of Tukey's 1-parameter \dQuote{biweight}.
}
\usage{
smoothWgt(x, c, h)
}
\arguments{
\item{x}{numeric vector of abscissa values}
\item{c}{\dQuote{cutoff}, a typically positive number.}
\item{h}{\dQuote{bandwidth}, a positive number.}
}
\details{
Let \eqn{w(x;c,h) := }\code{smoothWgt(x, c, h)}. Then,
\deqn{% "FIXME": rather use amsmath package \cases{.}
w(x; c,h) := 0 \ \ \ \ \ \mathrm{if}\ |x| \ge c + h/2,}{%
w(x; c,h) := 0 if |x| >= c + h/2,}
\deqn{
w(x; c,h) := 1 \ \ \ \ \ \mathrm{if}\ |x| \le c - h/2,}{%
w(x; c,h) := 1 if |x| <= c - h/2,}
\deqn{
w(x; c,h) := \bigl((1 - |x| - (c-h/2))^2\bigr)^2 \ \mathrm{if}\ c-h/2 < |x| < c+h/2,}{%
w(x; c,h) := (1 - (|x| - (c-h/2))^2)^2 if c-h/2 < |x| < c+h/2.}
\code{smoothWgt()} is \emph{scale invariant} in the sense that
\deqn{w(\sigma x; \sigma c, \sigma h) = w(x; c, h),}{%
w(S x; S c, S h) = w(x; c, h),} when \eqn{\sigma > 0}{S > 0}.
}
\value{
a numeric vector of the same length as \code{x} with weights between
zero and one. Currently all \code{\link{attributes}} including
\code{\link{dim}} and \code{\link{names}} are dropped.
}
%% \references{ TODO: Write a small vignette !
%% }
\author{Martin Maechler}
\seealso{
\code{\link{Mwgt}(.., psi = "bisquare")} of which \code{smoothWgt()}
is a generalization, and
\code{\link{Mwgt}(.., psi = "optimal")} which looks similar for larger
\code{c} with its constant one part around zero, but also has only
one parameter.
}
\examples{
## 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))
}
\keyword{arith}
\keyword{robust}
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