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\name{Mpsi}
\alias{Mchi}
\alias{Mpsi}
\alias{Mwgt}
\alias{MrhoInf}
\alias{.Mchi}
\alias{.Mpsi}
\alias{.Mwgt}
\alias{.Mwgt.psi1}
\alias{.MrhoInf}
\title{Psi / Chi / Wgt / Rho Functions for *M-Estimation}
\description{
Compute Psi / Chi / Wgt / Rho functions for M-estimation,
i.e., including MM, etc.
%% TODO: More, notably definitions ... but they are all nicely in the
%% vignette.. How can we link from here to there ???
\code{MrhoInf(x)} computes \eqn{\rho(\infty)}{rho(Inf)}, i.e., the
normalizing or scaling constant for the transformation
from \eqn{\rho(\cdot)}{rho(.)} to
\eqn{\tilde\rho(\cdot)}{rho~(.)}, where the latter, aka as
\eqn{\chi()}{chi()} fulfills \eqn{\tilde\rho(\infty) = 1}{rho~(Inf) = 1}
which makes only sense for \dQuote{redescending} psi functions, i.e.,
not for \code{"huber"}.
\code{Mwgt(x, *)} computes \eqn{\psi(x)/x} (fast and numerically accurately).
}
\usage{
Mpsi(x, cc, psi, deriv = 0)
Mchi(x, cc, psi, deriv = 0)
Mwgt(x, cc, psi)
MrhoInf(cc, psi)
.Mwgt.psi1(psi, cc = .Mpsi.tuning.default(psi))
}
\arguments{
\item{x}{numeric (\dQuote{abscissa} values) vector, possibly with
\code{\link{attributes}} such as \code{\link{dim}} or
\code{\link{names}}, etc. These are preserved for the
\code{M*()} functions (but not the \code{.M()} ones).}
\item{cc}{numeric tuning constant, for some \code{psi} of length
\eqn{> 1}.}
\item{psi}{a string specifying the psi / chi / rho / wgt function;
either \code{"huber"}, or one of the same possible specifiers as for
\code{psi} in \code{\link{lmrob.control}}, i.e. currently,
\code{"bisquare"}, \code{"lqq"}, \code{"welsh"}, \code{"optimal"},
\code{"hampel"}, or \code{"ggw"}.}
\item{deriv}{an integer, specifying the \emph{order} of derivative to
consider; particularly, \code{Mpsi(x, *, deriv = -1)} is the
principal function of \eqn{\psi()}{psi()}, typically denoted
\eqn{\rho()}{rho()} in the literature.}
%% FIXME: mention that deriv = 2 is *partially* implemented
}
\details{
Theoretically, \code{Mchi()} would not be needed explicitly as it can be computed
from \code{Mpsi()} and \code{MrhoInf()}, namely, by
\preformatted{Mchi(x, *, deriv = d) == Mpsi(x, *, deriv = d-1) / MrhoInf(*)}
for \eqn{d = 0, 1, 2} (and \sQuote{*} containing \code{par, psi}, and
equality is in the sense of \code{\link{all.equal}(x,y, tol)} with a
small \code{tol}.
Similarly, \code{Mwgt} would not be needed strictly, as it could be
defined via \code{Mpsi}), but the explicit definition takes care of
0/0 and typically is of a more simple form.
For experts, there are slightly even faster versions,
\code{.Mpsi()}, \code{.Mwgt()}, etc.
\code{.Mwgt.psi1()} mainly a utility for \code{\link{nlrob}()},
returns a \emph{\code{\link{function}}} with similar semantics as
\code{\link[MASS]{psi.hampel}}, \code{\link[MASS]{psi.huber}}, or
\code{\link[MASS]{psi.bisquare}} from package \pkg{MASS}. Namely,
a function with arguments \code{(x, deriv=0)}, which for
\code{deriv=0} computes \code{Mwgt(x, cc, psi)} and otherwise computes
\code{Mpsi(x, cc, psi, deriv=deriv)}.
}
\value{
a numeric vector of the same length as \code{x}, with corresponding
function (or derivative) values.
}
\references{
See the vignette about %% \link{ .. } ????
\dQuote{\eqn{\psi}{psi}-Functions Available in Robustbase}.
%% ../inst/doc/psi_functions.Rnw
}
\author{
Manuel Koller, notably for the original C implementation;
tweaks and speedup via \code{\link{.Call}} by Martin Maechler.
}
\seealso{
\code{\link{psiFunc}} and the \code{\linkS4class{psi_func}} class, both
of which provide considerably more on the \R side, but are less
optimized for speed.
}
\examples{
x <- seq(-5,7, by=1/8)
matplot(x, cbind(Mpsi(x, 4, "biweight"),
Mchi(x, 4, "biweight"),
Mwgt(x, 4, "biweight")), type = "l")
abline(h=0, v=0, lty=2, col=adjustcolor("gray", 0.6))
hampelPsi
(ccHa <- hampelPsi @ xtras $ tuningP $ k)
psHa <- hampelPsi@psi(x)
% FIXME: interesting as long as hampelPsi does not use Mpsi(... "hampel") !
## using Mpsi():
Mp.Ha <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(all.equal(Mp.Ha, psHa, tolerance = 1e-15))
psi.huber <- .Mwgt.psi1("huber")
if(getRversion() >= "3.0.0")
stopifnot(identical(psi.huber, .Mwgt.psi1("huber", 1.345),
ignore.env=TRUE))
curve(psi.huber(x), -3, 5, col=2, ylim = 0:1)
curve(psi.huber(x, deriv=1), add=TRUE, col=3)
## and show that this is indeed the same as MASS::psi.huber() :
x <- runif(256, -2,3)
stopifnot(all.equal(psi.huber(x), MASS::psi.huber(x)),
all.equal( psi.huber(x, deriv=1),
as.numeric(MASS::psi.huber(x, deriv=1))))
## and how to get MASS::psi.hampel():
psi.hampel <- .Mwgt.psi1("Hampel", c(2,4,8))
x <- runif(256, -4, 10)
stopifnot(all.equal(psi.hampel(x), MASS::psi.hampel(x)),
all.equal( psi.hampel(x, deriv=1),
as.numeric(MASS::psi.hampel(x, deriv=1))))
## M*() preserving attributes :
x <- matrix(x, 32, 8, dimnames=list(paste0("r",1:32), col=letters[1:8]))
comment(x) <- "a vector which is a matrix"
px <- Mpsi(x, cc = ccHa, psi = "hampel")
stopifnot(identical(attributes(x), attributes(px)))
}
\keyword{robust}
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