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\name{adjOutlyingness}
\alias{adjOutlyingness}
\title{Compute Skewness-adjusted Multivariate Outlyingness}
\description{
For an \eqn{n \times p}{n * p} data matrix (or data frame) \code{x},
compute the \dQuote{\emph{outlyingness}} of all \eqn{n} observations.
Outlyingness here is a generalization of the Donoho-Stahel
outlyingness measure, where skewness is taken into account via the
medcouple, \code{\link{mc}()}.
}
\usage{
adjOutlyingness(x, ndir = 250, clower = 3, cupper = 4,
alpha.cutoff = 0.75, coef = 1.5, qr.tol = 1e-12)
}
\arguments{
\item{x}{a numeric \code{\link{matrix}} or \code{\link{data.frame}}.}
\item{ndir}{positive integer specifying the number of directions that
should be searched.}
\item{clower, cupper}{the constant to be used for the lower and upper
tails, in order to transform the data towards symmetry.}
%%% MM: How can I modify it to the *NON*-adjusted case ?
\item{alpha.cutoff}{number in (0,1) specifying the quantiles
\eqn{(\alpha, 1-\alpha)} which determine the \dQuote{outlier}
cutoff.}
\item{coef}{positive number specifying the factor with which the
interquartile range (\code{\link{IQR}}) is multiplied to determine
\sQuote{boxplot hinges}-like upper and lower bounds.}
\item{qr.tol}{positive tolerance to be used for \code{\link{qr}} and
\code{\link{solve.qr}} for determining the \code{ndir} directions,
each determined by a random sample of \eqn{p} (out of \eqn{n})
observations.}
}
\note{
The result is \emph{random} as it depends on the sample of
\code{ndir} directions chosen.
}
\details{
\bold{FIXME}: Details in the comment of the Matlab code;
also in the reference(s).
%% SEE /u/maechler/R/MM/STATISTICS/robust/MC/mcmatl/adjoutlyingness.m
%% ---- which has notes about input/output etc of the corresponding
%% Matlab code
The method as described can be useful as preprocessing in
FASTICA (\url{http://www.cis.hut.fi/projects/ica/fastica/};
see also the \R package \pkg{fastICA}.
}
\value{
a list with components
\item{adjout}{numeric of \code{length(n)} giving the adjusted
outlyingness of each observation.}
\item{cutoff}{cutoff for \dQuote{outlier} with respect to the adjusted
outlyingnesses, and depending on \code{alpha.cutoff}.}
\item{nonOut}{logical of \code{length(n)}, \code{TRUE} when the
corresponding observation is \bold{non}-outlying with respect to the
cutoff and the adjusted outlyingnesses.}
}
\references{
Brys, G., Hubert, M., and Rousseeuw, P.J. (2005)
A Robustification of Independent Component Analysis;
\emph{Journal of Chemometrics}, \bold{19}, 1--12.
For the up-to-date reference, please consult
\url{http://wis.kuleuven.be/stat/robust.html}
}
\author{Guy Brys; help page and improvements by Martin Maechler}
\seealso{the adjusted boxplot, \code{\link{adjbox}} and the medcouple,
\code{\link{mc}}.
}
\examples{
## An Example with bad condition number and "border case" outliers
if(FALSE) {## Not yet ok, because of bug in adjOutl
dim(longley)
set.seed(1) ## result is random %% and there's a bug - FIXME! -- try set.seed(3)
ao1 <- adjOutlyingness(longley)
## which are not outlying ?
table(ao1$nonOut) ## all of them
stopifnot(all(ao1$nonOut))
}
## An Example with outliers :
dim(hbk)
set.seed(1)
ao.hbk <- adjOutlyingness(hbk)
str(ao.hbk)
hist(ao.hbk $adjout)## really two groups
table(ao.hbk$nonOut)## 14 outliers, 61 non-outliers:
## outliers are :
which(! ao.hbk$nonOut) # 1 .. 14 --- but not for all random seeds!
## here, they are the same as found by (much faster) MCD:
cc <- covMcd(hbk)
stopifnot(all(cc$mcd.wt == ao.hbk$nonOut))
## This is revealing (about 1--2 cases, where outliers are *not* == 1:14
## but needs almost 1 [sec] per call:
if(interactive()) {
for(i in 1:30) {
print(system.time(ao.hbk <- adjOutlyingness(hbk)))
if(!identical(iout <- which(!ao.hbk$nonOut), 1:14)) {
cat("Outliers:\n"); print(iout)
}
}
}
}
\keyword{robust}
\keyword{multivariate}
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