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\name{covMcd}
\alias{covMcd}
\alias{print.mcd}
%
\title{Robust Location and Scatter Estimation via MCD}
\concept{High breakdown point}
\description{
Compute a robust multivariate location and scale estimate with a high
breakdown point, using the \sQuote{Fast MCD} (Minimum Covariance
Determinant) estimator.
}
\usage{
covMcd(x, cor = FALSE, alpha = 1/2, nsamp = 500, nmini = 300, seed = NULL,
trace = FALSE, use.correction = TRUE, control = rrcov.control())
}
\arguments{
\item{x}{a matrix or data frame. }
\item{cor}{should the returned result include a correlation matrix?
Default is \code{cor = FALSE}}.
\item{alpha}{numeric parameter controlling the size of the subsets
over which the determinant is minimized, i.e., \code{alpha*n}
observations are used for computing the determinant. Allowed values
are between 0.5 and 1 and the default is 0.5.}
\item{nsamp}{number of subsets used for initial estimates or \code{"best"}
or \code{"exact"}. Default is \code{nsamp = 500}. For
\code{nsamp = "best"} exhaustive enumeration is done, as long as the
number of trials does not exceed 100'000 (\code{= nLarge}). For \code{"exact"},
exhaustive enumeration will be attempted however many samples are
needed. In this case a warning message may be displayed saying
that the computation can take a very long time.}
\item{nmini}{for large \eqn{n}, the algorithm splits the data into
maximally \eqn{k_{\mathrm{rep}} = 5}{krep = 5} subsets of size
\code{nmini}. %% FIXME? -- more accurately and in detail ?!
The original algorithm had \code{nmini = 300} hard coded.}
\item{seed}{initial seed for random generator, see \code{\link{rrcov.control}}.}
\item{trace}{logical (or integer) indicating if intermediate results
should be printed; defaults to \code{FALSE}; values \eqn{\ge 2}{>= 2}
also produce print from the internal (Fortran) code.}
\item{use.correction}{ whether to use finite sample correction
factors; defaults to \code{TRUE}.}
\item{control}{a list with estimation options - this includes those
above provided in the function specification, see
\code{\link{rrcov.control}} for the defaults. If \code{control} is
supplied, the parameters from it will be used. If parameters are
passed also in the invocation statement, they will override the
corresponding elements of the control object.}
}
\details{
The minimum covariance determinant estimator of location and scatter
implemented in \code{covMcd()} is similar to \R function
\code{\link[MASS]{cov.mcd}()} in \pkg{MASS}. The MCD method looks for
the \eqn{h (> n/2)} (\eqn{h = h(\alpha,n,p) =}
\code{\link{h.alpha.n}(alpha,n,p)}) observations (out of \eqn{n})
whose classical covariance matrix has the lowest possible determinant.
The raw MCD estimate of location is then the average of these \eqn{h} points,
whereas the raw MCD estimate of scatter is their covariance matrix,
multiplied by a consistency factor and a finite sample correction factor
(to make it consistent at the normal model and unbiased at small samples).
The implementation of \code{covMcd} uses the Fast MCD algorithm of
Rousseeuw and Van Driessen (1999) to approximate the minimum
covariance determinant estimator.
Both rescaling factors (consistency and finite sample) are returned
also in the vector \code{raw.cnp2} of length 2. Based on these raw
MCD estimates, a reweighting step is performed which increases the
finite-sample eficiency considerably - see Pison et al.~(2002). The
rescaling factors for the reweighted estimates are returned in the
vector \code{cnp2} of length 2. Details for the computation of the
finite sample correction factors can be found in Pison et al. (2002).
The finite sample corrections can be suppressed by setting
\code{use.correction = FALSE}.
}
\author{Valentin Todorov \email{valentin.todorov@chello.at}, based on
work written for S-plus by Peter Rousseeuw and Katrien van Driessen
from University of Antwerp.% no E-mails for spam-protection
}
\value{
An object of class \code{"mcd"} which is basically a
\code{\link{list}} with components
\item{center}{the final estimate of location.}
\item{cov}{the final estimate of scatter.}
\item{cor}{the (final) estimate of the correlation matrix (only if
\code{cor = TRUE}).}
\item{crit}{the value of the criterion, i.e. the determinant.}
\item{best}{the best subset found and used for computing the raw
estimates, with \code{\link{length}(best) == quan =
\link{h.alpha.n}(alpha,n,p)}.}
\item{mah}{mahalanobis distances of the observations using the final
estimate of the location and scatter.}
\item{mcd.wt}{weights of the observations using the final estimate of
the location and scatter.}
\item{cnp2}{a vector of length two containing the consistency
correction factor and the finite sample correction factor of
the final estimate of the covariance matrix.}
\item{raw.center}{the raw (not reweighted) estimate of location.}
\item{raw.cov}{the raw (not reweighted) estimate of scatter.}
\item{raw.mah}{mahalanobis distances of the observations based on the
raw estimate of the location and scatter.}
\item{raw.weights}{weights of the observations based on the raw
estimate of the location and scatter.}
\item{raw.cnp2}{a vector of length two containing the consistency
correction factor and the finite sample correction factor of
the raw estimate of the covariance matrix.}
\item{X}{the input data as numeric matrix, without \code{\link{NA}}s.}
\item{n.obs}{total number of observations.}
\item{alpha}{the size of the subsets over which the determinant is
minimized (the default is \eqn{(n+p+1)/2}).}
\item{quan}{the number of observations, \eqn{h}, on which the MCD is
based. If \code{quan} equals \code{n.obs}, the MCD is the classical
covariance matrix.}
\item{method}{character string naming the method (Minimum Covariance
Determinant).}
\item{call}{the call used (see \code{\link{match.call}}).}
}
\references{
P. J. Rousseeuw and A. M. Leroy (1987)
\emph{Robust Regression and Outlier Detection.} Wiley.
P. J. Rousseeuw and K. van Driessen (1999)
A fast algorithm for the minimum covariance determinant estimator.
\emph{Technometrics} \bold{41}, 212--223.
Pison, G., Van Aelst, S., and Willems, G. (2002),
Small Sample Corrections for LTS and MCD,
\emph{Metrika}, \bold{55}, 111-123.
}
\seealso{
\code{\link[MASS]{cov.mcd}} from package \pkg{MASS};
\code{\link{covOGK}} as cheaper alternative for larger dimensions.
}
\examples{
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
covMcd(hbk.x)
## the following three statements are equivalent
c1 <- covMcd(hbk.x, alpha = 0.75)
c2 <- covMcd(hbk.x, control = rrcov.control(alpha = 0.75))
## direct specification overrides control one:
c3 <- covMcd(hbk.x, alpha = 0.75,
control = rrcov.control(alpha=0.95))
c1
}
\keyword{robust}
\keyword{multivariate}
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