\HeaderA{cov.rob}{Resistant Estimation of Multivariate Location and Scatter}{cov.rob}
\aliasA{cov.mcd}{cov.rob}{cov.mcd}
\aliasA{cov.mve}{cov.rob}{cov.mve}
\keyword{robust}{cov.rob}
\keyword{multivariate}{cov.rob}
\begin{Description}\relax
Compute a multivariate location and scale estimate with a high
breakdown point -- this can be thought of as estimating the mean and
covariance of the \code{good} part of the data. \code{cov.mve} and
\code{cov.mcd} are compatibility wrappers.
\end{Description}
\begin{Usage}
\begin{verbatim}
cov.rob(x, cor = FALSE, quantile.used = floor((n + p + 1)/2),
        method = c("mve", "mcd", "classical"),
        nsamp = "best", seed)

cov.mve(...)
cov.mcd(...)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a matrix or data frame.

\item[\code{cor}] should the returned result include a correlation matrix?

\item[\code{quantile.used}] the minimum number of the data points regarded as \code{good} points.

\item[\code{method}] the method to be used -- minimum volume ellipsoid, minimum
covariance determinant or classical product-moment. Using
\code{cov.mve} or \code{cov.mcd} forces \code{mve} or \code{mcd}
respectively.

\item[\code{nsamp}] the number of samples or \code{"best"} or \code{"exact"} or
\code{"sample"}.
If \code{"sample"} the number chosen is \code{min(5*p, 3000)}, taken
from Rousseeuw and Hubert (1997). If \code{"best"} exhaustive
enumeration is done up to 5000 samples: if \code{"exact"}
exhaustive enumeration will be attempted however many samples are needed.

\item[\code{seed}] the seed to be used for random sampling: see \code{\LinkA{RNGkind}{RNGkind}}. The
current value of \code{.Random.seed} will be preserved if it is set.

\item[\code{...}] arguments to \code{cov.rob} other than \code{method}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
For method \code{"mve"}, an approximate search is made of a subset of
size \code{quantile.used} with an enclosing ellipsoid of smallest volume; in
method \code{"mcd"} it is the volume of the Gaussian confidence
ellipsoid, equivalently the determinant of the classical covariance
matrix, that is minimized. The mean of the subset provides a first
estimate of the location, and the rescaled covariance matrix a first
estimate of scatter. The Mahalanobis distances of all the points from
the location estimate for this covariance matrix are calculated, and
those points within the 97.5\% point under Gaussian assumptions are
declared to be \code{good}. The final estimates are the mean and rescaled
covariance of the \code{good} points.

The rescaling is by the appropriate percentile under Gaussian data; in
addition the first covariance matrix has an \emph{ad hoc} finite-sample
correction given by Marazzi.

For method \code{"mve"} the search is made over ellipsoids determined
by the covariance matrix of \code{p} of the data points. For method
\code{"mcd"} an additional improvement step suggested by Rousseeuw and
van Driessen (1999) is used, in which once a subset of size
\code{quantile.used} is selected, an ellipsoid based on its covariance
is tested (as this will have no larger a determinant, and may be smaller).
\end{Details}
\begin{Value}
A list with components

\begin{ldescription}
\item[\code{center}] the final estimate of location.

\item[\code{cov}] the final estimate of scatter.

\item[\code{cor}] (only is \code{cor = TRUE}) the estimate of the correlation
matrix.

\item[\code{sing}] message giving number of singular samples out of total

\item[\code{crit}] the value of the criterion on log scale. For MCD this is
the determinant, and for MVE it is proportional to the volume.

\item[\code{best}] the subset used. For MVE the best sample, for MCD the best
set of size \code{quantile.used}.

\item[\code{n.obs}] total number of observations.

\end{ldescription}
\end{Value}
\begin{References}\relax
P. J. Rousseeuw and A. M. Leroy (1987) 
\emph{Robust Regression and Outlier Detection.}
Wiley.

A. Marazzi (1993) 
\emph{Algorithms, Routines and S Functions for Robust Statistics.}
Wadsworth and Brooks/Cole. 

P. J. Rousseeuw and B. C. van Zomeren (1990) Unmasking
multivariate outliers and leverage points, 
\emph{Journal of the American Statistical Association}, \bold{85}, 633--639.

P. J. Rousseeuw and K. van Driessen (1999) A fast algorithm for the
minimum covariance determinant estimator. \emph{Technometrics}
\bold{41}, 212--223.

P. Rousseeuw and M. Hubert (1997) Recent developments in PROGRESS. In
\emph{L1-Statistical Procedures and Related Topics }
ed Y. Dodge, IMS Lecture Notes volume \bold{31}, pp. 201--214.
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{lqs}{lqs}}
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
set.seed(123)
cov.rob(stackloss)
cov.rob(stack.x, method = "mcd", nsamp = "exact")
\end{ExampleCode}
\end{Examples}

