

   RReessiissttaanntt EEssttiimmaattiioonn ooff MMuullttiivvaarriiaattee LLooccaattiioonn aanndd SSccaatttteerr

        cov.rob(x, cor=FALSE, quantile.used=floor((n + p + 1)/2),
                method=c("mve", "mcd", "classical"), nsamp="best", seed)
        cov.mve(x, cor=FALSE, quantile.used=floor((n + p + 1)/2),
                nsamp="best", seed)
        cov.mcd(x, cor=FALSE, quantile.used=floor((n + p + 1)/2),
                nsamp="best", seed)

   AArrgguummeennttss::

          x: a matrix or data frame.

        cor: should the returned result include a correlation
             matrix?

   quantile.used: the minimum number of the data points
             regarded as `good' points.

     method: the method to be used - minimum volume ellipsoid,
             minimum covariance determinant or classical prod-
             uct-moment. Using `cov.mve' or `cov.mcd' forces
             `mve' or `mcd' respectively.

      nsamp: the number of samples or `"best"' or `"exact"' or
             `"sample"'.  If `"sample"' the number chosen is
             `min(5*p, 3000)', taken from Rousseeuw and Hubert
             (1997). If `"best"' exhaustive enumeration is done
             up to 5000 samples: if `"exact"' exhaustive enu-
             meration will be attempted however many samples
             are needed.

       seed: the seed to be used for random sampling: see
             `RNGkind'. The current value of `.Random.seed'
             will be preserved if it is set.

   DDeessccrriippttiioonn::

        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 `good' part
        of the data. `cov.mve' and `cov.mcd' are compatibility
        wrappers.

   DDeettaaiillss::

        For method `"mve"', an approximate search is made of a
        subset of size `quantile.used' with an enclosing ellip-
        soid of smallest volume; in method `"mcd"' it is the
        volume of the Gaussian confidence ellipsoid, equiva-
        lently the determinant of the classical covariance
        matrix, that is minimized. The mean of the subset pro-
        vides 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 calcu-
        lated, and those points within the 97.5% point under
        Gaussian assumptions are declared to be `good'. The
        final estimates are the mean and rescaled covariance of
        the `good' points.

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

        For method `"mve"' the search is made over ellipsoids
        determined by the covariance matrix of `p' of the data
        points. For method `"mcd"' an additional improvement
        step suggested by Rousseeuw and van Driessen (1997) is
        used, in which once a subset of size `quantile.used' is
        selected, an ellipsoid based on its covariance is
        tested (as this will have no larger a determinant, and
        may be smaller).

   VVaalluuee::

        A list with components

     center: the final estimate of location.

        cov: the final estimate of scatter.

        cor: (only is `cor=TRUE') the estimate of the correla-
             tion matrix.

       sing: message giving number of singular samples out of
             total

       crit: the value of the criterion on log scale. For MCD
             this is the determinant, and for MVE it is propor-
             tional to the volume.

       best: the subset used. For MVE the best sample, for MCD
             the best set of size `quantile.used'.

      n.obs: total number of observations.

   AAuutthhoorr((ss))::

        B.D. Ripley

   RReeffeerreenncceess::

        P. J. Rousseeuw and A. M. Leroy (1987) Robust Regres-
        sion and Outlier Detection.  Wiley.

        A. Marazzi (1993) 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, Journal of
        the American Statistical Association, 85, 633-639.

        P. J. Rousseeuw and K. van Driessen (1997) A fast algo-
        rithm for the minimum covariance determinant estimator.
        Technical Report, Department of Mathematics and Com-
        puter Science, Universitaire Instelling Antwerpen.

   SSeeee AAllssoo::

        `lqs'

   EExxaammpplleess::

        data(stackloss)
        .Random.seed <- 1:4
        cov.rob(stackloss)
        cov.rob(stack.x, method="mcd", nsamp="exact")

