robustbase (version 0.5-0-1)

covMcd: Robust Location and Scatter Estimation via MCD

Description

Compute a robust multivariate location and scale estimate with a high breakdown point, using the Fast MCD (Minimum Covariance Determinant) estimator.

Usage

covMcd(x, cor = FALSE, alpha = 1/2, nsamp = 500, seed = NULL,
       trace = FALSE, use.correction = TRUE, control = rrcov.control())

Arguments

x

a matrix or data frame.

cor

should the returned result include a correlation matrix? Default is cor = FALSE

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized, i.e., alpha*n observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.5.

nsamp

number of subsets used for initial estimates or "best" or "exact". Default is nsamp = 500. For nsamp = "best" exhaustive enumeration is done, as long as the number of trials does not exc

seed

initial seed for random generator, see rrcov.control.

trace

logical (or integer) indicating if intermediate results should be printed; defaults to FALSE; values $\ge 2$ also produce print from the internal (Fortran) code.

use.correction

whether to use finite sample correction factors; defaults to TRUE.

control

a list with estimation options - this includes those above provided in the function specification, see rrcov.control for the defaults. If control is supplied, the paramete

Value

• An object of class "mcd" which is basically a list with components
• center
  the final estimate of location.
• cov
  the final estimate of scatter.
• cor
  the (final) estimate of the correlation matrix (only if cor = TRUE).
• crit
  the value of the criterion, i.e. the determinant.
• best
  the best subset found and used for computing the raw estimates, with length(best) == quan = h.alpha.n(alpha,n,p).
• mah
  mahalanobis distances of the observations using the final estimate of the location and scatter.
• mcd.wt
  weights of the observations using the final estimate of the location and scatter.
• 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.
• raw.center
  the raw (not reweighted) estimate of location.
• raw.cov
  the raw (not reweighted) estimate of scatter.
• raw.mah
  mahalanobis distances of the observations based on the raw estimate of the location and scatter.
• raw.weights
  weights of the observations based on the raw estimate of the location and scatter.
• 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.
• X
  the input data as numeric matrix, without NAs.
• n.obs
  total number of observations.
• alpha
  the size of the subsets over which the determinant is minimized (the default is $(n+p+1)/2$).
• quan
  the number of observations, $h$, on which the MCD is based. If quan equals n.obs, the MCD is the classical covariance matrix.
• method
  character string naming the method (Minimum Covariance Determinant).
• call
  the call used (see match.call).

concept

High breakdown point

Details

The minimum covariance determinant estimator of location and scatter implemented in covMcd() is similar to Rfunction cov.mcd() in MASS. The MCD method looks for the $h (> n/2)$ ($h = h(\alpha,n,p) =$ h.alpha.n(alpha,n,p)) observations (out of $n$) whose classical covariance matrix has the lowest possible determinant. The raw MCD estimate of location is then the average of these $h$ 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 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 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 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 use.correction = FALSE.

References

P. J. Rousseeuw and A. M. Leroy (1987) Robust Regression and Outlier Detection. Wiley. P. J. Rousseeuw and K. van Driessen (1999) A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212--223. Pison, G., Van Aelst, S., and Willems, G. (2002), Small Sample Corrections for LTS and MCD, Metrika, 55, 111-123.

See Also

cov.mcd from package MASS; 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