covMcd {robustbase}R Documentation

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, nmini = 300, 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 exceed 100'000 (= nLarge). For "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.

nmini

for large n, the algorithm splits the data into maximally krep = 5 subsets of size nmini. The original algorithm had nmini = 300 hard coded.

seed

initial seed for random generator, see rrcov.control.

trace

logical (or integer) indicating if intermediate results should be printed; defaults to FALSE; values >= 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 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 covMcd() is similar to R function cov.mcd() in MASS. The MCD method looks for the h (> n/2) (h = h(α,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.

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).

Author(s)

Valentin Todorov valentin.todorov@chello.at, based on work written for S-plus by Peter Rousseeuw and Katrien van Driessen from University of Antwerp.

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

[Package robustbase version 0.8-0 Index]