Robust Location and Scatter Estimation via MCD

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the ‘Fast MCD’ (Minimum Covariance Determinant) estimator.

Usage

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CovMcd(x,
       raw.only=FALSE, alpha=control@alpha, nsamp=control@nsamp,
       scalefn=control@scalefn, maxcsteps=control@maxcsteps,
       initHsets=NULL, save.hsets=FALSE,
       seed=control@seed, trace=control@trace,
       use.correction=control@use.correction,
       control=CovControlMcd(), ...)

Arguments

x

a matrix or data frame.

raw.only

should only the “raw” estimate be returned.

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", "exact" or "deterministic". Default is nsamp = 500. For nsamp="best" exhaustive enumeration is done, as long as the number of trials does not exceed 5000. For "exact", exhaustive enumeration will be attempted however many samples are needed. In this case a warning message will be displayed saying that the computation can take a very long time.

For "deterministic", the deterministic MCD is computed; as proposed by Hubert et al. (2012) it starts from the h most central observations of six (deterministic) estimators.

scalefn

function to compute a robust scale estimate or character string specifying a rule determining such a function, see rrcov.control.

maxcsteps

maximal number of concentration steps in the deterministic MCD; should not be reached.

initHsets

NULL or a K x h integer matrix of initial subsets of observations of size h (specified by the indices in 1:n).

save.hsets

(for deterministic MCD) logical indicating if the initial subsets should be returned as initHsets.

seed

starting value for random generator. Default is seed = NULL

trace

whether to print intermediate results. Default is trace = FALSE

use.correction

whether to use finite sample correction factors. Default is use.correction=TRUE

control

a control object (S4) of class CovControlMcd-class containing estimation options - same as these provided in the function specification. If the control object 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.

...

potential further arguments passed to robustbase's covMcd.

Details

This function computes the minimum covariance determinant estimator of location and scatter and returns an S4 object of class CovMcd-class containing the estimates. The implementation of the function is similar to the existing R function covMcd() which returns an S3 object. The MCD method looks for the h (> n/2) 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). Both rescaling factors 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 efficiency 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. The implementation in rrcov uses the Fast MCD algorithm of Rousseeuw and Van Driessen (1999) to approximate the minimum covariance determinant estimator.

Value

An S4 object of class CovMcd-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at

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.

M. Hubert, P. Rousseeuw and T. Verdonck (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21(3), 618–637.

Pison, G., Van Aelst, S., and Willems, G. (2002), Small Sample Corrections for LTS and MCD, Metrika, 55, 111-123.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1–47. URL http://www.jstatsoft.org/v32/i03/.

See Also

cov.mcd from package MASS

Examples

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data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMcd(hbk.x)
cD <- CovMcd(hbk.x, nsamp = "deterministic")
summary(cD)

## the following three statements are equivalent
c1 <- CovMcd(hbk.x, alpha = 0.75)
c2 <- CovMcd(hbk.x, control = CovControlMcd(alpha = 0.75))
## direct specification overrides control one:
c3 <- CovMcd(hbk.x, alpha = 0.75,
             control = CovControlMcd(alpha=0.95))
c1

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