covMcd {robustbase}

Robust Location and Scatter Estimation via MCD

Description

Compute the Minimum Covariance Determinant (MCD) estimator, a robust multivariate location and scale estimate with a high breakdown point, via the ‘Fast MCD’ or ‘Deterministic MCD’ (“DetMcd”) algorithm.

Usage

covMcd(x, cor = FALSE, raw.only = FALSE,
       alpha =, nsamp =, nmini =, kmini =,
       scalefn =, maxcsteps =,
       initHsets = NULL, save.hsets = FALSE, names = TRUE, 
       seed =, tolSolve =, trace =,
       use.correction =, wgtFUN =, control = rrcov.control())

Arguments

x

a matrix or data frame.

cor

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

raw.only

should only the “raw” estimate be returned, i.e., no (re)weighting step be performed; default is false.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized; roughly alpha*n, (see ‘Details’ below) 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 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.

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.

nmini, kmini

for n \ge 2 \times n_0, n_0 := \code{nmini}, the algorithm splits the data into maximally kmini (by default 5) subsets, of size approximately, but at least nmini. When nmini*kmini < n, the initial search uses only a subsample of size nmini*kmini. The original algorithm had nmini = 300 and kmini = 5 hard coded.

scalefn

for the deterministic MCD: function to compute a robust scale estimate or character string specifying a rule determining such a function. The default, currently "hrv2012", uses the recommendation of Hubert, Rousseeuw and Verdonck (2012) who recommend Qn for n < 1000 and scaleTau2 for larger n. Alternatively, scalefn = "v2014", uses that rule with cutoff n = 5000.

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.

names

logical; if true (as by default), several parts of the result have a names or dimnames respectively, derived from data matrix x.

seed

initial seed for random generator, like .Random.seed, see rrcov.control.

tolSolve

numeric tolerance to be used for inversion (solve) of the covariance matrix in mahalanobis.

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.

wgtFUN

a character string or function, specifying how the weights for the reweighting step should be computed. Up to April 2013, the only option has been the original proposal in (1999), now specified by wgtFUN = "01.original" (or via control). Since robustbase version 0.92-3, Dec.2014, other predefined string options are available, though experimental, see the experimental .wgtFUN.covMcd object.

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(\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 (.MCDcons(p, h/n)) and (if use.correction is true) a finite sample correction factor (.MCDcnp2(p, n, alpha)), to make it consistent at the normal model and unbiased at small samples. Both rescaling factors (consistency and finite sample) are returned in the length-2 vector raw.cnp2.

The implementation of covMcd uses the Fast MCD algorithm of Rousseeuw and Van Driessen (1999) to approximate the minimum covariance determinant estimator.

Based on these raw MCD estimates, (unless argument raw.only is true), a reweighting step is performed, i.e., V <- cov.wt(x,w), where w are weights determined by “outlyingness” with respect to the scaled raw MCD. Again, a consistency factor and (if use.correction is true) a finite sample correction factor (.MCDcnp2.rew(p, n, alpha)) are applied. The reweighted covariance is typically considerably more efficient than the raw one, see Pison et al. (2002).

The two rescaling factors for the reweighted estimates are returned in cnp2. Details for the computation of the finite sample correction factors can be found in Pison et al. (2002).

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 logarithm of the determinant. Previous to Nov.2014, it contained the determinant itself which can under- or overflow relatively easily.

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), starting with "Deterministic" when nsamp="deterministic".

iBest

(for the deterministic MCD) contains indices from 1:6 denoting which of the (six) initial subsets lead to the best set found.

n.csteps

(for the deterministic MCD) for each of the initial subsets, the number of C-steps executed till convergence.

call

the call used (see match.call).

Author(s)

Valentin Todorov [e-mail], based on work written for S-plus by Peter Rousseeuw and Katrien van Driessen from University of Antwerp.

Visibility of (formerly internal) tuning parameters, notably wgtFUN(): Martin Maechler

References

Rousseeuw, P. J. and Leroy, A. M. (1987) Robust Regression and Outlier Detection. Wiley.

Rousseeuw, P. J. and van Driessen, K. (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.

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

See Also

cov.mcd from package MASS; covOGK as cheaper alternative for larger dimensions.

BACON and covNNC, from package robustX;

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
set.seed(17)
(cH <- covMcd(hbk.x))
cH0 <- covMcd(hbk.x, nsamp = "deterministic")
with(cH0, stopifnot(quan == 39,
     iBest == c(1:4,6), # 5 out of 6 gave the same
     identical(raw.weights, mcd.wt),
     identical(which(mcd.wt == 0), 1:14), all.equal(crit, -1.045500594135)))

## 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

## Martin's smooth reweighting:

## List of experimental pre-specified wgtFUN() creators:
## Cutoffs may depend on  (n, p, control$beta) :
str(.wgtFUN.covMcd)

cMM <- covMcd(hbk.x, wgtFUN = "sm1.adaptive")

ina <- which(names(cH) == "call")
all.equal(cMM[-ina], cH[-ina]) # *some* differences, not huge (same 'best'):
stopifnot(all.equal(cMM[-ina], cH[-ina], tol = 0.2))