rrcov (version 1.7-2)

CovMrcd: Robust Location and Scatter Estimation via Minimum Regularized Covariance Determonant (MRCD)

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the Minimum Regularized Covariance Determonant (MRCD) estimator.

Usage

CovMrcd(x,
       alpha=control@alpha, 
       h=control@h,
       maxcsteps=control@maxcsteps,
       initHsets=NULL, save.hsets=FALSE,
       rho=control@rho,
       target=control@target,
       maxcond=control@maxcond,
       trace=control@trace,
       control=CovControlMrcd())

Value

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

Arguments

x

a matrix or data frame.

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.

h

the size of the subset (can be between ceiling(n/2) and n). Normally NULL and then it h will be calculated as h=ceiling(alpha*n). If h is provided, alpha will be calculated as alpha=h/n.

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.

rho

regularization parameter. Normally NULL and will be estimated from the data.

target

structure of the robust positive definite target matrix: a) "identity": target matrix is diagonal matrix with robustly estimated univariate scales on the diagonal or b) "equicorrelation": non-diagonal target matrix that incorporates an equicorrelation structure (see (17) in paper). Default is target="identity"

maxcond

maximum condition number allowed (see step 3.4 in algorithm 1). Default is maxcond=50

trace

whether to print intermediate results. Default is trace = FALSE

control

a control object (S4) of class CovControlMrcd-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.

Author

Kris Boudt, Peter Rousseeuw, Steven Vanduffel and Tim Verdonk. Improved by Joachim Schreurs and Iwein Vranckx. Adapted for rrcov by Valentin Todorov

Details

This function computes the minimum regularized covariance determinant estimator (MRCD) of location and scatter and returns an S4 object of class CovMrcd-class containing the estimates. Similarly like the MCD method, MRCD looks for the \(h (> n/2)\) observations (out of \(n\)) whose classical covariance matrix has the lowest possible determinant, but replaces the subset-based covariance by a regularized covariance estimate, defined as a weighted average of the sample covariance of the h-subset and a predetermined positive definite target matrix. The Minimum Regularized Covariance Determinant (MRCD) estimator is then the regularized covariance based on the h-subset which makes the overall determinant the smallest. A data-driven procedure sets the weight of the target matrix (rho), so that the regularization is only used when needed.

References

Kris Boudt, Peter Rousseeuw, Steven Vanduffel and Tim Verdonck (2020) The Minimum Regularized Covariance Determinant estimator, Statistics and Computing, 30, pp 113--128 tools:::Rd_expr_doi("10.1007/s11222-019-09869-x").

Mia Hubert, Peter Rousseeuw and Tim Verdonck (2012) A deterministic algorithm for robust location and scatter. Journal of Computational and Graphical Statistics 21(3), 618--637.

Todorov V & Filzmoser P (2009), An Object Oriented Framework for Robust Multivariate Analysis. Journal of Statistical Software, 32(3), 1--47. tools:::Rd_expr_doi("10.18637/jss.v032.i03").

See Also

CovMcd

Examples

## The result will be (almost) identical to the raw MCD
##  (since we do not do reweighting of MRCD)
##
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
c0 <- CovMcd(hbk.x, alpha=0.75, use.correction=FALSE)
cc <- CovMrcd(hbk.x, alpha=0.75)
cc$rho
all.equal(c0$best, cc$best)
all.equal(c0$raw.center, cc$center)
all.equal(c0$raw.cov/c0$raw.cnp2[1], cc$cov/cc$cnp2)

summary(cc)

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

if (FALSE) {

##  This is the first example from Boudt et al. (2020). The first variable is 
##  the dependent one, which we remove and remain with p=226 NIR absorbance spectra 

data(octane)

octane <- octane[, -1]    # remove the dependent variable y

n <- nrow(octane)
p <- ncol(octane)

##  Compute MRCD with h=33, which gives approximately 15 percent breakdown point.
##  This value of h was found by Boudt et al. (2020) using a data driven approach, 
##  similar to the Forward Search of Atkinson et al. (2004). 
##  The default value of h would be 20 (i.e. alpha=0.5) 

out <- CovMrcd(octane, h=33) 
out$rho

## Please note that in the paper is indicated that the obtained rho=0.1149, however,
##  this value of rho is obtained if the parameter maxcond is set equal to 999 (this was 
##  the default in an earlier version of the software, now the default is maxcond=50). 
##  To reproduce the result from the paper, change the call to CovMrcd() as follows 
##  (this will not influence the results shown further):

##  out <- CovMrcd(octane, h=33, maxcond=999) 
##  out$rho

robpca = PcaHubert(octane, k=2, alpha=0.75, mcd=FALSE)
(outl.robpca = which(robpca@flag==FALSE))

# Observations flagged as outliers by ROBPCA:
# 25, 26, 36, 37, 38, 39

# Plot the orthogonal distances versus the score distances:
pch = rep(20,n); pch[robpca@flag==FALSE] = 17
col = rep('black',n); col[robpca@flag==FALSE] = 'red'
plot(robpca, pch=pch, col=col, id.n.sd=6, id.n.od=6)

## Plot now the MRCD mahalanobis distances
pch = rep(20,n); pch[!getFlag(out)] = 17
col = rep('black',n); col[!getFlag(out)] = 'red'
plot(out, pch=pch, col=col, id.n=6)
}