S Estimates of Multivariate Location and Scatter
Description
Computes SEstimates of multivariate location and scatter based on Tukey's biweight function using a fast algorithm similar to the one proposed by SalibianBarrera and Yohai (2006) for the case of regression. Alternativley, the Ruppert's SURREAL algorithm, bisquare or Rocke type estimation can be used.
Usage
1 2 3 4 5 
Arguments
x 
a matrix or data frame. 
bdp 
a numeric value specifying the required
breakdown point. Allowed values are between

arp 
a numeric value specifying the asympthotic
rejection point (for the Rocke type S estimates),
i.e. the fraction of points receiving zero
weight (see Rocke (1996)). Default is 
eps 
a numeric value specifying the
relative precision of the solution of the Sestimate
(bisquare and Rocke type). Default is to 
maxiter 
maximum number of iterations allowed
in the computation of the Sestimate (bisquare and Rocke type).
Default is 
nsamp 
the number of random subsets considered. The default is different for the different methods:
(i) for 
seed 
starting value for random generator. Default is 
trace 
whether to print intermediate results. Default is 
tolSolve 
numeric tolerance to be used for inversion
( 
scalefn 

maxisteps 
maximal number of concentration steps in the deterministic Sestimates; should not be reached. 
initHsets 
NULL or a K x n integer matrix of initial
subsets of observations of size (specified by the indices in

save.hsets 
(for deterministic Sestimates) logical indicating if the
initial subsets should be returned as 
method 
Which algorithm to use: 'sfast'=C implementation of FASTS, 'surreal'=SURREAL,
'bisquare', 'rocke'. The method 'suser' currently calls the R implementation of FASTS
but in the future will allow the user to supply own 
control 
a control object (S4) of class 
t0 
optional initial HBDP estimate for the center 
S0 
optional initial HBDP estimate for the covariance matrix 
initcontrol 
optional control object to be used for computing the initial HBDP estimates 
Details
Computes multivariate Sestimator of location and scatter. The computation will be performed by one of the following algorithms:
 FASTS
An algorithm similar to the one proposed by SalibianBarrera and Yohai (2006) for the case of regression
 SURREAL
Ruppert's SURREAL algorithm when
method
is set to 'surreal' BISQUARE
Bisquare SEstimate with
method
set to 'bisquare' ROCKE
Rocke type SEstimate with
method
set to 'rocke'
Except for the last algorithm, ROCKE, all other use Tukey biweight loss function.
The tuning parameters used in the loss function (as determined by bdp) are
returned in the slots cc
and kp
of the result object. They can be computed
by the internal function .csolve.bw.S(bdp, p)
.
Value
An S4 object of class CovSestclass
which is a subclass of the
virtual class CovRobustclass
.
Author(s)
Valentin Todorov valentin.todorov@chello.at, Matias SalibianBarrera matias@stat.ubc.ca and Victor Yohai vyohai@dm.uba.ar. See also the code from Kristel Joossens, K.U. Leuven, Belgium and Ella Roelant, Ghent University, Belgium.
References
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.
M. Hubert, P. Rousseeuw, D. Vanpaemel and T. Verdonck (2015) The DetS and DetMM estimators for multivariate location and scatter. Computational Statistics and Data Analysis 81, 64–75.
H.P. Lopuhaä (1989) On the Relation between Sestimators and Mestimators of Multivariate Location and Covariance. Annals of Statistics 17 1662–1683.
D. Ruppert (1992) Computing S Estimators for Regression and Multivariate Location/Dispersion. Journal of Computational and Graphical Statistics 1 253–270.
M. SalibianBarrera and V. Yohai (2006) A fast algorithm for Sregression estimates, Journal of Computational and Graphical Statistics, 15, 414–427.
R. A. Maronna, D. Martin and V. Yohai (2006). Robust Statistics: Theory and Methods. Wiley, New York.
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/.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41  library(rrcov)
data(hbk)
hbk.x < data.matrix(hbk[, 1:3])
cc < CovSest(hbk.x)
cc
## summry and different types of plots
summary(cc)
plot(cc)
plot(cc, which="dd")
plot(cc, which="pairs")
plot(cc, which="xydist")
## the following four statements are equivalent
c0 < CovSest(hbk.x)
c1 < CovSest(hbk.x, bdp = 0.25)
c2 < CovSest(hbk.x, control = CovControlSest(bdp = 0.25))
c3 < CovSest(hbk.x, control = new("CovControlSest", bdp = 0.25))
## direct specification overrides control one:
c4 < CovSest(hbk.x, bdp = 0.40,
control = CovControlSest(bdp = 0.25))
c1
summary(c1)
plot(c1)
## Use the SURREAL algorithm of Ruppert
cr < CovSest(hbk.x, method="surreal")
cr
## Use Bisquare estimation
cr < CovSest(hbk.x, method="bisquare")
cr
## Use Rocke type estimation
cr < CovSest(hbk.x, method="rocke")
cr
## Use Deterministic estimation
cr < CovSest(hbk.x, method="sdet")
cr
