CovOgk {rrcov}

Robust Location and Scatter Estimation - Ortogonalized Gnanadesikan-Kettenring (OGK)

Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the pairwise algorithm proposed by Marona and Zamar (2002) which in turn is based on the pairwise robust estimator proposed by Gnanadesikan-Kettenring (1972).

Usage

CovOgk(x, niter = 2, beta = 0.9, control)

Arguments

x

a matrix or data frame.

niter

number of iterations, usually 1 or 2 since iterations beyond the second do not lead to improvement.

beta

coverage parameter for the final reweighted estimate

control

a control object (S4) of class CovControlOgk-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. The control object contains also functions for computing the robust univariate location and dispersion estimate mrob and for computing the robust estimate of the covariance between two random variables vrob.

Details

The method proposed by Marona and Zamar (2002) allowes to obtain positive-definite and almost affine equivariant robust scatter matrices starting from any pairwise robust scatter matrix. The default robust estimate of covariance between two random vectors used is the one proposed by Gnanadesikan and Kettenring (1972) but the user can choose any other method by redefining the function in slot vrob of the control object CovControlOgk. Similarly, the function for computing the robust univariate location and dispersion used is the tau scale defined in Yohai and Zamar (1998) but it can be redefined in the control object.

The estimates obtained by the OGK method, similarly as in CovMcd are returned as 'raw' estimates. To improve the estimates a reweighting step is performed using the coverage parameter beta and these reweighted estimates are returned as 'final' estimates.

Value

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

Note

If the user does not specify a scale and covariance function to be used in the computations or specifies one by using the arguments smrob and svrob (i.e. the names of the functions as strings), a native code written in C will be called which is by far faster than the R version.

If the arguments mrob and vrob are not NULL, the specified functions will be used via the pure R implementation of the algorithm. This could be quite slow.

See CovControlOgk for details.

Author(s)

Valentin Todorov [e-mail] and Kjell Konis [e-mail]

References

Maronna, R.A. and Zamar, R.H. (2002) Robust estimates of location and dispersion of high-dimensional datasets; Technometrics 44(4), 307–317.

Yohai, R.A. and Zamar, R.H. (1998) High breakdown point estimates of regression by means of the minimization of efficient scale JASA 86, 403–413.

Gnanadesikan, R. and John R. Kettenring (1972) Robust estimates, residuals, and outlier detection with multiresponse data. Biometrics 28, 81–124.

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

See Also

CovMcd, CovMest

Examples

data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovOgk(hbk.x)

## the following three statements are equivalent
c1 <- CovOgk(hbk.x, niter=1)
c2 <- CovOgk(hbk.x, control = CovControlOgk(niter=1))

## direct specification overrides control one:
c3 <- CovOgk(hbk.x, beta=0.95,
             control = CovControlOgk(beta=0.99))
c1

x<-matrix(c(1,2,3,7,1,2,3,7), ncol=2)
##  CovOgk(x)   - this would fail because the two columns of x are exactly collinear.
##              In order to fix it, redefine the default 'vrob' function for example
##              in the following way and pass it as a parameter in the control
##              object.
cc <- CovOgk(x, control=new("CovControlOgk",
                            vrob=function(x1, x2, ...)
                            {
                                r <- .vrobGK(x1, x2, ...)
                                if(is.na(r))
                                    r <- 0
                                r
                            })
)
cc