CovMest: Constrained M-Estimates of Location and Scatter
In armstrtw/rrcov: Scalable Robust Estimators with High Breakdown Point

Description Usage Arguments Details Value Note Author(s) References See Also Examples

Computes constrained M-Estimates of multivariate location and scatter based on the translated biweight function (‘t-biweight’) using a High breakdown point initial estimate as defined by Rocke (1996). The default initial estimate is the Minimum Volume Ellipsoid computed with CovMve. The raw (not reweighted) estimates are taken and the covariance matrix is standardized to determinant 1.

1 2	CovMest(x, r = 0.45, arp = 0.05, eps=1e-3, maxiter=120, control, t0, S0, initcontrol)

`x`	a matrix or data frame.
`r`	required breakdown point. Allowed values are between `(n - p)/(2 * n)` and 1 and the default is 0.45
`arp`	asympthotic rejection point, i.e. the fraction of points receiving zero weight (see Rocke (1996)). Default is `0.05`.
`eps`	a numeric value specifying the relative precision of the solution of the M-estimate. Defaults to `1e-3`
`maxiter`	maximum number of iterations allowed in the computation of the M-estimate. Defaults to 120
`control`	a control object (S4) of class `CovControlMest-class` containing estimation options - same as these provided in the fucntion 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.
`t0`	optional initial high breakdown point estimates of the location. If not supplied MVE will be used.
`S0`	optional initial high breakdown point estimates of the scatter. If not supplied MVE will be used.
`initcontrol`	optional control object - of class CovControl - specifing the initial high breakdown point estimates of location and scatter. If not supplied MVE will be used.

Rocke (1996) has shown that the S-estimates of multivariate location and scatter in high dimensions can be sensitive to outliers even if the breakdown point is set to be near 0.5. To mitigate this problem he proposed to utilize the translated biweight (or t-biweight) method with a standardization step consisting of equating the median of rho(d) with the median under normality. This is then not an S-estimate, but is instead a constrained M-estimate. In order to make the smooth estimators to work, a reasonable starting point is necessary, which will lead reliably to a good solution of the estimator. In CovMest the MVE computed by CovMve is used, but the user has the possibility to give her own initial estimates.

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

The psi, rho and weight functions for the M estimation are encapsulated in a virtual S4 class PsiFun from which a PsiBwt class, implementing the translated biweight (t-biweight), is dervied. The base class PsiFun contains also the M-iteration itself. Although not documented and not accessibale directly by the user these classes will form the bases for adding other functions (biweight, LWS, etc.) as well as S-estimates.

Valentin Todorov valentin.todorov@chello.at,

(some code from C. Becker - http://www.sfb475.uni-dortmund.de/dienst/de/content/struk-d/bereicha-d/tpa1softw-d.html)

D.L.Woodruff and D.M.Rocke (1994) Computable robust estimation of multivariate location and shape on high dimension using compound estimators, Journal of the American Statistical Association, 89, 888–896.

D.M.Rocke (1996) Robustness properties of S-estimates of multivariate location and shape in high dimension, Annals of Statistics, 24, 1327-1345.

D.M.Rocke and D.L.Woodruff (1996) Identification of outliers in multivariate data Journal of the American Statistical Association, 91, 1047–1061.

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/.

covMcd, Cov-class, CovMve, CovRobust-class, CovMest-class

library(rrcov)
data(hbk)
hbk.x <- data.matrix(hbk[, 1:3])
CovMest(hbk.x)

## the following four statements are equivalent
c0 <- CovMest(hbk.x)
c1 <- CovMest(hbk.x, r = 0.45)
c2 <- CovMest(hbk.x, control = CovControlMest(r = 0.45))
c3 <- CovMest(hbk.x, control = new("CovControlMest", r = 0.45))

## direct specification overrides control one:
c4 <- CovMest(hbk.x, r = 0.40,
             control = CovControlMest(r = 0.25))
c1
summary(c1)
plot(c1)