Robust Location and Scatter Estimation via MVE

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Description

Computes a robust multivariate location and scatter estimate with a high breakdown point, using the ‘MVE’ (Minimum Volume Ellipsoid) estimator.

Usage

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CovMve(x, alpha = 1/2, nsamp = 500, seed = NULL, trace = FALSE, control)

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.

nsamp

number of subsets used for initial estimates or "best" or "exact". Default is nsamp = 500. For nsamp="best" exhaustive enumeration is done, as long as the number of trials does not exceed 5000. For "exact", exhaustive enumeration will be attempted however many samples are needed. In this case a warning message will be displayed saying that the computation can take a very long time.

seed

starting value for random generator. Default is seed = NULL

trace

whether to print intermediate results. Default is trace = FALSE

control

a control object (S4) of class CovControlMve-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.

Details

This function computes the minimum volume ellipsoid estimator of location and scatter and returns an S4 object of class CovMve-class containing the estimates.

The approximate estimate is based on a subset of size alpha*n with an enclosing ellipsoid of smallest volume. The mean of the best found subset provides the raw estimate of the location, and the rescaled covariance matrix is the raw estimate of scatter. The rescaling of the raw covariance matrix is by median(dist)/qchisq(0.5, p) and this scale factor is returned in the slot raw.cnp2. Currently no finite sample corrction factor is applied. The Mahalanobis distances of all observations from the location estimate for the raw covariance matrix are calculated, and those points within the 97.5 under Gaussian assumptions are declared to be good. The final (reweightd) estimates are the mean and rescaled covariance of the good points. The reweighted covariance matrix is rescaled by 1/pgamma(qchisq(alpha, p)/2, p/2 + 1)/alpha (see Croux and Haesbroeck, 1999) and this scale factor is returned in the slot cnp2.

The search for the approximate solution is made over ellipsoids determined by the covariance matrix of p+1 of the data points and applying a simple but effective improvement of the subsampling procedure as described in Maronna et al. (2006), p. 198. Although there exists no formal proof of this improvement (as for MCD and LTS), simulations show that it can be recommended as an approximation of the MVE.

Value

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

Note

Main reason for implementing the MVE estimate was that it is the recommended initial estimate for S estimation (see Maronna et al. (2006), p. 199) and will be used by default in CovMest (after removing the correction factors from the covariance matrix and rescaling to determinant 1).

Author(s)

Valentin Todorov valentin.todorov@chello.at and Matias Salibian-Barrera matias@stat.ubc.ca

References

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

C. Croux and G. Haesbroeck (1999). Influence function and efficiency of the minimum covariance determinant scatter matrix estimator. Journal of Multivariate Analysis, 71, 161–190.

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/.

See Also

cov.mve from package MASS

Examples

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

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

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