rrcov: CovSest – R documentation

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CovSest

S Estimates of Multivariate Location and Scatter

Description

Computes S-Estimates of multivariate location and scatter based on Tukey's biweight function using a fast algorithm similar to the one proposed by Salibian-Barrera and Yohai (2006) for the case of regression. Alternativley, the Ruppert's SURREAL algorithm, bisquare or Rocke type estimation can be used.

Usage

CovSest(x, bdp = 0.5, arp = 0.1, eps = 1e-5, maxiter = 120,
        nsamp = 500, seed = NULL, trace = FALSE, tolSolve = 1e-14,
        scalefn, maxisteps=200, initHsets = NULL, save.hsets = FALSE,
        method = c("sfast", "surreal", "bisquare", "rocke", "suser", "sdet"), control,
        t0, S0, initcontrol)

Arguments

`x`	a matrix or data frame.
`bdp`	a numeric value specifying the required breakdown point. Allowed values are between `(n - p)/(2 * n)` and 1 and the default is `bdp=0.5`.
`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 `arp=0.1`.
`eps`	a numeric value specifying the relative precision of the solution of the S-estimate (bisquare and Rocke type). Default is to `eps=1e-5`.
`maxiter`	maximum number of iterations allowed in the computation of the S-estimate (bisquare and Rocke type). Default is `maxiter=120`.
`nsamp`	the number of random subsets considered. The default is different for the different methods: (i) for `sfast` it is `nsamp = 20`, (ii) for `surreal` it is `nsamp = 600*p` and (iii) for `bisquare` or `rocke` it is `nsamp = 500`.
`seed`	starting value for random generator. Default is `seed = NULL`.
`trace`	whether to print intermediate results. Default is `trace = FALSE`.
`tolSolve`	numeric tolerance to be used for inversion (`solve`) of the covariance matrix in `mahalanobis`.
`scalefn`	`function` to compute a robust scale estimate or character string specifying a rule determining such a function. Used for computing the "deterministic" S-estimates (`method="sdet"`). If `scalefn` is missing or is `NULL`, the function is selected depending on the data set size, following the recomendation of Hubert et al. (2012) - `Qn` if `n <= 1000` and `scaleTau2` otherwise.
`maxisteps`	maximal number of concentration steps in the deterministic S-estimates; 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 `1:n`).
`save.hsets`	(for deterministic S-estimates) logical indicating if the initial subsets should be returned as `initHsets`.
`method`	Which algorithm to use: 'sfast'=C implementation of FAST-S, 'surreal'=SURREAL, 'bisquare', 'rocke'. The method 'suser' currently calls the R implementation of FAST-S but in the future will allow the user to supply own `rho` function. The method 'sdet' invokes the deterministic algorihm of Hubert et al. (2012).
`control`	a control object (S4) of class `CovControlSest-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 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 S-estimator of location and scatter. The computation will be performed by one of the following algorithms:

FAST-S: An algorithm similar to the one proposed by Salibian-Barrera and Yohai (2006) for the case of regression
SURREAL: Ruppert's SURREAL algorithm when method is set to 'surreal'
BISQUARE: Bisquare S-Estimate with method set to 'bisquare'
ROCKE: Rocke type S-Estimate 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 CovSest-class which is a subclass of the virtual class CovRobust-class.

Author(s)

Valentin Todorov valentin.todorov@chello.at, Matias Salibian-Barrera 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 S-estimators and M-estimators 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. Salibian-Barrera and V. Yohai (2006) A fast algorithm for S-regression 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

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

rrcov

Scalable Robust Estimators with High Breakdown Point

v1.5-5

GPL (>= 2)

Authors

Valentin Todorov [aut, cre] (<https://orcid.org/0000-0003-4215-0245>)

Initial release

2020-07-31