robustbase (version 0.93-6)

adjOutlyingness: Compute (Skewness-adjusted) Multivariate Outlyingness

Description

For an \(n \times p\) data matrix (or data frame) x, compute the “outlyingness” of all \(n\) observations. Outlyingness here is a generalization of the Donoho-Stahel outlyingness measure, where skewness is taken into account via the medcouple, mc().

Usage

adjOutlyingness(x, ndir = 250, p.samp = p, clower = 4, cupper = 3,
                alpha.cutoff = 0.75, coef = 1.5,
                qr.tol = 1e-12, keep.tol = 1e-12,
                only.outlyingness = FALSE, maxit.mult = max(100, p),
                trace.lev = 0)

Arguments

x

a numeric matrix or data.frame, which must be of full rank \(p\).

ndir

positive integer specifying the number of directions that should be searched.

p.samp

the sample size to use for finding good random directions, must be at least p. The default, p had been hard coded previously.

clower, cupper

the constant to be used for the lower and upper tails, in order to transform the data towards symmetry. You can set clower = 0, cupper = 0 to get the non-adjusted, i.e., classical (“central” or “symmetric”) outlyingness. In that case, mc() is not used.

alpha.cutoff

number in (0,1) specifying the quantiles \((\alpha, 1-\alpha)\) which determine the “outlier” cutoff. The default, using quartiles, corresponds to the definition of the medcouple (mc), but there is no stringent reason for using the same alpha for the outlier cutoff.

coef

positive number specifying the factor with which the interquartile range (IQR) is multiplied to determine ‘boxplot hinges’-like upper and lower bounds.

qr.tol

positive tolerance to be used for qr and solve.qr for determining the ndir directions, each determined by a random sample of \(p\) (out of \(n\)) observations. Note that the default \(10^{-12}\) is rather small, and qr's default = 1e-7 may be more appropriate.

keep.tol

positive tolerance to determine which of the sample direction should be kept, namely only those for which \(\|x\| \cdot \|B\|\) is larger than keep.tol.

only.outlyingness

logical indicating if the final outlier determination should be skipped. In that case, a vector is returned, see ‘Value:’ below.

maxit.mult

integer factor; maxit <- maxit.mult * ndir will determine the maximal number of direction searching iterations. May need to be increased for higher dimensional data, though increasing ndir may be more important.

trace.lev

an integer, if positive allows to monitor the direction search.

Value

If only.outlyingness is true, a vector adjout, otherwise, as by default, a list with components

adjout

numeric of length(n) giving the adjusted outlyingness of each observation.

cutoff

cutoff for “outlier” with respect to the adjusted outlyingnesses, and depending on alpha.cutoff.

nonOut

logical of length(n), TRUE when the corresponding observation is non-outlying with respect to the cutoff and the adjusted outlyingnesses.

Details

FIXME: Details in the comment of the Matlab code; also in the reference(s).

The method as described can be useful as preprocessing in FASTICA (http://www.cis.hut.fi/projects/ica/fastica/; see also the R package fastICA.

References

Brys, G., Hubert, M., and Rousseeuw, P.J. (2005) A Robustification of Independent Component Analysis; Journal of Chemometrics, 19, 1--12.

Hubert, M., Van der Veeken, S. (2008) Outlier detection for skewed data; Journal of Chemometrics 22, 235--246.

For the up-to-date reference, please consult http://wis.kuleuven.be/stat/robust

See Also

the adjusted boxplot, adjbox and the medcouple, mc.

Examples

# NOT RUN {
## An Example with bad condition number and "border case" outliers

dim(longley)
set.seed(1) ## result is random!
ao1 <- adjOutlyingness(longley)
## which are outlying ?
which(!ao1$nonOut) ## one: "1948" - for this seed! (often: none)
all(ao1$nonOut[-2]) # TRUE typically (R's own BLAS, Lapack, ..)

## An Example with outliers :

dim(hbk)
set.seed(1)
ao.hbk <- adjOutlyingness(hbk)
str(ao.hbk)
hist(ao.hbk $adjout)## really two groups
table(ao.hbk$nonOut)## 14 outliers, 61 non-outliers:
## outliers are :
which(! ao.hbk$nonOut) # 1 .. 14   --- but not for all random seeds!

## here, they are the same as found by (much faster) MCD:
cc <- covMcd(hbk)
stopifnot(all(cc$mcd.wt == ao.hbk$nonOut))

## This is revealing: About 1--2 cases, where outliers are *not* == 1:14
##  but needs almost 1 [sec] per call:
if(interactive()) {
  for(i in 1:30) {
    print(system.time(ao.hbk <- adjOutlyingness(hbk)))
    if(!identical(iout <- which(!ao.hbk$nonOut), 1:14)) {
	 cat("Outliers:\n"); print(iout)
    }
  }
}

## "Central" outlyingness: *not* calling mc()  anymore, since 2014-12-11:
trace(mc)
out <- capture.output(
  oo <- adjOutlyingness(hbk, clower=0, cupper=0)
)
untrace(mc)
stopifnot(length(out) == 0)

## A rank-deficient case
T <- tcrossprod(data.matrix(toxicity))
try(adjOutlyingness(T, maxit. = 20, trace.lev = 2)) # fails and recommends:
T. <- fullRank(T)
aT <- adjOutlyingness(T.)
plot(sort(aT$adjout, decreasing=TRUE), log="y")
plot(T.[,9:10], col = (1:2)[1 + (aT$adjout > 10000)])
## .. (not conclusive; directions are random, more 'ndir' makes a difference!)
# }