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,
                IQRtype = 7,
                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,
                mcReflect = n

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.

Arguments

x: a numeric \(n \times p\) 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.
IQRtype: a number from 1:9, denoting type of empirical quantile computation for the IQR(). The default 7 corresponds to quantile's and IQR's default. MM has evidence that type=6 would be a bit more stable for small sample size.
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.

mcReflect: passed as doReflect to mc().
mcScale: passed as doScale to mc().
mcMaxit: passed as maxit to mc().
mcEps1: passed as eps1 to mc(); the default is slightly looser (100 larger) than the default for mc().
mcEps2: passed as eps2 to mc().
mcTrace: passed as trace.lev to mc().

Author

Guy Brys; help page and improvements by Martin Maechler

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://research.ics.aalto.fi/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; tools:::Rd_expr_doi("10.1002/cem.1123").

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

Examples

Run this code

## An Example with bad condition number and "border case" outliers

dim(longley) # 16 x 7  // set seed, as result is random :
set.seed(31)
ao1 <- adjOutlyingness(longley, mcScale=FALSE)
## which are outlying ?
which(!ao1$nonOut) ## for this seed, two: "1956", "1957"; (often: none)
## For seeds 1:100, we observe (Linux 64b)
if(FALSE) {
  adjO <- sapply(1:100, function(iSeed) {
            set.seed(iSeed); adjOutlyingness(longley)$nonOut })
  table(nrow(longley) - colSums(adjO))
}
## #{outl.}:  0  1  2  3
## #{cases}: 74 17  6  3


## 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!)

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