huberize: Huberization - Bringing Outliers In
In robustbase: Basic Robust Statistics

huberize

R Documentation

Huberization – Bringing Outliers In

Description

Huberization (named after Peter Huber's M-estimation algorithm for location originally) replaces outlying values in a sample x by their respective boundary: when x_j < c_1 it is replaced by c_1 and when x_j > c_2 it is replaced by c_2. Consequently, values inside the interval [c_1, c_2] remain unchanged.

Here, c_j = M \pm c\cdot s where s := s(x) is the robust scale estimate Qn(x) if that is positive, and by default, M is the robust huber estimate of location \mu (with tuning constant k).

In the degenerate case where Qn(x) == 0, trimmed means of abs(x - M) are tried as scale estimate s, with decreasing trimming proportions specified by the decreasing trim vector.

Usage

huberize(x, M = huberM(x, k = k)$mu, c = k,
         trim = (5:1)/16,
         k = 1.5,
         warn0 = getOption("verbose"), saveTrim = TRUE)

Arguments

`x`	numeric vector which is to be huberized.
`M`	a number; defaulting to `huberM(x, k)`, the robust Huber M-estimator of location.
`c`	a positive number, the tuning constant for huberization of the sample `x`.
`trim`	a decreasing vector of trimming proportions in `[0, 0.5]`, only used to trim the absolute deviations from `M` in case `Qn(x)` is zero.
`k`	used if `M` is not specified as huberization center `M`, and so, by default is taken as Huber's M-estimate `huberM(x, k)`.
`warn0`	`logical` indicating if a warning should be signalled in case `Qn(x)` is zero and the trimmed means for all trimming proportions `trim` are zero as well.
`saveTrim`	a `logical` indicating if the last tried `trim[j]` value should be stored if `Qn(x)` was zero.

Details

In regular cases, s = Qn(x) is positive and used to huberize values of x outside [M - c*s, M + c*s].
In degenerate cases where Qn(x) == 0, we search for an s > 0 by trying the trimmed mean s := mean(abs(x-M), trim = trim[j]) with less and less trimming (as the trimming proportions trim[] must decrease). If even the last, trim[length(trim)], leads to s = 0, a warning is printed when warn0 is true.

Value

a numeric vector as x; in case Qn(x) was zero and saveTrim is true, also containing the (last) trim proportion used (to compute the scale s) as attribute "trim" (see attr(), attributes).

Note

For the use in mc() and similar cases where mainly numerical stabilization is necessary, a large c = 1e12 will lead to no huberization, i.e., all y == x for y <- huberize(x, c) for typical non-degenerate samples.

Author(s)

Martin Maechler

Examples

## For non-degenerate data and large c, nothing is huberized,
## as there are *no* really extreme outliers :
set.seed(101)
x <- rnorm(1000)
stopifnot(all.equal(x, huberize(x, c=100)))
## OTOH, the "extremes" are shrunken towards the boundaries for smaller c:
xh <- huberize(x, c = 2)
table(x != xh)
## 45 out of a 1000:
table(xh[x != xh])# 26 on the left boundary -2.098 and 19 on the right = 2.081
## vizualization:
stripchart(x); text(0,1, "x {original}", pos=3); yh <- 0.9
stripchart(xh, at = yh, add=TRUE, col=2)
text(0, yh, "huberize(x, c=2)",   col=2, pos=1)
arrows( x[x!=xh], 1,
       xh[x!=xh], yh, length=1/8, col=adjustcolor("pink", 1/2))

robustbase documentation built on Nov. 1, 2024, 3 p.m.