dirOutl  R Documentation 
Computes the directional outlyingness of p
dimensional points z
relative to a p
dimensional dataset x
. For each multivariate point z_i
, its directional outlyingness relative to x
is defined as its maximal univariate directional outlyingness measured over all directions. To obtain the univariate directional outlyingness in the direction v
, the dataset x
is projected on v
, and the robustly skewadjusted standardized distance of v'z_i
to the median of the projected data points x
v
is computed. This is done through the estimation of 2 scales, one on each side of the median, using a 1step Mestimator of scale.
dirOutl(x, z = NULL, options = list())
x 
An 
z 
An optional 
options 
A list of available options:

The directional outlyingness (DO) of multivariate data was introduced in Rousseeuw et al. (2018). It extends the StahelDonoho outlyingness towards skewed distributions.
Depending on the dimension p
, different approximate algorithms are implemented. The affine invariant algorithm can only be used when n > p
. It draws ndir
times at random p
observations from x
and considers the direction orthogonal to the hyperplane spanned by these p
observations. At most p
out of n
directions can be considered. The orthogonal invariant version can be applied to highdimensional data. It draws ndir
times at random 2
observations from x
and considers the direction through these two observations. Here, at most 2 out of n
directions can be considered. Finally, the shift invariant version randomly draws ndir
vectors from the unit sphere.
The resulting DO values are invariant to affine transformations, rotations and shifts respectively provided that the seed
is kept fixed at different runs of the algorithm. Note that the DO values are guaranteed to increase when more directions are considered provided the seed is kept fixed, as this ensures that the random directions are generated in a fixed order.
An observation from x
and z
is flagged as an outlier if its DO exceeds a cutoff value. This cutoff value is determined using the procedure in Rousseeuw et al. (2018). First, the logarithm of the DO values is taken to render their distribution more symmetric, after which a normal approximation yields a cutoff on these values. The cutoff is then transformed back by applying the exponential function.
It is first checked whether the data lie in a subspace of dimension smaller than p
. If so, a warning is given, as well as the dimension of the subspace and a direction which is orthogonal to it. Furthermore, the univariate directional outlyingness of the projected points x
v
is illdefined when the scale in its denominator becomes zero. This can happen when many observations collapse. In these cases the algorithm will stop and give a warning. The returned values then include the direction v
as well as an indicator specifying which of the observations of x
belong to the hyperplane orthogonal to v
.
A list with components:
outlyingnessX 
Vector of length 
outlyingnessZ 
Vector of length 
cutoff 
Points whose directional outlyingness exceeds this cutoff can be considered as outliers with respect to 
flagX 
Observations of 
flagZ 
Points of 
singularSubsets 
When the input parameter type is equal to 
dimension 
When the data 
hyperplane 
When the data 
inSubspace 
When a direction 
J. Raymaekers and P. Rousseeuw
Rousseeuw P.J., Raymaekers J., Hubert M. (2018). A measure of directional outlyingness with applications to image data and video. Journal of Computational and Graphical Statistics, 27, 345–359.
dprojdepth
, dprojmedian
, adjOutl
, outlyingness
# Compute the directional outlyingness of a twodimensional dataset.
# Outliers are plotted in red.
data(geological)
BivData < geological[c("MnO", "MgO")]
Result < dirOutl(x = BivData)
IndOutliers < which(!Result$flagX)
plot(BivData, pch = 16, col = "grey60")
points(BivData[IndOutliers, ], pch = 16, col = "red")
# The number of directions may be specified through
# the option list. The resulting directional outlyingness
# is monotone increasing in the number of directions.
Result1 < dirOutl(x = BivData, options = list(ndir = 50))
Result2 < dirOutl(x = BivData, options = list(ndir = 100))
which(Result2$outlyingnessX  Result1$outlyingnessX < 0)
# This is however not the case when the seed is changed
Result1 < dirOutl(x = BivData, options = list(ndir = 50))
Result2 < dirOutl(x = BivData, options = list(ndir = 100, seed = 950))
plot(Result2$outlyingnessX  Result1$outlyingnessX,
xlab = "Index", ylab = "Difference in DO")
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