Computes several measures of functional outlyingness for multivariate functional data.
fOutl(x, z = NULL, type = "fAO", alpha = 0, time = NULL,
diagnostic = FALSE, distOptions = NULL)A list with the following components:
Vector of length \(n\) containing the
functional outlyingness of every curve
from x.
Vector of length \(m\) containing the
functional outlyingness of every curve
from z.
Vector of weights according to the input parameter
alpha.
An \(n\) by \(t\) matrix containing the multivariate
outlyingness of each observation of x at each point.
Only provided if the input parameter diagnostic is
set to TRUE.
An \(m\) by \(t\) matrix containing the multivariate
outlyingness of each observation of z at each point.
Only provided if the input parameter diagnostic is
set to TRUE.
An \(n\) by \(t\) matrix flagging local outlyingness for
x. Only provided if the input parameter diagnostic
is set to TRUE.
The \((i,j)\)th element takes value 1 if curve
\(x_i\) is outlying at time point \(j\).
An \(m\) by \(t\) matrix flagging local outlyingness for
z. Only provided if the input parameter diagnostic
is set to TRUE.
The \((i,j)\)th element takes value 1 if curve
\(z_i\) is outlying at time point \(j\).
Vector containing the indices of the time points for which an exact fit is detected.
A three dimensional \(t\) by \(n\) by \(p\) array, with \(t\) the number of observed time points, \(n\) the number of functional observations and \(p\) the number of measurements for every functional observation at every time point.
An optional three-dimensional \(t\) by \(m\) by \(p\)
array, containing the observations for which to compute the
functional outlyingness with respect to x.
If z is not specified, it is set equal to x.
The time points of z should correspond to those of
x.
The outlyingness measure used in the computations.
One of the following options: "fAO", "fSDO",
"fDO" or "fbd".
Defaults to "fAO".
Specifies the weights at every cross-section.
When alpha = 0, uniform weights are used. Otherwise
alpha should be a weight vector of length \(t\).
Defaults to 0.
If the measurements are not equidistant,
a sorted numeric vector containing a set of time points.
Defaults to 1:t.
If set to TRUE, the output contains some additional
components:
crossDists: an \(n\) by \(t\) matrix containing the
multivariate outlyingness of each observation at each time
point
locOutl: output containing flags for local outlyingness
(see "Value" for more details)
Defaults to FALSE.
A list of options to pass to the function
computing the cross-sectional distances.
See adjOutl, outlyingness, dirOutl, or
bagdistance.
P. Segaert
The functional outlyingness of a multivariate curve with respect to a
given set of multivariate curves is defined as the weighted average of its
multivariate outlyingness at each time point (Hubert et al., 2015).
The functional outlyingness can be computed in all dimensions \(p\) using
the adjusted outlyingness (fAO), the directional outlyingness (fDO),
the Stahel-Donoho outlyingness (fSDO) or the bagdistance (fbd).
When the data array z is specified, the functional outlyingness and diagnostic
information for the data array x is also returned whenever the underlying
outlyingness routine allows it. For more information see the specific routines
listed in the section "See Also".
In some situations, additional diagnostics are available to flag outlying time
points. At each time point, observations from the data array x are marked
if they are flagged as outliers. The observations from the data array x
are marked if their scaled outlyingness is larger than a prescribed cutoff value
from the chi-square distribution. For more details see the respective
outlyingness routines.
It is possible that at certain time points a part of the algorithm can not be executed due to e.g. exact fits. In that case the weight of that particular time point is set to zero. A warning is issued at the end of the algorithm to signal these time points. Furthermore the output contains an extra argument giving the indices of the time points where problems occured.
Hubert M., Rousseeuw P.J., Segaert P. (2015). Multivariate functional outlier detection (with rejoinder). Statistical Methods and Applications, 24, 177--202.
Hubert M., Rousseeuw P.J., Segaert P. (2017). Multivariate and functional classification using depth and distance. Advances in Data Analysis and Classification, 11, 445--466.
bagdistance, outlyingness, adjOutl, dirOutl, fom
data(octane)
Data <- octane
# When the option diagnostic is set to TRUE, a crude diagnostic
# to detect outliers can be extracted from the local outlyingness
# indicators.
Result <- fOutl(x = Data, type = "fAO", diagnostic = TRUE)
matplot(Data[,,1], type = "l", col = "black", lty = 1)
for (i in 1:dim(Data)[2]) {
if(sum(Result$locOutlZ[i, ]) > 0) {
obsData <- matrix(Data[,i,1], nrow = 1)
obsData[!Result$locOutlZ[i,]] <- NA
obsData <- rbind(obsData, obsData)
matpoints(t(obsData), col = "red", pch = 15)
}
}
# For more advanced outlier detection techniques, see the
# fom routine.
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