Computes the multivariate functional depth for multivariate functional data.
mfd(x, z = NULL, type = "hdepth", alpha = 0, time = NULL, diagnostic = FALSE,
depthOptions = NULL)A list with the following components:
Vector of length \(n\) containing the MFD depth
of every curve from x.
Vector of length \(m\) containing the MFD depth
of every curve from z.
Vector of weights according to the input parameter
alpha.
An \(n\) by \(t\) matrix containing the multivariate
depth of each observation of x at each time point.
Only
provided if the input parameter diagnostic is set to
TRUE.
An \(m\) by \(t\) matrix containing the multivariate
depth of each observation of z at each time 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.
Vector containing the indices of the time points for which the bagplot could not be computed.
Vector containing the indices of the time points for which the cross-sectional \(\alpha\)-depth contours could not be computed.
Vector containing the indices of the time points for which the volume of the cross-sectional \(\alpha\)-depth contours could not be computed.
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
multivariate functional depth 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 depth used in the computations.
One of the following options: "hdepth",
"projdepth", "sprojdepth", "dprojdepth",
"sdepth".
Defaults to "hdepth".
Specifies the weights at every cross-section.
When alpha = 0, uniform weights are used. Weights
following equation (2) in Claeskens et al. (2014) are obtained
by setting alpha to a number smaller than the maximal
depth at any time point. The weights are then proportional to
the volume of the \(\alpha\)-depth regions at each
cross-section. 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:
crossDepths: an \(n\) by \(t\) matrix containing the
multivariate depth 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 depths.
See hdepth, projdepth, sprojdepth,
dprojdepth or sdepth.
P. Segaert and M. Hubert
The multivariate functional depth of a multivariate curve with respect to a given set of multivariate curves is defined as the weighted average of its multivariate depth at each time point (Claeskens et al., 2014). The MFD can be computed in all dimensions \(p\) using halfspace depth, projection depth, skewness-adjusted projection depth and directional projection depth. For \(p<=2\) also simplicial depth is available.
When the data array z is specified, the MFD depth and diagnostic
information for the data array x is also returned whenever the underlying
depth routine allows it. For more information see the specific depth routines
listed in the section "See Also".
For the weight vector, three options are available: uniform weights,
user-defined weights or weights proportional to the volume of the
\(\alpha\)-depth contour at each time point.
The \(\alpha\)-depth contours are computed using the depthContour function.
In some situations, additional diagnostics are available to flag outlying time
points, as described in Hubert et al. (2012). At each time point, observations
from the data array x are marked if they are flagged as outliers.
When using any of the projection depth measures, this flag is automatically
returned by the corresponding functions. When using halfspace depth,
the diagnostic is only available for bivariate curves. The observations
from the data array x are marked if they are flagged as outliers by the
bagplot, or similarly if their bagdistance is larger than 3 at that time
point. This can be seen as a measure of local outlyingness. The option is not
available for simplicial depth.
A heatmap of the cross-sectional depth values can be drawn by setting diagnostic
to TRUE and passing the results to fHeatmap.
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.
Claeskens G., Hubert M., Slaets L., Vakili K. (2014). Multivariate functional halfspace depth. Journal of the American Statistical Association, 109, 411--423.
Hubert M., Claeskens G., De Ketelaere B., Vakili K. (2012). A new depth-based approach for detecting outlying curves. In Proceedings of COMPSTAT 2012, edited by A. Colubi, K. Fokianos, G. Gonzalez-Rodriguez, E.J. Kontoghiorghes, 329--340.
depthContour, hdepth, projdepth,
sprojdepth, dprojdepth, sdepth,
fHeatmap
data(octane)
Result <- mfd(x = octane, alpha = 0.125, diagnostic = TRUE)
Plot <- fHeatmap(rowValues = Result$MFDdepthZ,
cellValues = Result$crossdepthZ,
type = "depth",
legend.title = "HD")
Plot
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