mfdmedian {mrfDepth}

Multivariate functional median for functional data

Description

Computes the multivariate functional median, an estimate for the central tendency of multivariate functional data.

Usage

  mfdmedian(x, type = "hdepth", crossDepthsX = NULL,
            depthOptions = NULL, centerOption = "maxdepth")

Arguments

x

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.

type

The depth used in the computations. One of the following options: "hdepth", "projdepth", "sprojdepth", "dprojdepth", "sdepth".
Defaults to "hdepth".

crossDepthsX

Depth values at each time point. Can be used to save computing time.

depthOptions

A list of options to pass to the function that computes the cross-sectional depths.

centerOption

When equal to "maxdepth" the functional median equals at each time point the point with cross-sectional maximal depth. When type is equal to "projdepth", also a weighted center of gravity can be computed based on Huber weights (see projmedian). Then centerOption should be set to "Huber". Defaults to "maxdepth".

Details

The multivariate functional median of a multivariate functional data set is defined as the multivariate curve connecting the cross-sectional multivariate depth medians (Claeskens et al., 2014). The MFD median can be computed in all dimensions p using halfspace depth, projection depth, skewness-adjusted projection depth or directional projection depth. The simplicial depth can only be used for p \le 2.

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 estimate for the center will be set to NaN. 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.

Value

A list with the following component:

MFDmedian

An t by p matrix containing the estimated central curve.

IndFlagExactFit

Vector containing the indices of the time points for which an exact fit is detected.

Author(s)

P. Segaert, M. Hubert

References

Claeskens G., Hubert M., Slaets L., Vakili K. (2014). Multivariate functional halfspace depth. Journal of the American Statistical Association, 109, 411–423.

See Also

hdepth, projdepth, sprojdepth, dprojdepth, sdepth, mfd

Examples

# Consider a bivariate functional sample.
data(characterA)
Data <- characterA[, 1:50, ]
Result <- mfdmedian(Data)

# Plot the data and the functional median
par(mfrow = c(1,2))
matplot(Data[, , 1], type = "l", col = "black", lty = 1, ylab = "x-coordinate")
matlines(Result$MFDmedian[, 1], type = "l", col = "red", lwd = 2)
matplot(Data[, , 2], type = "l", col = "black", lty = 1, ylab = "y-coordinate")
matlines(Result$MFDmedian[, 2], type = "l", col = "red", lwd = 2)
par(mfrow = c(1,1))

# Other depth functions such as the adjusted outlyingness may also 
# be used to determine the cross-sectional depth median. 
# In this case the depth median is computed by the 
# sprojmedian routine. 
# Optional arguments used by the sprojmedian routine may be specified
# using the depthOptions. For example one might choose the
# "Rotation" option with 300 directions. 
Result <- mfdmedian(Data, type = "sprojdepth",
                    depthOptions = list(type = "Rotation",
                                        ndir = 300))
par(mfrow = c(1,2))
matplot(Data[, , 1], type = "l", col = "black", lty = 1, ylab = "x-coordinate")
matlines(Result$MFDmedian[, 1], type = "l", col = "red", lwd = 2)
matplot(Data[, , 2], type = "l", col = "black", lty = 1, ylab = "y-coordinate")
matlines(Result$MFDmedian[, 2], type = "l", col = "red", lwd = 2)
par(mfrow = c(1,1))

# If the user already placed a call to the mfd routine
# with the diagnostic options set to TRUE, the 
# mfdmedian can easily be obtained by passing the cross-sectional 
# depths. This considerably saves computing time.  
tResult <- mfd(x = Data, type = "sprojdepth", diagnostic = TRUE)
Result <- mfdmedian(Data, type = "sprojdepth",
                    crossDepthsX = tResult$crossdepthX,
                    )
  
# Univariate curves should also be represented as arrays
Data.x <- Data[, , 1, drop = FALSE]
Result <- mfdmedian(Data.x)
matplot(Data.x[ , , 1], type = "l", col = "black", lty = 1, ylab = "x-coordinate")
matlines(Result$MFDmedian[, 1], type = "l", col = "red", lwd = 2)