mrfDepth (version 1.0.3)

projmedian: Location estimates based on projection depth.

Description

Computes a projection depth base location estimate of a \(p\)-dimensional dataset x.

Usage

projmedian(x, projection.depths = NULL, options = NULL)

Arguments

x

An \(n\) by \(p\) data matrix with observations in the rows and variables in the columns.

projection.depths

Vector containing the projection depths of the points in x.

options

A list of options to pass to the projdepth routine. See projdepth for more details.

Value

A list with components:

max

The point of x with maximal projection depth. If multiple points have maximum depth, their center of gravity is returned

gravity

The center of gravity for the 50 percent points with highest projection depth. Also called the center of gravity of the bag.

Huber

The weighted center of gravity of all points. The weights are defined by the Huber function with parameter alpha = 1 / (1+sqrt(qchisq(0.95, p))) Observations for which the projection depth is more than alpha receive weight 1, other points receive weight ( sqrt(qchisq(0.95, p)) / outlyingness )^2.

Details

The algorithm depends on the function projdepth to calculate the projection depths of the dataset x. If the projection depths of the points \(x_i\) have already been calculated they can be passed as an optional argument to save computation time. If not, projections depths will be calculated and the user can pass a list with options to the projdepth 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.

References

Zuo Y. (2003). Projection based depth functions and associated medians. The Annals of Statistics, 31, 1460--1490.

See Also

outlyingness, projdepth, adjOutl, sprojdepth, sprojmedian

Examples

# Compute a location estimate of a simple two-dimensional dataset.
if (requireNamespace("robustbase", quietly = TRUE)) {
  BivData <- log(robustbase::Animals2)
} else {
  BivData <- matrix(rnorm(120), ncol = 2)
  BivData <- rbind(BivData, matrix(c(6,6, 6, -2), ncol = 2))
}

result <- projmedian(x = BivData)
plot(BivData)
points(result$max, col = "red", pch = 15)
points(result$gravity, col = "blue", pch = 16)
points(result$Huber, col = "orange", pch = 17)

# Options for the underlying projdepth routine may be passed 
# using the options argument. 
result <- projmedian(x = BivData,
                     options = list(type = "Affine",
                                    ndir = 10,
                                    stand = "MedMad",
                                    h = nrow(BivData)
                                    )
                    )
plot(BivData)
points(result$max, col = "red", pch = 15)
points(result$gravity, col = "blue", pch = 16)
points(result$Huber, col = "orange", pch = 17)

# One may also calculate the depths of the points in the data
# separately. This avoids having to recompute the depths when these
# are previously calculated. 
depth.result <- projdepth(x = BivData)
result <- projmedian(x = BivData, 
                     projection.depths = depth.result$depthX)