TukeyKMedian {TukeyRegion}

Computation of the Tukey median set and Tukey median

Description

Computes the Tukey median set and its barycenter, the Tukey median, starting from the region with dpeth 1 iteratively increasing it, according to the algorithm by Fojtik et al. (2022).

Usage

TukeyKMedian(data, algMedian = "upwards", method = "bfs",
  trgFacets = TRUE, retHalfspaces = FALSE, retHalfspacesNR = FALSE,
  retInnerPoint = FALSE, retVertices = TRUE, retFacets = TRUE,
  retVolume = FALSE, retBarycenter = TRUE, verbosity = 0)

Arguments

data

data set for which the Tukey median shall be computed, a matrix having > 2 columns and more rows than columns.

algMedian

the algorithm used to compute the Tukey median, a string containing bsbarydepth, or "cutintwo", or "downwards", or "upwards", see ‘Details’, "bsbarydepth" by default.

method

the method to use to compute the Tukey region, a string containing "bfs", or "cmb", or "bf", see TukeyRegion, "bfs" by default.

trgFacets

whether to triangulate facets, logical, FALSE by default. In this case no facet ridges are plotted, see ‘Value’.

retHalfspaces

whether to return all found halfspaces, logical, FALSE by default.

retHalfspacesNR

whether to return non-redundant halfspaces, logical, FALSE by default.

retInnerPoint

whether to return inner point, logical, FALSE by default.

retVertices

whether to return vertices, logical, TRUE by default.

retFacets

whether to return facets, logical, TRUE by default.

retVolume

whether to return volume, logical, FALSE by default.

retBarycenter

whether to return the region's barycenter, logical, FALSE by default.

verbosity

level of details to print during execution, integer, from 0 (= print nothing) to 2 (= print all details).

Details

The function computes the Tukey median set, i.e. the region with the highest depth value, for n points in the Euclidean d-variate space contained in data.

It also computes this set's barycenter, which is the Tukey median. Four search algorithms are implemented: Algorithm bsbarydepth is the most efficient, it is the bisection algorithm starting with the lower bound as the maximum of the theoretical minimum and the depth of the componentwise median, and updating lower bound with the depth of the barycenter of the last found region. Algorithm "cutintwo" sequntially cuts the range of remaining depths into two parts starting with the range from 1 to the upper bound obtained by Liu, Luo, and Zuo (2016). Algorithm "downwards" is checking each depth value decrementally with step 1 starting with the upper bound obtained by Liu, Luo, and Zuo (2016) until the first existing region is found. Algorithm "upwards" is checking each depth value incrementally until the first non-existing region is found.

The main goal of the function is to provide the polytope (Tukey median set) and its barycenter (Tukey median); the settings can be adjusted though. After the median depth is found, the TukeyRegion function is called.

See function TukeyRegion for details regarding the output.

Value

The function returns an object of class TukeyRegion with fields specified by ret...-flags in the arguments:

data

the input data set.

depth

chosen depth level.

numRegions

number of time the depth region has been computed.

halfspacesFound

whether at least one of the determining Tukey region halfspaces has been found.

halfspaces

if requested, halfspaces defining the Tukey region by their intersection, a matrix with number of columns equal to space dimension and where each row corresponds to a halfspace defined by three point numbers in data.

innerPointFound

a logical indicating whether an inner point of the region has been found. If FALSE then the region of the given depth does not exist. If the field is absent then the inner point has not been requested by the input arguments.

innerPoint

coordinates of a point inside of the Tukey region. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

halfspacesNR

non-redundant halfspaces (i.e. those containing Tukey region's facets), a matrix with number of columns equal to space dimension and where each row corresponds to a halfspace defined by three point numbers in data. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

vertices

vertices of the Tukey region, a matrix with number of columns equal to space dimension and where each row represents vertex coordinates. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments. If field facets is returned, this field is returned as well.

triangulated

a logical repeating the trgFacets input argument. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

facets

facets of the Tukey region. If input argument trgFacets is set, then this is a list where each element is an array enumerating numbers of the rows in field vertices, their number for each facet can vary. If input argument trgFacets is unset, then this is a matrix with each row corresponding to a triangulated facet, and no facets' ridges reconstruction is performed, so it cannot be visualized. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

volume

volume of the Tukey region. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

barycenter

the barycenter of the Tukey region. If the field is absent then either no halfspaces or no inner point have been found or facet computation has not been requested by the input arguments.

numRidges

number of used ridges (for computing the last region).

Author(s)

Pavlo Mozharovskyi <[e-mail]>

References

Liu, X., Luo, S., and Zuo, Y. (2020). Some results on the computing of Tukey's halfspace median. Statistical Papers, 61, 303-316.

Liu, X., Mosler, K., and Mozharovskyi, P. (2019). Fast computation of Tukey trimmed regions and median in dimension p > 2. Journal of Computational and Graphical Statistics, 28, 682-697.

Vit Fojtik, Petra Laketa, Pavlo Mozharovskyi, and Stanislav Nagy (2022). On exact computation of Tukey depth central regions. arXiv:2208.04587.

See Also

TukeyRegion, depth.wm

Examples

# Load required packages
require(TukeyRegion)
require(MASS)
# Generate data
set.seed(1)
X <- mvrnorm(100, rep(0, 3),
  matrix(c(1, 1, 1, 1, 2, 2, 1, 2, 4), nrow = 3))
# Compute the Tukey median
Tm <- TukeyKMedian(X)
summary(Tm)
# Visualize the Tukey median
plot(Tm)