TukeyKRegions: Computation of the Tukey Region

In TukeyRegion: Tukey Region and Median

Computation of the Tukey Region

Description

Computes the Tukey depth trimmed regions for all depth levels from 1 to K.

Usage

TukeyKRegions(data, maxDepth, method = "bfs",
  trgFacets = FALSE, checkInnerPoint = TRUE,
  retHalfspaces = TRUE, retHalfspacesNR = FALSE,
  retInnerPoint = FALSE, retVertices = FALSE,
  retFacets = FALSE, retVolume = FALSE, retBarycenter = FALSE,
  verbosity = 0L)

Arguments

data	data set for which the Tukey region shall be computed, a matrix having > 2 columns and more rows than columns.
maxDepth	depth level until which Tukey regions to compute, an integer between 1 and the half of the number of rows in data.
method	the method to use to compute the Tukey region, a string containing "bfs", or "cmb", or "bf", see ‘Details’, "bfs" by default.
trgFacets	whether to triangulate facets, logical, FALSE by default. In this case no facet ridges are plotted, see ‘Value’.
checkInnerPoint	whether to check correctness of the inner point in case it is provided, logical, TRUE by default.
retHalfspaces	whether to return all found halfspaces, logical, TRUE 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, FALSE by default.
retFacets	whether to return facets, logical, FALSE 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 regions (upper-level set of the Tukey depth function) for n points in the Euclidean d-variate space contained in data at the depth values specified in the argument maxDepth. This function calls iteratively function TukeyRegion for depth levels from 1 to maxDepth, where each time the initial set of ridges conincides with all the ridges found at the previous step (see Fojtik et al., 2022).

Due to the nature of the function, arguments halfspaces and/or innerPoint cannot be provided here anymore.

Value

The function returns a list of objects of class TukeyRegion with fields specified by ret...-flags in the arguments:

data	the input data set.
depth	chosen depth level.
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.

Author(s)

Pavlo Mozharovskyi <[e-mail]>

References

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 TukeyMedian

Examples

# Load required packages
require(TukeyRegion)
require(MASS)
# Generate data
set.seed(1)
X <- mvrnorm(500, rep(0, 3),
  matrix(c(1, 0.25, 0.25, 0.25, 1, 0.25, 0.25, 0.25, 1), nrow = 3))
# Compute the Tukey region
Trs <- TukeyKRegions(X, 5, "bfs",
  retFacets = TRUE, retVolume = TRUE, retBarycenter = TRUE)
for (i in 1:5){
  summary(Trs[[i]])
  cat("\n")
}

TukeyRegion documentation built on April 18, 2023, 1:10 a.m.