ddalpha (version 1.3.16)

depth.halfspace: Calculate Halfspace Depth

Description

Calculates the exact or random Tukey (=halfspace, location) depth (Tukey, 1975) of points w.r.t. a multivariate data set.

Usage

depth.halfspace(x, data, exact, method, num.directions = 1000, seed = 0)

Value

Numerical vector of depths, one for each row in x; or one depth value if x is a numerical vector.

Arguments

x

Matrix of objects (numerical vector as one object) whose depth is to be calculated; each row contains a \(d\)-variate point. Should have the same dimension as data.

data

Matrix of data where each row contains a \(d\)-variate point, w.r.t. which the depth is to be calculated.

exact

The type of the used method. The default is exact=F, which leads to approximate computation of the Tukey depth. For exact=F, method="Sunif.1D" is used by default. If exact=T, the Tukey depth is computed exactly, with method="recursive" by default.

method

For exact=F, if method="Sunif.1D" (by default), the Tukey depth is computed approximately by being minimized over univariate projections (see Details below).

For exact=T, the Tukey depth is calculated as the minimum over all combinations of \(k\) points from data (see Details below). In this case parameter method specifies \(k\), with possible values \(1\) for method="recursive" (by default), \(d-2\) for method="plane", \(d-1\) for method="line".

The name of the method may be given as well as just parameter exact, in which case the default method will be used.

num.directions

Number of random directions to be generated (for method="Sunif.1D"). The algorithmic complexity is linear in the number of observations in data, given the number of directions.

seed

The random seed. The default value seed=0 makes no changes (for method="Sunif.1D").

Details

For exact=F, if method="Sunif.1D", the Tukey depth is computed approximately using the random Tukey depth method proposed by Cuesta-Albertos and Nieto-Reyes (2008). Here the depth is determined as the minimum univariate Tukey depth of the - on lines in several directions - projected data. The directions are distributed uniformly on the \((d-1)\)-sphere; the same direction set is used for all points.

For exact=T, the Tukey depth is computed exactly as the minimum of the sum of the depths in two orthogonal complementary affine subspaces, which dimensions add to \(d\): one of the subspaces (combinatorial) is the \(k\)-dimensional hyperplane through (a point from) x and \(k\) points from data, another one is its orthogonal complement (see Dyckerhoff and Mozharovskyi, 2016 for the detailed description of the algorithmic framework). The algorithm then minimizes the depth over all combinations of \(k\) points, in which the depth in the orthogonal complements is computed using an exact algorithm. In this case, parameter method specifies the dimensionality \(k\) of the combinatorial space. The implemented (reasonable) algorithms (and corresponding names) are: \(k=1\) (or method="recursive"), \(k=d-2\) (or method="plane"), and \(k=d-1\) (or method="line").

References

Cuesta-Albertos, J.A. and Nieto-Reyes, A. (2008). The random Tukey depth. Computational Statistics and Data Analysis 52 4979--4988.

Dyckerhoff, R. and Mozharovskyi, P. (2016). Exact computation of the halfspace depth. Computational Statistics and Data Analysis 98 19--30.

Rousseeuw, P.J. and Ruts, I. (1996). Algorithm AS 307: Bivariate location depth. Journal of the Royal Statistical Society. Seriec C (Applied Statistics) 45 516--526.

Tukey, J.W. (1974). Mathematics and the picturing of data. In: Proceeding of the International Congress of Mathematicians, Vancouver, 523--531.

See Also

depth.Mahalanobis for calculation of Mahalanobis depth.

depth.projection for calculation of projection depth.

depth.simplicial for calculation of simplicial depth.

depth.simplicialVolume for calculation of simplicial volume depth.

depth.spatial for calculation of spatial depth.

depth.zonoid for calculation of zonoid depth.

depth.potential for calculation of data potential.

Examples

# 3-dimensional normal distribution
data <- mvrnorm(200, rep(0, 3), 
                matrix(c(1, 0, 0,
                         0, 2, 0, 
                         0, 0, 1),
                nrow = 3))
x <- mvrnorm(10, rep(1, 3), 
             matrix(c(1, 0, 0,
                      0, 1, 0, 
                      0, 0, 1),
             nrow = 3))
              
# default - random Tukey depth
depths <- depth.halfspace(x, data)
cat("Depths: ", depths, "\n")

# default exact method - "recursive"
depths <- depth.halfspace(x, data, exact = TRUE)
cat("Depths: ", depths, "\n")

# method "line"
depths <- depth.halfspace(x, data, method = "line")
cat("Depths: ", depths, "\n")