depth.simplicial: Calculate Simplicial Depth

Description

Calculates the simplicial depth of points w.r.t. a multivariate data set.

Usage

depth.simplicial(x, data, exact = F, k = 0.05, 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: exact=F (by default) implies the approximative algorithm, considering k simplices, exact=T implies the exact algorithm.
k: Number ( $k > 1$ ) or portion (if $0 < k < 1$ ) of simplices that are considered if exact=F. If $k > 1$ , then the algorithmic complexity is polynomial in $d$ but is independent of the number of observations in data, given $k$ . If $0 < k < 1$ , then the algorithmic complexity is exponential in the number of observations in data, but the calculation precision stays approximately the same.
seed: the random seed. The default value seed=0 makes no changes.

Details

Calculates simplicial depth. Simplicial depth is counted as a probability that a point lies in a simplex, built on $d + 1$ data points.

References

Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. Journal of the American Statistical Association 91 862--872.

Liu, R. Y. (1990). On a notion of data depth based on random simplices. The Annals of Statistics 18 405--414.

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.

Examples

Run this code

# 3-dimensional normal distribution
data <- mvrnorm(20, 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))

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

#approximative
depths <- depth.simplicial(x, data, exact = FALSE, k = 0.2)
cat("Depths: ", depths, "\n")

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