simplicial_depth: Liu Simplicial Depth

In depthR: Multivariate Depth Functions for General Dimension

Liu Simplicial Depth

Description

Computes the simplicial depth of one or more query points with respect to a reference distribution estimated from data, using an adaptive Monte Carlo approximation with parallel computation.

Usage

simplicial_depth(
  x,
  data,
  tol = 0.05,
  batch_size = 200L,
  min_batches = 3L,
  max_batches = 20L,
  seed = 42L
)

Arguments

x	Numeric matrix of query points (m x d), or a numeric vector of length d for a single point.
data	Numeric matrix of reference data (n x d). Must have at least d+1 rows.
tol	Relative standard error tolerance for the adaptive stopping rule. Sampling stops when the standard error of the depth estimate drops below tol times the estimate itself. Default 0.05.
batch_size	Number of random simplices sampled per batch. Default 200.
min_batches	Minimum number of batches before checking convergence. Default 3.
max_batches	Maximum number of batches regardless of convergence. Acts as a hard cap on computation time. Default 20.
seed	Integer random seed for reproducibility. Default 42.

Details

Simplicial depth is the probability that a random simplex formed by d+1 points drawn from the data contains the query point. It is a genuine multivariate generalization of the median with strong geometric intuition and no distributional assumptions.

The deepest point — the simplicial median — is a robust estimator of location that reduces to the univariate median when d=1.

Value

Numeric vector of depth values in [0, 1], one per query point. Higher values indicate greater centrality.

References

Liu, R. Y. (1990). On a notion of data depth based on random simplices. Annals of Statistics, 18(1), 405–414.

Zuo, Y. & Serfling, R. (2000). General notions of statistical depth function. Annals of Statistics, 28(2), 461–482.

Examples

set.seed(42)
data <- matrix(rnorm(500), nrow = 100, ncol = 5)
x    <- matrix(rnorm(25),  nrow = 5,   ncol = 5)

# Basic usage
simplicial_depth(x, data)

# Via compute_depth for full depth object
dd <- compute_depth(data, depth_fn = simplicial_depth)
median(dd)
outliers(dd)
plot(dd)

depthR documentation built on June 26, 2026, 5:07 p.m.