Skip to content

Simplicial depth

A multivariate centrality measure equal to the number or probability of sample-generated d-simplices whose convex hull contains a query point.

Version
v1 · 2026-09-28 · History
Domain-specific #
12050
Domain group
Formal Sciences
Origin domain
Experimental Design & Statistics
Subdomains
Data Depth, Nonparametric Multivariate Statistics → Experimental Design & Statistics

Core Idea

Simplicial depth measures how centrally a point sits relative to multivariate data by counting sample simplices that contain it. In d dimensions, each simplex is the convex hull of d+1 sample observations; in the plane, triangles replace one-dimensional intervals.

For n observations, a raw count can be divided by the number of d+1 subsets to obtain an empirical probability. The distributional version asks for the probability that a random d+1 tuple has a convex hull containing the query point.

The statistic is affine invariant and robust to a bounded fraction of outliers, but depth contours need not behave like nested radial distance: centrally symmetric distributions may have nonunique deepest points and depth need not decline monotonically along every ray. Boundary weighting and computational approximation must be reported.

Structural Signature

Sig role-phrases:

  • ambient dimension. Fixes d and therefore the d+1 vertices in each candidate simplex. Constitutive geometry. If altered: Wrong dimension changes both hulls and normalization.
  • sample or distribution. Supplies points from which simplices are drawn. Constitutive reference population. If altered: Changing the reference changes the depth landscape.
  • query point. Provides the location whose centrality is assessed. Constitutive target. If altered: Depth is point-relative, not one sample-wide scalar.
  • containment convention. Tests membership in closed, open, or boundary-half-weighted convex hulls. Identity-bearing event. If altered: Boundary treatment changes tied/discrete cases.
  • count or probability. Aggregates containment over all sample simplices or random tuples and normalizes when required. Constitutive output. If altered: Raw count and probability are not comparable without n and convention.

What It Is Not

  • Not convex-hull membership alone. Depth counts many containing simplices.
  • Not distance from the mean. No mean or Euclidean radius defines it.
  • Not halfspace depth. The containment objects differ.
  • Not always uniquely maximized. Symmetry does not guarantee one deepest point.

Scope of Application

The measure is used in robust multivariate statistics and computational geometry for center estimation, depth contours, and approximate depth queries.

  • Robust location. Uses a deepest point as representative.
  • Outlier analysis. Ranks points by centrality.
  • Depth contours. Maps multivariate center-to-periphery structure.
  • Computational geometry. Counts containing simplices efficiently.
  • Dynamic approximation. Supports query structures under sample insertions.

Clarity

A result should state dimension, sample size, raw or normalized form, open/closed boundary rule, degeneracy handling, and exact or approximate algorithm. Values from different n or conventions cannot be compared casually.

Manages Complexity

The statistic summarizes a combinatorial number of geometric relations into one centrality value. Affine invariance and robustness are gained at substantial exact computational cost, motivating dimension-specific and approximate algorithms.

Abstract Reasoning

  1. Fix the reference sample/distribution and ambient dimension.
  2. Enumerate or sample d+1-tuples and construct their convex hulls.
  3. Apply one explicit boundary-containment convention to the query point.
  4. Aggregate to a count or normalized probability with uncertainty if approximated.
  5. Interpret maxima and contours without assuming uniqueness or radial monotonicity.

Knowledge Transfer

Convex-hull containment transfers between empirical and distributional settings, but normalization, boundary ties, and computational guarantees do not. ‘Deep in a cluster’ is only analogous without simplex probabilities.

Examples

Canonical

In the plane, enumerate every triangle formed by three sample points and count those whose closed triangle contains p; divide by the number of triples for normalized empirical simplicial depth.

Mapped back: ambient dimension → d=2; sample or distribution → n planar observations; query point → p; containment convention → closed triangles; count or probability → count divided by n choose 3.

Applied / In Practice

An epsilon-net data structure approximates depth queries for a fixed or insertion-updated sample in higher dimensions, reporting an error bound rather than claiming the exact combinatorial count.

Mapped back: ambient dimension → declared d; sample or distribution → fixed or insertion-updated points; query point → incoming query; containment convention → same simplex event; count or probability → bounded approximation.

Structural Tensions

T1: robust invariance vs. geometric intuition. Affine invariance supports coordinate changes while contours can violate radial expectations. Diagnostic: Which property is needed for interpretation?

T2: exact count vs. computational scale. Combinatorial enumeration is precise while approximation enables repeated queries. Diagnostic: What error is acceptable for the decision?

T3: closed containment vs. boundary stability. Full boundary weight is simple while discrete ties can behave poorly. Diagnostic: Which boundary convention is reported?

Structural–Framed Character

Simplicial depth is structural. Convex geometry and probability define it; boundary conventions and algorithms frame implementation. Its portable skeleton is Centrality, related rather than an asserted strict parent because this node is one specialized depth measure. Evaluation and practice dependence are low; disciplinary convention fixes ties; vocabulary travels under affine geometry; informal depth is metaphor. Its character: centrality measured by participation in random or sample simplices.

Structural Core vs. Domain Accent

Skeletal core. Rank a target by how often it lies inside structures generated from reference observations.

Domain-bound accent. Euclidean dimension, convex hulls, d-simplices, samples, probability, and robust depth define the statistic.

Why not prime. Containment-based centrality travels, but simplicial depth is a specific multivariate measure.

  • Centrality. Depth ranks points from center toward periphery.
  • Sampling. Distributional depth is a random-simplex containment probability.
  • No strict DAG edge is added.

Neighborhood in Abstraction Space

Simplicial depth sits in a moderately populated region (49th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Domain-Specific Measurement Parameters (36 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Halfspace depth. Tell: Are containing halfspaces or sample simplices counted?
  • Spherical depth. Tell: Are random balls from pairs or simplices from d+1 points used?
  • Convex-hull membership. Tell: Is one hull tested or all sample simplices aggregated?
  • Mahalanobis distance. Tell: Is centrality probability/geometric or covariance-distance based?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Simplicial_depth (revision 1335587829).
  • Preserved source candidate: http://www.cccg.ca/proceedings/2004/49.pdf
  • Preserved source candidate: http://www.cccg.ca/proceedings/2001/ouyang-24534.ps.gz

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.