Simplicial depth¶
A multivariate centrality measure equal to the number or probability of sample-generated d-simplices whose convex hull contains a query point.
Core Idea¶
Simplicial depth ranks a query point by the number or probability of d-dimensional simplices, formed from d+1 sample points, whose convex hull contains it. The statistic is affine invariant and robust but need not have a unique deepest point. For n observations, a raw count can be divided by the number of d+1 subsets to obtain an empirical probability. For n observations, a raw count can be divided by the number of d+1 subsets to obtain an empirical probability.
Scope of Application¶
The measure is used in robust multivariate statistics and computational geometry for center estimation, depth contours, and approximate depth queries. Use it with dimension, reference sample or distribution, raw/normalized output, open/closed boundary rule, degeneracy handling, and exact or approximate error guarantee explicit.
- 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. The closest near miss sets the boundary: Spherical depth is the nearest miss: it uses random closed balls from pairs rather than d-simplices from d+1 points.
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. The central robust invariance–geometric intuition tradeoff is this: Affine invariance supports coordinate changes while contours can violate radial expectations. A second exact count–computational scale tension matters because Combinatorial enumeration is precise while approximation enables repeated queries. The closed containment–boundary stability tension adds that Full boundary weight is simple while discrete ties can behave poorly.
Abstract Reasoning¶
Use three linked moves: fix the reference sample/distribution and ambient dimension; enumerate or sample d+1-tuples and construct their convex hulls; apply one explicit boundary-containment convention to the query point. As a collapse test, the case exits when containment is not simplex-based, normalization is inconsistent, or boundary convention is unstated in a degenerate sample. A fourth check is to aggregate to a count or normalized probability with uncertainty if approximated. A final check is to 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. No canonical parent prime is currently asserted; broader structural comparisons remain related-prime analogies until separately adjudicated in the DAG. Depth ranks points from center toward periphery. Distributional depth is a random-simplex containment probability.
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
- M-Estimator — 0.87
- ARGUS distribution — 0.87
- Newton–Okounkov body — 0.87
- Cuzick–Edwards Test — 0.85
- MAP estimator — 0.85
Computed from structural-signature embeddings · 2026-10-08