[Paper Review] Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support

Giacomo Francisci, Alicia Nieto-Reyes+1 more|arXiv (Cornell University)|Sep 6, 2019

Advanced Statistical Methods and Models30 references4 citations

TL;DR

This paper introduces two novel generalizations of simplicial depth that do not vanish outside the convex hull of the distribution support—via simplex enlargement or distribution enlargement—enabling meaningful depth assignment to all points. The proposed estimators are uniformly consistent, asymptotically normal, and outperform classical depth methods in classification tasks, especially when outliers lie beyond convex hulls.

ABSTRACT

The simplicial depth, like other relevant multivariate statistical data depth functions, vanishes right outside the convex hull of the support of the distribution with respect to which the depth is computed. This is problematic when it is required to differentiate among points outside the convex hull of the distribution support, with respect to which the depth is computed, based on their depth values. We provide the first proposal for simplicial depth which do not vanish right outside the convex hull of the distribution. The properties of the proposal and of the corresponding estimator are studied theoretically and by means of Monte Carlo simulations and analysis of datasets.

Motivation & Objective

• Address the limitation of classical simplicial depth vanishing outside the convex hull of the distribution support, which hinders classification of outlying points.
• Develop a generalization of simplicial depth that assigns positive depth values to points outside the convex hull, enabling discrimination among such points.
• Ensure theoretical consistency and robustness of the proposed depth estimators under mild regularity conditions.
• Demonstrate superior performance in supervised classification tasks, particularly when training and test distributions have non-overlapping convex hulls.
• Provide a practical framework for selecting the key parameter σ in the generalized depth estimators using cross-validation and empirical evaluation.

Proposed method

• Introduce a σ-enlarged simplex in the definition of simplicial depth, where the simplex vertices are stochastically perturbed to include points outside the original convex hull.
• Define the σ-simplicial depth as the probability that a point lies within a simplex formed by vertices drawn from a distribution that is a location-scale transformation of the original.
• Use U-statistics to estimate the generalized depth functions, ensuring uniform consistency and asymptotic normality of the empirical processes.
• Propose two variants: one based on enlarging the simplex (σ-simplicial depth) and another by enlarging the underlying distribution via affine combinations of independent random variables.
• Establish theoretical properties such as continuity in σ, inheritance of symmetry and continuity under affine combinations, and monotonicity of depth trimmed regions.
• Apply cross-validation to select the optimal σ parameter that minimizes misclassification rates in classification tasks.

Experimental results

Research questions

• RQ1 Can simplicial depth be generalized to assign positive depth values to points outside the convex hull of the distribution support, thereby enabling classification of such points?
• RQ2 How do the proposed generalizations of simplicial depth behave under varying values of the enlargement parameter σ, and what is the continuity and consistency of the resulting estimators?
• RQ3 Do the generalized depth functions inherit key theoretical properties of statistical depth, such as maximality at the center and monotonicity, under mild regularity conditions?
• RQ4 How do the proposed estimators compare to existing depth functions—especially Tukey depth variants—that do not vanish outside the convex hull in terms of classification accuracy?
• RQ5 What is the optimal strategy for selecting the σ parameter in practice, and how does it affect misclassification rates in real and simulated datasets?

Key findings

• The proposed σ-simplicial depth and distribution-enlarged depth estimators do not vanish outside the convex hull of the distribution support, enabling depth-based classification of all points.
• The simplex-enlarged σ-simplicial depth estimator consistently outperformed other methods in classification tasks, achieving the lowest median misclassification rate across four real datasets.
• Cross-validation for σ selection yielded results only slightly worse than the best-case scenario, indicating robustness and practical utility of the method.
• The distribution-enlarged depth estimator satisfies more theoretical depth properties (e.g., monotonicity, connected depth regions) under mild conditions, offering a stronger theoretical foundation.
• Monte Carlo simulations showed a U-shaped misclassification rate pattern with σ: decreasing initially, stabilizing, then increasing slowly, with optimal σ selected as the smallest value yielding minimal median error.
• In the Alzheimer’s disease dataset, 4 of 6 outsiders were correctly classified as mild or severe stage using the σ-simplicial depth, whereas classical simplicial depth assigned zero depth to all, rendering them unclassifiable.