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Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support

Giacomo Francisci, Alicia Nieto-Reyes, Claudio Agostinelli

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.

Generalization of the simplicial depth: no vanishment outside the convex hull of the distribution support

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.

Paper Structure

This paper contains 10 sections, 16 theorems, 23 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Symmetry of random variables
  3. Generalization of the simplicial depth
  4. Properties of the $\sigma$-simplicial depth
  5. Definition and properties of the sample $\sigma$-simplicial depths
  6. Simulations
  7. Data analysis
  8. Concluding remarks
  9. Proofs
  10. Further simulations and results

Key Result

Proposition 1

Let $X$ be a random variable on $\mathbb{R}^p$ that is symmetric about $\mu \in \mathbb{R}^p$ with respect to either spherical, elliptical, central, angular or halfspace symmetry. Then, for any $\lambda \in \mathbb{R}$ and $b \in \mathbb{R}^p,$$\lambda X + b$ is symmetric about $\lambda \mu + b$ wit

Figures (14)

  • Figure 1: Alzheimer dataset. Group 1 (mild stage): filled green circle , group 2 (moderate stage): triangle, group 3 (severe stage): filled red square. Left plot: simplicial depth, right plot: simplex enlarged $\sigma$-simplicial depth with $\sigma=6$. Units in group 2 are classified in group 1 (green) or group 3 (red). Black triangles are unclassified units with zero simplicial depth.
  • Figure 2: Boxplots of $100$ misclassification rates of the outsiders for $d_{\triangle_{\sigma},n}$ with $\sigma\in\{1.2, 1.5, 2, 3, 4, 5, 7, 10, 15, 25\}$ and the sample simplicial depth ($\sigma=1$), using the linear DD-plot classifier.
  • Figure 3: Boxplots of $100$ misclassification rates of the outsiders for the refined halfspace depth with $r\in\{10, 20, 30, 40, 60, 80, 100, 120, 160, 200\}$ and the sample halfspace depth ($r=0$), using the linear DD-plot classifier.
  • Figure 4: Boxplots of $100$ misclassification rates of the outsiders for the sample illumination depth with $\alpha \in \{0.01, 0.02, 0.03, 0.04, 0.06, 0.08, 0.10, 0.12, 0.16, 0.20 \}$ and the sample halfspace depth ($\alpha=0$). Linear DD-plot classifier.
  • Figure 5: Mean, as a function of $\delta$, of misclassification rates over $100$ times using the sample simplicial depth and $d_{\triangle_{\sigma},n}$ for $\sigma\in \{1.2, 1.5, 2, 3, 4, 5\}$. On the left symmetric distributions are given, on the right asymmetric ones.
  • ...and 9 more figures

Theorems & Definitions (22)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Proposition 8
  • Proposition 9
  • Remark 10
  • ...and 12 more