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Statistical depth and support medians for fuzzy data

Luis González-De La Fuente, Alicia Nieto-Reyes, Pedro Terán

TL;DR

The paper addresses how to define and relate median-like location estimators for fuzzy data via statistical depth. It develops two main median notions—support medians and depth medians—and proves deep connections between them using univariate and multivariate depth concepts adapted to fuzzy sets, including Tukey depth, projection depth, and fuzzy simplicial depths. Key results show that in the univariate fuzzy case, the 1-median coincides with the support median, and that depth medians under several depths coincide with support medians (e.g., $D_{FT}$, $D_{FS}$) or the univariate $L^1$-median, under appropriate assumptions; nondegeneracy yields a unique $D_{FP}$-median equal to $\mathrm{med}_{Si}(\mathcal{X})$. The work thus unifies medians and depth notions for fuzzy data, providing a practical framework for computing central tendency via depth while highlighting existence limitations in higher dimensions. It also lays groundwork for extending depth-based notions to fuzzy quantiles and encourages further computational methods for the fuzzy setting.

Abstract

Statistical depth functions order the elements of a space with respect to their centrality in a probability distribution or dataset. Since many depth functions are maximized in the real line by the median, they provide a natural approach to defining median-like location estimators for more general types of data (in our case, fuzzy data). We analyze the relationships between depth-based medians, medians based on the support function, and some notions of a median for fuzzy data in the literature. We take advantage of specific depth functions for fuzzy data defined in our former papers: adaptations of Tukey depth, simplicial depth, $L^1$-depth and projection depth.

Statistical depth and support medians for fuzzy data

TL;DR

The paper addresses how to define and relate median-like location estimators for fuzzy data via statistical depth. It develops two main median notions—support medians and depth medians—and proves deep connections between them using univariate and multivariate depth concepts adapted to fuzzy sets, including Tukey depth, projection depth, and fuzzy simplicial depths. Key results show that in the univariate fuzzy case, the 1-median coincides with the support median, and that depth medians under several depths coincide with support medians (e.g., , ) or the univariate -median, under appropriate assumptions; nondegeneracy yields a unique -median equal to . The work thus unifies medians and depth notions for fuzzy data, providing a practical framework for computing central tendency via depth while highlighting existence limitations in higher dimensions. It also lays groundwork for extending depth-based notions to fuzzy quantiles and encourages further computational methods for the fuzzy setting.

Abstract

Statistical depth functions order the elements of a space with respect to their centrality in a probability distribution or dataset. Since many depth functions are maximized in the real line by the median, they provide a natural approach to defining median-like location estimators for more general types of data (in our case, fuzzy data). We analyze the relationships between depth-based medians, medians based on the support function, and some notions of a median for fuzzy data in the literature. We take advantage of specific depth functions for fuzzy data defined in our former papers: adaptations of Tukey depth, simplicial depth, -depth and projection depth.
Paper Structure (10 sections, 14 theorems, 90 equations, 2 figures)

This paper contains 10 sections, 14 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Zadeh's extension principle
  4. Metrics in the fuzzy setting
  5. Fuzzy random variables
  6. Depth functions for fuzzy sets
  7. Definition of support medians and depth medians
  8. Main results
  9. Proofs
  10. Discussion

Key Result

Proposition 3.2

Let $\mathcal{X}\in L^{0}[\mathcal{F}_{c}(\mathbb{R})].$ Then, $A_\alpha\subseteq(\text{med}_{Gr})_\alpha$ for every $A\in\mathop{\mathrm{Med}}\nolimits_s(\mathcal{X})$ and $\alpha\in [0,1].$

Figures (2)

  • Figure 1: Visual representation of the set of fuzzy sets maximizing $D_{FT}(\cdot;\mathcal{X})$. Every fuzzy set 'contained' in the blue area has maximal depth.
  • Figure 2: Illustration (in blue and red) of the set of medians of $\mathcal{X}$. The areas where the left and right slope of a median must be contained are represented in different colors.

Theorems & Definitions (30)

  • Definition 3.1
  • Proposition 3.2
  • Example 3.3
  • Definition 3.4
  • Proposition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Proposition 4.6
  • ...and 20 more