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Projection depth and $L^r$-type depths for fuzzy random variables

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

TL;DR

This paper adapt projection depth and $L^{r}$-type depth to the fuzzy setting, studying their properties and illustrating their behaviour with a real data example.

Abstract

Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt projection depth and $L^{r}$-type depth to the fuzzy setting, studying their properties and illustrating their behaviour with a real data example.

Projection depth and $L^r$-type depths for fuzzy random variables

TL;DR

This paper adapt projection depth and -type depth to the fuzzy setting, studying their properties and illustrating their behaviour with a real data example.

Abstract

Statistical depth functions are a standard tool in nonparametric statistics to extend order-based univariate methods to the multivariate setting. Since there is no universally accepted total order for fuzzy data (even in the univariate case) and there is a lack of parametric models, a fuzzy extension of depth-based methods is very interesting. In this paper, we adapt projection depth and -type depth to the fuzzy setting, studying their properties and illustrating their behaviour with a real data example.
Paper Structure (20 sections, 20 theorems, 91 equations, 3 figures, 1 table)

This paper contains 20 sections, 20 theorems, 91 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Fuzzy sets
  4. Arithmetics and Zadeh's extension principle
  5. Metrics between fuzzy sets
  6. Fuzzy random variables
  7. Symmetry and depth: semilinear and geometric notions
  8. Banach spaces
  9. Projection depth and its properties
  10. Definition
  11. Properties
  12. $L^{r}$-type depths and their properties
  13. Definitions
  14. Properties
  15. Affine invariance
  16. ...and 5 more sections

Key Result

Proposition 3.2

Let $\mathcal{J} = \{\text{I}_{\{x\}} : x\in\mathbb R^p\}$. For any random vector $X$ on $\mathbb{R}^{p}$ and any $x\in\mathcal{J},$

Figures (3)

  • Figure 1: Display of the dataset Trees.
  • Figure 2: Display of the dataset Trees. Color is assigned based on the projection depth (left panel) and on the $1$-natural and $2$-natural depths (right panel) of each fuzzy set in the empirical distribution. Colors range from red (high depth) to blue (low depth).
  • Figure 3: Display of the dataset Trees. Color is assigned based on the $(1,5)$-location depth (left) and $(1,10)$-location depth (right), ranging from red (high depth) to blue (low depth).

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Proposition 3.5
  • Definition 4.1
  • Definition 4.2
  • Definition 4.3
  • Definition 4.4
  • ...and 37 more