Simplicial depths for fuzzy random variables

TL;DR

This paper extends the concept of simplicial depth to fuzzy random variables by introducing pseudosimplices and proposing three generalizations, supported by theoretical analysis and empirical illustrations.

Contribution

It introduces three new definitions of simplicial depth for fuzzy sets and analyzes their properties, advancing the statistical depth framework for fuzzy data.

Findings

The proposed depths are theoretically characterized.

Empirical studies demonstrate their practical behavior.

Applications to synthetic and real data validate the methods.

Abstract

The recently defined concept of a statistical depth function for fuzzy sets provides a theoretical framework for ordering fuzzy sets with respect to the distribution of a fuzzy random variable. One of the most used and studied statistical depth function for multivariate data is simplicial depth, based on multivariate simplices. We introduce a notion of pseudosimplices generated by fuzzy sets and propose three plausible generalizations of simplicial depth to fuzzy sets. Their theoretical properties are analyzed and the behavior of the proposals illustrated through a study of both synthetic and real data.