Statistical depth in abstract metric spaces

TL;DR

This paper extends the concept of statistical depth to general metric spaces, providing a flexible and adaptable measure suitable for complex modern data types, with demonstrated applications in real data analysis.

Contribution

It introduces a new depth measure for metric spaces, broadening the applicability of depth-based analysis to diverse data structures.

Findings

The proposed depth measure is adaptable to various metric spaces.

Real data analyses demonstrate the method's flexibility and effectiveness.

The measure retains desirable statistical properties in different contexts.

Abstract

The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid development of technology, in particular the advent of `Big Data', we extend here that concept to general metric spaces, propose a natural depth measure and explore its properties as a statistical depth function. Working in a general metric space allows the depth to be tailored to the data at hand and to the ultimate goal of the analysis, a very desirable property given the polymorphic nature of modern data sets. This flexibility is thoroughly illustrated by several real data analyses.