On Some Functional Characterizations of (Fuzzy) Set-Valued Random Elements

Abstract

One of the most common spaces to model imprecise data through (fuzzy) sets is that of convex and compact (fuzzy) subsets in $\mathbb {R}^p$. The properties of compactness and convexity allow the identification of such elements by means of the so-called support function, through an embedding into a functional space. This embedding satisfies certain valuable properties, however it is not always intuitive. Recently, an alternative functional representation has been considered for the analysis of imprecise data based on the star-shaped sets theory. The alternative representation admits an easier interpretation in terms of ‘location’ and ‘imprecision’, as a generalized idea of the concepts of mid-point and spread of an interval. A comparative study of both functional representations is made, with an emphasis on the structures required for a meaningful statistical analysis from the ontic perspective.