A new framework for the statistical analysis of set-valued random elements

• Published in: International journal of approximate reasoning Vol. 92; pp. 279 - 294
• Main Authors: González-Rodríguez, Gil, Ramos-Guajardo, Ana Belén, Colubi, Ana, Blanco-Fernández, Ángela
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2018
• Subjects: Center-radial representation; Functional characterizations; Imprecise data; Set-valued data; Star-shaped set; Support function; Functional characterizations; Support function; Star-shaped set; Set-valued data; Center-radial representation; Imprecise data
• ISSN: 0888-613X; 1873-4731
• DOI: 10.1016/j.ijar.2017.10.025

The space of nonempty convex and compact (fuzzy) subsets of Rp, Kc(Rp), has been traditionally used to handle imprecise data. Its elements can be characterized via the support function, which agrees with the usual Minkowski addition, and naturally embeds Kc(Rp) into a cone of a separable Hilbert space. The support function embedding holds interesting properties, but it lacks of an intuitive interpretation for imprecise data. As a consequence, it is not easy to identify the elements of the image space that correspond to sets in Kc(Rp). Moreover, although the Minkowski addition is very natural when p=1, if p>1 the shapes which are obtained when two sets are aggregated are apparently unrelated to the original sets, because it tends to convexify. An alternative and more intuitive functional representation will be introduced in order to circumvent these difficulties. The imprecise data will be modeled by using star-shaped sets on Rp. These sets will be characterized through a center and the corresponding polar coordinates, which have a clear interpretation in terms of location and imprecision, and lead to a natural directionally extension of the Minkowski addition. The structures required for a meaningful statistical analysis from the so-called ontic perspective are introduced, and how to determine the representation in practice is discussed. •The support function is shown to lack of an intuitive interpretation when imprecise information is considered.•As an alternative the imprecise data will be modelled by using star-shaped sets.•Star-shaped sets will be characterized through a center (location) and the corresponding polar coordinates (imprecision).•The structures required for a meaningful statistical analysis from the so-called ontic perspective are introduced.•How to determine the representation in practice is discussed.