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

作者：Gonzalez Rodriguez Gil; Belen Ramos Guajardo Ana; Colubi Ana^*; Blanco Fernandez Angela

来源：International Journal of Approximate Reasoning, 2018, 92: 279-294.

DOI：10.1016/j.ijar.2017.10.025

• Imprecise data
• Set-valued data
• Support function
• Star-shaped set
• Functional characterizations
• Center-radial representation

The space of nonempty convex and compact (fuzzy) subsets of R-P, k(c)(R-P), 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 k(c)(R-P) 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 k(c)(R-P). 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 convexity. 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 R-P. 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.

• 出版日期2018-1