Consonant Random Sets: Structure and Properties

Abstract

Abstract. In this paper, we investigate consonant random sets from the point of view of lattice theory. We introduce a new definition of consonancy and study its relationship with possibility measures as upper probabilities. This allows us to improve a number of results from the literature. Finally, we study the suitability of consonant random sets as models of the imprecise observation of random variables.

On the connection between probability boxes and possibility measures

“…Another way of relating possibility measures and p-boxes goes via random sets (see for instance [19] and [9]). Possibility measures on ordered spaces can also be obtained via upper probabilities of random sets (see for instance [6,] and [21]).…”

Generalized representation theorem and its application to the construction of fuzzy sets: Existence and uniqueness

“…The consonant basic belief assignment or mass function based on nested family of focal sets was adopted by Aregui and Denoeux [2], Lawry and González-Rodríguez [8] and Quost et al [13]. We also mention that the existence and uniqueness for the construction of fuzzy sets were also investigated by Miranda et al [9][10][11] in relation with random set theory. However, the universal set (U, d) taken by Miranda et al was assumed to be a metric space that is also endowed with a topology induced by the metric d. For example, in Miranda et al [11,Theorem 5.3], the random set was assumed to be compact-valued (in this case, the universal set U was assumed to be Polish) or closed-valued (in this case, the universal set U was assumed to be compact).…”

“…The main result Herencia [6,Theorem 4.3] presented the relationship between fuzzy sets and graded sets, where the 0-level set was also implicitly assumed as the whole universal set U. The main focus of this paper is to treat the 0-level set according to the topology for U, where the 0-level set is not necessarily the whole universal set U. Nested families are frequently considered in possibility and random set theories (see, for instance Alvarez [1], Baudrit et al [3], Dubois et al [5] and Miranda et al [9][10][11]). For example, the concept of consonance is based on nested family.…”