On Convergences of Estimates Concerned with Fuzzy Random Data

Abstract

In this paper, the author investigates the convergence properties of estimators concerned with expectations for a class of fuzzy random sets, where the fuzzy random set is considered as a model of the capricious vague perception of a crisp phenomenon or a crisp random phenomenon.First, the class of fuzzy sets, which has been proposed by author, is refined from the practical point of view. Secondly, using the refined class of fuzzy sets, fuzzy random sets as vague perceptions of crisp phenomena and their extended one as vague perceptions of random phenomena are also refined. The expectations of fuzzy random sets are also reviewed. Finally applying the standard strong law of large numbers(SLLN) for the random elements in a separable Banach space, the convergence property of estimators for expectations of random fuzzy sets is examined. Keywords: fuzzy random sets, capricious vague perception, expectation, strong law of large numbers Metric Spaces of Fuzzy SetsThe class of fuzzy sets adopted in this paper is inspired by the one originally proposed by Kwakernaak[1], Kruse and Meyer[2], Kruse, Gebhardt and Klawonn [3], and the revised version of previously proposed by the author(see e.g., [4,5,6,7,8].) Let I be the open interval between 0 and 1, i.e., I = (0, 1) andĪ = [0, 1]. A fuzzy set U(u o ) as the vague perception of u o ∈ R n is defined by the triplewithwhere R n is the n-dimensional Euclidean space called the basic space; s U is the predicate, i.e., s U : R n → S with S the "universe of discourse" defined by a set of statements, which assigns a propositionto each element u ∈ R n [9]; and [ U(u o )] is the family of subsets of R n called the set representation of U(u o ), which satisfieswhereare the strong cut(strong level set) and the level set of U(u o ) at the level α(see e.g., fukuda [7,8]). The crisp point u o in R n , the vague perception of which gives the fuzzy setHereafter, a fuzzy set U(u o ) will be abbreviated as U, when no confusion occurs or no special interest in the original point u o has been concerned. Definition 1. The family of fuzzy sets is denoted by F cc (R n ), whose element U = (R n , [ U], s U ) satisfies the following conditions:where K cc (R n ) is the family of nonempty compact and convex subsets of R n .(ii) [ U] 1 defined byis the element of K cc (R n ).does not equal to L α U only at the finite points of α, i.e.,at most only at α ∈ I K = {α 1 , α 2 , · · · , α K }The support of U is defined byThen, the subclass F b cc (R n ) of F cc (R n ) is defined as follows.

On Two-Dimensional Fuzzy Random Data as Vague Perceptions of Random Phenomena

“…Let p be a point in [1, +∞), F (p) cc (R 2 ) be the family of fuzzy sets given in [7], and let U M (ω) ∈ F (p) cc (R 2 ) be a FRS given by…”

On Two-Dimensional Fuzzy Random Data as Vague Perceptions of Crisp Phenomena

“…Proceedings of the 48th ISCIE International Symposium on Stochastic Systems Theory and Its Applications Fukuoka, Nov. [4][5]2016 where (…”

“…In recent years, motivated by the importance for treating the data exhibiting both vagueness and randomness, fuzzy random variables or more generally fuzzy random sets have been intensively investigated by many researchers with various definitions [1,2,3,4,5,6,7,8,9,10].…”

Simulation Studies for a Class of Fuzzy Random Data

“…Hence, the fuzzy set obtained as a vague perception of a crisp phenomenon may be some kind of 'fuzzy random set', i.e., it is a function of the generating point of some sample space. Using the class of fuzzy sets F (p) cc (R n ), the elementary fuzzy random set is described as follows [1,2]: Definition 1. Let (Ω, A, P u o ) be an elementary probability space, where Ω = {ω 1 , ω 2 , • • • , ω M }; A be a σ-algebra given by the subsets of Ω; and P u o is a probability measure such that P u o (ω i ) > 0 for each i = 1, 2, • • • , M. Then, an elementary fuzzy random set as a vague perception of the original point u o ∈ R n is defined by U(u o , ω) = (R n , [ U(u o , ω)], s U(u o ,ω) ) ∈ F (p) cc (R n )…”

“…Then, the extended fuzzy random set as a capricious vague perception of the original random point u o is defined on (Ω, A, P) = (Ω 1 × Ω 2 , A 1 ⊗ A 2 , P 1 × P 2 ) and given as follows: Definition 3. An extended fuzzy random set U(ω) on (Ω, A, P) obtained as the capricious vague perception of an original random point u o (ω (2)…”