On Statistical Moments for Capricious Vague Perceptions

Abstract

In this paper, the author investigates the statistical moments such as expectations and variances 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 random sets, which has been proposed by author[1, 2], where the vague perception of a crisp phenomenon fluctuates slightly but randomly by the state of a capricious person's mind, is refined and its expectation and variance are introduced. Secondly, the refined class of fuzzy random sets is extended to the models for the capricious vague perceptions of crisp random phenomena, and their expectations and variances are investigated from the viewpoint of the multivalued logic.

On Convergences of Estimates Concerned with Fuzzy Random Data

“…are 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 set…”

“…The 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 set…”

On Second Statistical Moments for Capricious Vague Perceptions

“…

In this paper, the author investigates the statistical moments such as expectations and variances 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 random sets, which has been proposed by author [1][2][3], where the vague perception of a crisp phenomenon fluctuates slightly but randomly by the state of a capricious person's mind, is refined and its expectation and variance are introduced.Secondly, the refined class of fuzzy random sets is extended to the models for the capricious vague perceptions of crisp random phenomena, and their expectations and variances are investigated from the viewpoint of the multi-valued logic.

…”

“…First, the class of fuzzy random sets, which has been proposed by author [1][2][3], where the vague perception of a crisp phenomenon fluctuates slightly but randomly by the state of a capricious person's mind, is refined and its expectation and variance are introduced.…”