Probabilistic Approach for Vague Perceptions of Complex Phenomena

Fukuda, Tokuo

doi:10.5687/sss.2012.101

Cited by 5 publications

(12 citation statements)

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“…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]. )…”

Section: Metric Spaces Of Fuzzy Setsmentioning

confidence: 99%

On Convergences of Estimates Concerned with Fuzzy Random Data

Fukuda

2014

Stochastic Systems Theory and its Applications (SSS)

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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.

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Section: Metric Spaces Of Fuzzy Setsmentioning

confidence: 99%

On Convergences of Estimates Concerned with Fuzzy Random Data

Fukuda

2014

Stochastic Systems Theory and its Applications (SSS)

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“…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. The proposed fuzzy random set is a revised one proposed by author last year [1][2][3].…”

Section: Fuzzy Random Sets As Models Of Capricious Vague Perceptions mentioning

confidence: 99%

“…For this purpose, a new class of fuzzy random sets is firstly introduced, considering that the vague perception of a crisp phenomenon fluctuates slightly but randomly due to the state of a capricious person's mind. The class of fuzzy random sets proposed in this paper is a revised one introduced in [1,2]. Secondly, the proposed fuzzy random set is generalized as the vague perception of a crisp random phenomenon.…”

Section: Introductionmentioning

confidence: 99%

“…

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.

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confidence: 99%

“…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.…”

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