On a Probabilistic Approach for Capricious Vague Perceptions of Random Phenomena

Stochastic Systems Theory and its Applications (SSS)

2014

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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.

Section: Metric Spaces Of Fuzzy Setsmentioning

confidence: 99%

“…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…”

Section: Metric Spaces Of Fuzzy Setsmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

On Convergences of Estimates Concerned with Fuzzy Random Data

Stochastic Systems Theory and its Applications (SSS)

2014

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“…for some fixed p ∈ [0, ∞), where R 2 is the two-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 proposition s U (u) = u in U coincides with the original point u o ; (2) and [ U(u o , ω)] is the family of subsets of R 2 called the set representation of U(u o , ω), which satisfies…”

Section: Two-dimensional Efrssmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Stochastic Systems Theory and its Applications (SSS)

2017

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In this paper, the author investigates numerically a class of two-dimensional elementary fuzzy random sets(abbreviated as EFRSs), which is one case of those proposed by the author[1] as models of vague capricious perceptions of crisp non-random phenomena.First, the basic results concerned with two-dimensional EFRSs as vague capricious perceptions of crisp phenomena are reviewed. Expectations of two-dimensional EFRSs, their estimators, and distances between expectations and their estimators are also reviewed briefly.Finally, estimates for expectations of two-type of twodimensional EFRSs and their estimation errors are numerically examined by simulation studies.

On Second Statistical Moments for Capricious Vague Perceptions

2013

Transactions of ISCIE

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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.