Statistics with Vague Data

Kruse, Rudolf; Meyer, Klaus Dieter

doi:10.1007/978-94-009-3943-1

Cited by 413 publications

(59 citation statements)

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“…• The methodologies based on the notion of fuzzy random variable, introduced by Kwakernaak (1978Kwakernaak ( , 1979 and later formalized by Kruse and Meyer (1987 Skala (1975)], which was mostly prompted by the development by Kruse and Meyer of useful fuzzy methods and a software tool for statistical applications for the Siemens AG. The model refers to the epistemic perspective and fuzzy random variable is viewed as the fuzzy perception of an original nonfuzzy random variable.…”

Section: On the Analysis And Classification Of Fuzzy Datamentioning

confidence: 99%

Fuzzy data analysis and classification

D’Urso

Alvarez

2017

Adv Data Anal Classif

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In analyzing and classifying data from a statistical perspective, fuzzy sets and logic have become a valuable tool either to model and handle imprecise data or to establish flexible techniques to deal with precise data.From the very beginning of his 52 years-old theory, Professor Zadeh highlighted that "Probability theory/statistics and fuzzy logic should be viewed as complementary rather than competitive," and he anticipated and encouraged the materialization of such a complementarity. Nowadays, this assertion is a reality, as shown by the many related papers, specialized conferences, special sessions and tracks in general conferences, and so on.This special issue started in 2015, with the 50th anniversary of the seminal paper on fuzzy sets by Zadeh (1965), aiming to collect a sample of research papers about the current trends on the combination of fuzzy sets/logic and data analysis/classification. When this special issue was almost ready for publication, Zadeh unfortunately passed away at age 96 (February 4, 1921-September 6, 2017. We wish this special issue to be dedicated to Professor Zadeh, as a modest part of the many tributes that he will receive, and intending to show that fuzzy sets/logic and data analysis/classification can certainly work in synergy.

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Section: On the Analysis And Classification Of Fuzzy Datamentioning

confidence: 99%

Fuzzy data analysis and classification

D’Urso

Alvarez

2017

Adv Data Anal Classif

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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%

“…Then, the extended fuzzy random set as a capricious vague perception of the origina 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 6. 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) …”

Section: Vague Perception Of Random Phenonemamentioning

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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“…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

Fukuda

2017

Stochastic Systems Theory and its Applications (SSS)

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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.

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Statistics with Vague Data

Cited by 413 publications

References 0 publications

Fuzzy data analysis and classification

Fuzzy data analysis and classification

On Convergences of Estimates Concerned with Fuzzy Random Data

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

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