Metric Spaces of Fuzzy Sets: Theory and Applications

Diamond, Phil; Kloeden, Peter E.

doi:10.1142/2326

Cited by 603 publications

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“…For convenience, we give some definitions and introduce the necessary notations and lemmas (see e. g., monographs [16,17,18,19] and references contained therein) which will be used in the following sections.…”

Section: Preliminariesmentioning

confidence: 99%

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations

Wang¹,

Sun²,

Han³

2017

AAN

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In this paper, we investigate the existence and uniqueness of solution to boundary value problems for a class of nonlinear fuzzy factional differential equations involving the fuzzy gHfractional Caputo derivative. By means of the Schauder fixed point theorem in semi-linear spaces and integral inequality technique, some qualitative results of solutions are obtained. An example is provided from which new results are found.

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Section: Preliminariesmentioning

confidence: 99%

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations

Wang¹,

Sun²,

Han³

2017

AAN

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“…It is known that (F, δ ∞ H ) is a complete metric space while (F, d2) is not (cf. [4,Chapter 7]); in the following we even use the mid-spread metric D θ that we introduce later in this section. Let ν, η ∈ F, the Hukuhara difference ν ⊖H η is defined by…”

Section: For Anymentioning

confidence: 99%

A decomposition theorem for fuzzy set-valued random variables

Aletti

Bongiorno

2013

Fuzzy Sets and Systems

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In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two parts. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X ⊖H C exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C ′ chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize fuzzy random translations; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C ′ (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented.

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“…The support function is the way of defining an 2 L -metric on the space of normal compact convex fuzzy sets ) ( p C  F using the 2 Lmetric on the Hilbert space of squareintegrable functions…”

Section: Using Generalized Hukuhara Difference Instead Of Fuzzy Subtrmentioning

confidence: 99%

“…Square-integrable random variables are assumed, defined on a Hilbert space equipped with a suitable 2 L -metric that allows the projection theorem to be still valid. However, it cannot be properly applied as usually onto a subspace, but rather onto cones (i.e., subject to some constraints), due to the lack of a general additive inverse in the space of fuzzy variables, which is only a semi-linear space.…”

Section: Introductionmentioning

confidence: 99%

Fuzzy Time Series Estimation and Prediction: Criticism, Suitable New Methods and Experimental Evidence

Georgescu¹

2010

SIC

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This paper is devoted to exploring suitable methods for modelling, estimating and forecasting fuzzy time series, when facing the problem of non-invertibility of the standard Minkovsky addition and multiplication in a fuzzy framework. Some generalized versions of Hukuhara difference, which allow the fuzzy estimation problem to be handled in some L 2 -type metric space, are first examined from a critical viewpoint. This leads us to propose a new estimation procedure, where the monolithic fuzzy model is broken in several more tractable crisp estimation subproblems, based upon a partial decoupling principle. Our aim is to produce fuzzy estimations with non-negative spreads, capable not only to help decomposing, but also to make the process invertible, by recomposing a nonstationary fuzzy time series from its components, such as trend, cycle, seasonality and the simulated residuals, all of them properly defined as LR-fuzzy sets. Computational Intelligence techniques such as wavelet decomposition and de-noising or nonlinear model fitting with wavelet networks are also addressed. Finally, the proposed methods are exemplified for a fuzzy time series with fuzzy daily temperatures (minimum, average and maximum values).

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Metric Spaces of Fuzzy Sets: Theory and Applications

Cited by 603 publications

References 0 publications

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations

Existence of Solutions to Boundary Value Problems for a Class of Nonlinear Fuzzy Fractional Differential Equations

A decomposition theorem for fuzzy set-valued random variables

Fuzzy Time Series Estimation and Prediction: Criticism, Suitable New Methods and Experimental Evidence

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