The fuzzy integral

Ralescu, Dan A.; Adams, Gregory T.

doi:10.1016/0022-247x(80)90101-8

Cited by 355 publications

(95 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although Sugeno's definition of a fuzzy integral is clearly a fuzzification of the Choquet integral [3], his integral also bears a formal resemblance to the Lebesgue integral, that plays a very important role in classical measure and probability theory. This was shown in 1980 by Ralescu and Adams [15] in an important paper which explores the relation between Sugeno's fuzzy integral and the Lebesgue integral. Weber has thoroughly studied integrals associated with special types of decomposable fuzzy measures [24,25,26].…”

Section: Introductionmentioning

confidence: 86%

“…Furthermore, since the items (ii) of the abovementioned theorems, and definitions, for that matter, are entirely original, we have only given the proofs of these items. The proofs of the items (i) can be found by dual analogy, or by the not always trivial extension of the proofs found in the literature [15,19,20]. Theorem 2 deserves some extra attention.…”

Section: In General We Shall Also Call These Integrals Seminormed Rementioning

confidence: 97%

“…This result is an analogon of a well-known theorem 4 in the theory of classical measures and integrals (see, for instance, [2]). This is a very interesting point, since Ralescu and Adams [15] have shown that an analogous theorem is not necessarily valid in general 5 for fuzzy integrals associated with arbitrary confidence measures.…”

Section: An Arbitrary Familiy Of Elements Of R and Let H Be An Arbitmentioning

confidence: 99%

“…The items (i) of definitions 2, 3 and 4 and of theorems 3 and 4 are generalizations-towards more general codomains for the mappings and towards more general types of measures-of definitions and theorems by Suárez García and GilÁlvarez [19], which in turn are based upon the work of Ralescu and Adams [15] and Sugeno [20]. Our introduction of semiconormed fuzzy integrals-generalizations of the semiconormed fuzzy integrals of Suárez García and GilÁlvarez-is on the other hand completely new and exposes the dual analogy between seminormed and semiconormed fuzzy integrals.…”

Section: In General We Shall Also Call These Integrals Seminormed Rementioning

confidence: 99%

See 3 more Smart Citations

Possibility and necessity integrals

Cooman

Kerre

1996

Fuzzy Sets and Systems

View full text Add to dashboard Cite

In this paper, we introduce seminormed and semiconormed fuzzy integrals associated with confidence measures. These confidence measures have a field of sets as their domain, and a complete lattice as their codomain. In introducing these integrals, the analogy with the classical introduction of Legesgue integrals is explored and exploited. It is amongst other things shown that our integrals are the most general integrals that satisfy a number of natural basic properties. In this way, our dual classes of fuzzy integrals constitute a significant generalization of Sugeno's fuzzy integrals.A large number of important general properties of these integrals is studied. Furthermore, and most importantly, the combination of seminormed fuzzy integrals and possibility measures on the one hand, and semiconormed fuzzy integrals and necessity measures on the other hand, is extensively studied. It is shown that these combinations are very natural, and have properties which are analogous to the combination of Lebesgue integrals and classical measures. Using these results, the very basis is laid for a unifying measure-and integral-theoretic account of possibility and necessity theory, in very much the same way as the theory of Lebesgue integration provides a proper framework for a unifying and formal account of probability theory.

show abstract

Section: Introductionmentioning

confidence: 86%

Section: In General We Shall Also Call These Integrals Seminormed Rementioning

confidence: 97%

Section: An Arbitrary Familiy Of Elements Of R and Let H Be An Arbitmentioning

confidence: 99%

Section: In General We Shall Also Call These Integrals Seminormed Rementioning

confidence: 99%

See 2 more Smart Citations

Possibility and necessity integrals

Cooman

Kerre

1996

Fuzzy Sets and Systems

View full text Add to dashboard Cite

show abstract

“…The topic of fuzzy integration was first discussed in [61]. Particularly, it has been studied by Kaleva [41], Goetschel and Voxman [37], Nanda [47], Ralescu and Adams [49], Wang [59], Bede and Gal [16] and others [32,33]. The fuzzy Riemann integral and its numerical integration was investigated by Wu in [60].…”

Section: Introductionmentioning

confidence: 99%

Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

Noeiaghdam¹,

Araghi²

2017

Journal of Fuzzy Set Valued Analysis

View full text Add to dashboard Cite

The aim of this work, is to evaluate the value of a fuzzy integral by applying the Newton-Cotes integration rules via a reliable scheme. In order to perform the numerical examples, the CADNA (Control of Accuracy and Debugging for Numerical Applications) library and the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method are applied based on the stochastic arithmetic. By using this method, the optimal number of points in the fuzzy numerical integration rules and the optimal approximate solution are obtained. Also, the accuracy of the fuzzy quadrature rules are discussed. An algorithm is given to illustrate the implementation of the method. In this case, the termination criterion is considered as the Hausdorff distance between two sequential results to be an informatical zero. Two sample fuzzy integrals are evaluated based on the proposed algorithm to show the importance and advantage of using the stochastic arithmetic in place of the floating-point arithmetic.

show abstract

Using genetic algorithms to determine nonnegative monotone set functions for information fusion in environments with random perturbation

Wang

et al. 1999

Int. J. Intell. Syst.

View full text Add to dashboard Cite

Due to some inherent interactions among diverse information sources, the classical weighted average method is not adequate for information fusion in many real problems. To describe the interactions, an intuitive and effective way is to use an appropriate nonadditive set function. Instead of the weighted average method, which is essentially the Lebesgue integral, we should thus use the Choquet integral or some other nonlinear integrals. To apply this alternative, more realistic approach to information fusion, we need to determine the nonadditive set function from given input-output data, viewing the nonlinear integral as a multi-input one-output system. In this paper, we employ an adaptive genetic algorithm to construct an approximate optimal nonnegative monotone set function from given input-output data in an environment with random perturbation. An example for diverse strengths of random perturbation is shown to demonstrate the efficiency of this algorithm.

show abstract

The fuzzy integral

Cited by 355 publications

References 1 publication

Possibility and necessity integrals

Possibility and necessity integrals

Finding optimal step of fuzzy Newton-Cotes integration rules by using the CESTAC method

Using genetic algorithms to determine nonnegative monotone set functions for information fusion in environments with random perturbation

Contact Info

Product

Resources

About